Inequalities Involving Functions and Their Integrals and Derivatives
Inequalities Involving Functions and Their Integrals and Derivatives
Inequalities Involving Functions and Their Integrals and Derivatives
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
Volume 53
Inequalities Involving Functions
and Their Integrals
and Derivatives
by
D. S. Mitrinovic
University of Belgrade,
Yugoslavia
J. E. Pecaric
University of Zagreb,
Yugoslavia
and
A.M.Fink
Iowa State University,
Ames, U.sA.
~l moil ... , Ii j'avait su comment en revenir, One service mathematics has rendered the
je n'y serais point aUe.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non-
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non-
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com-
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'(ftre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu-
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vi SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non-
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci-
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com-
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre-
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub-
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis-
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and! or
scien tific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
Inequalities are everywhere. Whole series of conferences are devoted to them. Indeed in my more
despondent moments, when struggling with one or another problem, I sometimes have the feeling
that mathematics (especially analysis) is all inequalities. It is not that I dislike them or do not
appreciate their usefulness; indeed the vast importance of inequalities is manifest and many, espe-
cially the more powerful ones, have great aesthetic appeal - for here, as in many other parts of
mathematics and science, power and elegance go together. But, on the other side, there are so many
inequalities and it is, by and large, such an unstructured mass of results. (This is, for that matter, an
aspect that this book addresses with success.)
Since the time of the classical book by Hardy, Littlewood and Polya of 1934 (recently reprinted
by Cambridge Univ. Press in their 'classics series'; a bargain!) the subject of this book, 'Differential
and Integral Inequalities', has grown by about 800%. The senior author of this book, D.S. Mitrino-
vic, has worked in inequalities all his life and has collccted results in the subsubject of inequalities
between functions and their integrals and derivatives, i.e. differential and integral inequalities, since
1950. Other books by him and his coworkers on different topics within 'Inequality Theory' have
also appeared in this series (p.S. Bullen, D.S. Mitrinovic, P.M. Vasic, Means and their Inequalities,
1987; D.S. Mitrinovic, J.E. Pecaric, V. Volenec, Recent Advances in Geometric Inequalities, 1989).
The present volume is devoted to the particular topic of differential and integral inequalities. It sys-
tematizes and surveys the whole subject (thUS making it possible for nonspecialists such as myself to
find the results needed) and, additionally, it describes typical uses of these inequalities.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
l'occasion de re"soudre des problclmes ... eUe right than other people, but to be wrong for
nous fait pressentir 1a solution. more sophisticated reasons.
H. Poincare David Butler
Preface ix
Organization of the book xiii
Notation xv
Dragoslav S. Mitrinovic
Smiljaniceva 38
11000 Belgrade, Yugoslavia
Josip E. Pecaric
Tehnoloski Fakultet
41000 Zagreb, Yugoslavia
A. M. Fink
Mathematics Department
Iowa State University
Ames, Iowa 50011, USA
ORGANIZATION OF THE BOOK
xiii
NOTATION
IIfllp = (J If(xWdx)l/P for 1 ~ p < 00, with 11J1l00 = esssup If(x)l, and
in this context, lip + lip' = 1 with the usual conventions that I' = 00,
and 00' = 1.
xv
xvi
References to other sections in the book are given in two different ways:
references to sections from the same chapter are indicated by the number
followed by a period, for example, if in Chapter III, section 10 we want to
refer the reader to section 2 of Chapter III we merely say "see 2." If the
section occurs in a different chapter we say "see 2. of Chapter III" .
(1.1)
(2.3)
(2.4)
C nl (+00, R) :s; 2n - l .
G. H. Hardy and J. E. Littlewood [3] at about the same time showed that the
constants Cnk( +00, R) and Cnk( +00, R+) exist but did not get good estimates.
3. In [4] E. Landau gave some estimates similar to his Theorems from 2. but
for a fixed interval [0,1]. (see also [7].)
4. Suppose that f( x) is defined for x > 0 (this involves no loss of generality) and
that f" (x) exists for x > o. If, as x - t +00, g( x) and h( x) are positive functions
of x, both increasing or decreasing steadily, and If(x)1 < g(x), 1f"(x)1 < hex),
then, as x - t +00,
Results from [8] are exposed in the book [9]. Here, we shall give results from
this important book. (These results are also exposed in [188] and we use this
exposi tion.)
(6.1)
J J J
(X) (X) (X)
Equality holds in (6.2) if and only if f(x) = Ae- xl2 sin (~ x - ~) == fA(X), and
in (6.1) if and only if f(x) = fA(Bx), where A,B are arbitrary constants.
PROOF: The implication (6.1) implies (6.2) is obvious (by GA inequality). Now,
we show that (6.2) implies (6.1). If (6.2) holds (with equality only as asserted),
then for t > 0 and F(x) = f(tx) we have
J J J
(X) (X) (X)
J J J
(X) (X) (X)
Moreover, equality holds only if f({3x) = fA(X) for arbitrary B since it is easy to
verify that if equality holds in (6.1) for a ~iven f it also holds for g( x) = f( kx)
with arbitrary k.
4 CHAPTER I
J
X
J
x
(6.3) [f2 - f'2 + f"2 - (f + f' + f"?] dx
o
= [J(O) + J'(0))2 - [f(X) + J'(X))2 .
Also,
J J f"
x x
(6.4) f'2 dx = f(X)J'(X) - f(O)J'(O) - f dx.
o 0
x
Since f,f" E L 2, it follows from Holder's inequality that lim J ff"
X-OO O
dx =
J f f" dx exists (finite). J f,2 dx < +00 since otherwise it would fol-
00 00
Hence also
o 0
x
low from (6.4) that f(X)f'(X) -+ +00 and so also J f J' dx = ~ [P(X) - P(O)] -+
o
+00. This is impossible since f E L 2 • Thus both integrals in (6.4) converge to a
finite limit as X -+ +00, and so also lim f(x)f'(x) exists. This limit must be
x-oo
zero since otherwise, as before, we could not have f E L 2 • But now, as X -+ 00
the integrals on the left side of (6.3) all exist (use Holder's inequality for J ff' dx
J
and f' f" dx), and so
00 00
J
00
J
00
The inequality (6.2) folows from (6.5), with strict inequality unless f satisfies
f + I' + I" = 0 andf(O) + 1'(0) = 0, that is unless f(x) = fA(X) as specified.
The corresponding theorem for the interval R is also due to Hardy, Littlewood
and P6lya [9, Th. 261), and is much easier to prove.
THEOREM 2. If f' is locally absolutely continuous and f, I" E L 2 (R), then
(6.6) l
( +001'2 dx ) 2
< l l
(+00f2 dx ) (+001"2 dx ) ,
with equality if and only if f = O. The (unit) constant is best possible.
PROOF: As in the proof of Theorem 1, f(X)f'(X) -+ 0 as X -+ ±oo. Hence by
Holder's inequality,
l
( +001'2 dx) 2
- -l
(+00 f I" dx
) 2
< l l
(+00 f2 dx ) (+001"2 dx ) ,
with strict inequality unless f" = kf, that is unless f(x) = AeC>X + Be-ax or
f(x) = Asin(ax + (3) for some constants A, B, a, (3. The only such f for which
f P dx is finite is f == O.
To prove that the constant is best possible, note that both sides of (6.5) have
the value 1 for the (non-admissible) function fo(x) = e- ixi . Given any n :::: 1 we
may smooth the corner of the graph of fo at x = 0 to obtain a function f n such
that f~ exists for all x and
Jf 12
n
dx>
-
1
1 - n'
-
REMARK 1: The above proofs are those of Hardy, Littlewood and P6lya [9) who
also gave two other proofs of (6.2) one of which was variational in character.
REMARK 2: By the same technique used in Theorem 2, one can show that if
l' is locally absolutely continuous and f E Lp(R), I" E Lp,(R), where p > 1,
p' = p/(p - 1), then
(6.7) £
( +001'2 dx ) < (+00
£, Ifl dx ) (+00
£, If"lP' dx )
P
lip lip'
6 CHAPTER I
unless f == o.
(cf. [9, Th. 269].)
REMARK 3: From [9, Theorems 272 and 276]: fED if f: R+ (or R) - t R,
f E AC1oc(R+) (or R), xf and f' both E £2(R+) (or £2 R)j then f E £2(R) (or
£2R) and
f(x) = Aexp[-px2]
7. The first relevant results for functions of two variables are given by T. Tchou-
Yun Tcheng [11].
Let function u(x, y) have continuous second partial derivatives on R2, and let
mo, m1 and m2 denote respectively supremum of
Then
(7.1)
then
(7.2)
8. An attack on the general problem for p = +00 was begun by G. E. Silov [12]
in 1937, who found
9. A. Gorny, unifying his results published in [13]' [14] and [15], but in a some-
what modified form, proved in [16] the following result:
If f is an n-times differentiable function on a closed interval I of length b, and
if If(x)1 ~ Mo and If(n)(x)1 ~ Mn, then for x E I and 0 < k < n,
where
M~ = max (Mn, Mon!b- n) .
where
(10.2)
_-
4+00l(_l
:- y )n+l
Kn - 7r 2v + 1
v=o
He also showed that 1 < Cnk( +00, R) < 7r /2 for all n and k, and that
Cn,n-l (+00, R) -+ 7r /2 and Cn1 ( +00, R) -+ 1 both as n -+ +00.
(11.1 )
1
f(x)- b-a! f(x) dx
b 1
~ ( 4+
(x-~)
(b-a)2
2) (b-a)M.
a
12. One of the earliest inequalities related to (6.2) is due to 1. Halperin and H.
Pitt [21] (here we used exposition of this result from [188]).
The inequality in question is
! ! !
x X x
(12.1) 1'2 dx ~ K(e) f2 dx +e 1"2 dx
o 0 0
valid for all e > 0 and locally absolutely continuous f with I" E L 2 (0, X). Here
P Q
K(e) = i + X2 with P = 1, Q = 12.
But this is also clearly true for e ~ o. Hence the discriminant of this quadratic
in e is less than or equal to zero. This gives C21 (2, R+) = "j2.
13. Let f( x) be a complex function of a real variable, defined over the whole real
line, which possesses n derivatives (the nth at least almost everywhere) and is
x
J
such that j<n-l)(x) = f(n)(t) dt. Then, if k is any integer for which 0 < k < n,
Kolmogorov's inequality (1.1) for p = +00, is valid and Cnk( +00, R) is defined
as in 10. The constant Cnk is the best possible, i.e. there is a (real) function for
which equality holds (see [22]).
(14.1)
15. A. M. Rodov (see [24] and [25]) gave a solution of the following general
problem: Given a finite set of integers i 1 , • •• ,in such that
is there a function I bounded, together with its n first derivative on R, such that
16. Some related results in the case when 1= [-1,1] are given in [26].
18. Inequalities
(18.1)
and
19. As we saw, the constant C nk ( +00, R) is explicitly obtained. But the related
problem for Cnk( +00, R+) seems more difficult. In 1955, A. P. Matorin showed
that ([30])
(19.1)
20. E. M. Stein [31] in 1957 used a convolution technique to show that in general
(20.1)
Ju. Ljubic also gives an algorithm for finding Cn k(2, R) in terms of computing
the zero of an explicit polynomial.
21. J. Dieudonne [33] gave some results related to results from 17.
22. W. Miiller [34] considered the inequality (12.1) with P = 48, Q = 24. The
best possible result, given in 12., is proved by R. M. Redheffer in [35].
23. S. B. Steckin [36] proved that in the case when I(x) ~ 0, we have C21 (1, R) =
1 and C21 (I,R+) = .;2. The same constants for p = +00, are given by V. M.
Olovyanisnikov [28] who also showed that C 32 ( +00, R+) = 2.31/ 3. Steckin also
proved that C 31 (+00, R+) = ~35/3. Moreover, he gave a result related to the
Kolmogorov inequality in Banach space.
°
I.e.
where
Cr = r(r + 1) sin r7r 22-(r/2) (21/(1-r) _ 1) r-1 .
r(l- r)(2 - r)7r
The constant C r is the best possible.
12 CHAPTER I
25. A. M. Pfeffer [38] considered inequality (18.1) for p = 2 where 1 ::; m < n,
c > 0, f E Cn[a, b], f{i)(a) = f{j)(b) (0::; i ::; n - 1). The proof of the existence
of a constant K(c) in (18.1) is given in [39] by S. Goldberg.
26. Some related results in which we have linear differential operators instead
of derivatives are considered in [40]-[42].
27. S. B. Steckin [43] in 1967 gave estimates for Cnk( +00, R+):
28. An elementary proof of (6.2) was also indicated by D. C. Benson [44] who
J J
applied the same method to other inequalities involving P dx, 1'2 dx, and
J 1"2 dx.
29. An extremal problem related to Kolmogorov's inequality for bounded func-
tions is considered in [45].
30. We gave the definition of a derivative of order r for 0 < r < 1 in 24. For
r> 1, if we put Dr(f,x) = f{r)(x), we can extend this definition by
31. Let 1 ::; p, r ::; +00 and let J be a finite or infinite interval of the real line.
For n a positive integer W;'r( J) denotes the set all functions y and J to the
real numbers such that y E LP( J), y{n-l) is absolutely continuous on compact
subintervals of J and y{n) E U(J).
The following result is proved by V. N. Gabusin [47]:
Let 1 ::; p, q, r ::; +00, J = R or J = R+, and let n, k be positive integers with
o ::; k < n. Let
v = 1- u.
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 13
Suppose
(31.1)
case °: ;
cases of (31.1) there exist the best possible constants K. This problem in the
k < n, n arbitrary, q = +00, r = p = 2 is solved in [50] for J = Rand
in [53] for J = R+, and in the case n = 2, k = 0,1; q = +00, p ~ 1, r = +00,
J = R in [50].
33. V. I. Berdysev [55] found that C21 (1, R) = 2 and C21 (1, R+) = 5/2.
34. R. Kallman and G. C. Rota [56] have generalized the inequality (2.3) to
a particular class of semi-groups. Let X be a B-space over the complex field,
{T(s)} a one-parameter semi-group of linear bounded transformations from X
to itself such that IIT(s)11 ::; 1 for all s ~ 0. Let A be the infinitesimal generator
of the semi-group, D[A] c X its domain of definition. Then for any f E D[A2],
Whence
s
IIfll )1/2
s = 2 ( IIA2fll
and the minimum value is
(34.2)
(34.9)
A direct proof of (34.4) and some applications of the previous results are given
in Hille's paper [61).
T. Kato [62) showed that (34.7) is valid whenever X is a Hilbert space. This
suggests that this inequality can be generalized for any space in which the ge-
ometry is sufficiently nearly Euclidean (in some appropriate sense). So, J. A. R.
Holbrook [63) proved:
Let (X, 11·11) is a real or complex normed linear space, and let
(34.10)
(34.11)
36. C. J. F. Upton [70] extended result from 13. to the cases when the sup
norm on R is replaced by one of the following: (1) Lp norm on R, (2) Lp norm
on R+, (3) Stepanov norm on almost periodic functions on R, (4) Weyl norm
on almost periodic functions on R, (5) Besicovitch norm on almost periodic
function on R, (6) Love's variation norm on almost periodic functions on R and
k-l
(7) Ilfll* = 2: IIf(r) 1100 + Ilf(k) 1100 on almost periodic functions on R, where 11·lla
r=O
is interpreted to be one of the norms (3)-(6).
(In fact, Upton also proved (20.1).)
37. V. V. Arestov [71] also considered the problem from 31. He used the result
from 14. in the proof of (31.1) for n = 2, p = +00, r ~ 1, q ~ 2r, u = 1 - v,
v = (1 - q-l )/(2 - r- 1 ). The best possible constant is
2Q, if J = [0,+(0),
K= {
2 v Q, if J = (-00,+00).
He also considered the case when n = 3, q = p = +00, 1 :S r < +00 for Rand
R+. (Q is defined as in 14.).
Let K;,r be subclass of W;'r such that Ilf(n)IILp :S 1, and E(N) be a set of
linear bounded operators S: Lp -+ Lq with a norm IISII = IIslIf; :S N. Define
(37.1)
(37.2)
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 17
for p,q E (1,+00), p-l +q-l = 1, f E LP(R+), f' absolutely continuous on all
compact subintervals of [R+), f" E U(R+) and J{ a positive number indepen-
dent of f but dependent of choice of p and q. Of course, (37.2) is an extension
of (6.1). The authors show that the best constant J{ in (37.2) is finite for all
(p, q), and they gave a characterization of this constant in terms of an equivalent
problem from the calculus of variations.
Everitt and Giertz [72] also obtained bounds for 1f(0)1 and If'(O)1 in terms of
IIfllp and 1If"lIq· .
38. The following consideration is given in [188].
In a series of papers (see e.g. [73],[75],[76]) W. N. Everitt has considered
generalizations of (6.1) and (6.2) which involve the ordinary linear differential
operator
(38.2)
Let ~(M) be the class of all f E L2 (R+) such that f' is locally absolutely con-
tinuous and M(f) E L 2(R+).
Suppose that the functions p, q satisfy the conditions (i) p' E C(R+) with
p(x) > 0 for x ;::: 0, and (ii) q E C(R+) with q(x) ;::: -d for x ;::: 0 and some
d ;::: O. In analogy with the relation between the inequalities (6.1), (6.2), it is
shown in [73] that a necessary and sufficient condition for the existence of a
positive constant J{ = J{(p, q) such that (38.2) holds for all f E ~(M) is that
there exists a constant k > 0 such that
J
00
for all f E ~(M) and all pER \ {O}. Moreover, in this case J{(p, q) = 4k- 2 • A
complete discussion of the possibility of such inequalities (38.2), (38.3) depends
on a spectral analysis of the second order equation M (y) = >..y and the fourth-
order equation (M - (3)2y = Ay. (Additional differentiability assumptions on p, q
are also required.) Here we only mention that when an inequality (38.2) holds
for all f E ~(M), then J{ ;::: 4, and that if q( x) ;::: 0 but q( x) ¢ 0 on [0,00), then
J{(p, q) = +00.
18 CHAPTER I
In the same way that the inequality (6.2) is easier than (6.1), the inequality
is much easiler than (38.2). Indeed we shall follow [73] and show that if ~(M), p,
and q are as above but with R+ replaced by R, then (38.4) holds for all f E ~(M),
with Ko(p, q) = 1. Moreover, with Ko = 1, inequality holds in (38.4) if and only
if f is a solution of the eigenvalue problem M(J) = >.f such that f E ~(M).
PROOF: The proof depends on the fact that
x
J J
X
[pf'2 + qf2] dx = pf' fl:: x + f M(J) dx.
-x -x
Now let X -t 00 and note that f M(J) E L(R) since f and M(J) are in L 2(R).
Hence
J J
00 00
follows. But now (38.4), with Ko = 1 and equality as asserted, follows from
Holder's inequality.
To prove (38.5) we deal with the case that X - t +00; the other limit will follow
in the same way. As above, we have
J
x
[pj'2 + (q + d)l] dx = p(X)j'(X)f(X) - p(O)f'(O)f(O)
o
J
x
+ [JM(J) + df2] dx.
o
As X - t 00, the integral on the right side converges since f M(J) E L(R) and
f E L 2 (R). However, so also does the integral on the left side coverage because
if not then, since q + d ~ 0, we would have both the integral diverging to +00,
and p(X)f(X)f'(X) - t +00. The latter is impossible since it would imply that
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 19
f(X), f'(X) have the same sign for sufficiently large X and this is impossible for
f E L 2 (R+). It follows from this discussion that we necessarily have
(a) pl/2 f' E L2(R+), Iqll/2 f E L 2(R+), and
(b) lim p(X)f'(X)f(X) = 0: exists (finite).
x-+ex>
Clearly 0: ~ 0 since 0: > 0 would again imply that f(X)f'(X) > 0 for
all sufficiently large X. We shall assume that 0: < 0 and obtain a contradic-
tion. Without loss of generality we may assume that f(x) > 0, f'(x) < 0 and
p(x)f(x)f'(x) < 0:/2 for x 2: Xo say. Hence for X> X o,
J J J
x x x
pl/2 f' f dx = p-l/2(pf f')dx < ~ p-l/2 dx,
Xo Xo Xo
or
J J
x x
0< p-l/2 dx < ~ (pl/2 f')f dx, (X> Xo).
Xo Xo
+ex>
Since f E L2 and pl/2 f' E L 2, it follows that J p-l/2 dx < 00. Hence
Xo
x
(c) P(x) = J p-l/2 dt -t R as x - t 00.
o
x
Let Q(x) = J(q + d + l)dt. Since q + d 2: 0, we have
o
(d) Q(x)2:x for x 2: o.
In addition to (a), (c), (d) we shall need the following general result which is
easy to prove (or see [9, Th. 223]):
x
(e) If 9 E L 2 (R+), then lim x- 1 / 2
x~oo
J0 9 dt = O.
We have
x x x
J pl/2 f' dx = J p-l/2pf' dx = P(X)p(X)f'(X) - J P(pf')' dx
o 0 0
J J
X X
= P(X)p(X)f'(X) + PM(f) dx - Pqf dx.
o 0
20 CHAPTER I
We shall now show that the product of [Q(X)]-1/2 with each of the three integrals
in the last equation tends to zero as X --t 00; since P(X) --t R> 0, it will follow
that
(38.6) lim [Q(x)]-1/2 p(x)f'(x) =
x-+co
o.
Using (d) and (e) with 9 = pl/2 q', we have
x X
[Q(X)]-1/2 / P M(f) dx ~ X- l / 2 / pl/2 f' dx --t o.
o 0
~ F(X,)+ V,
( x
P'(q + d + 1) dx
) 1/2 (X
V,(q + d + 1)1' dx
) 1/2
( x ) 1/2
~F(Xd+R[Q(X)-Q(Xl)]1/2 V,(q+d+l)f 2 dx
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 21
Given c; > 0, there exists Xl such that the last integral is ::; C;2 R- 2 for all X > Xl.
Hence
J
x
0< [Q(X)]-1/2 P(q + d + l)f dx ::; X- l / 2 F(Xt) + c; for X > Xl,
o
and since c; is arbitrary, it follows that
J
X
[Q(X)]-1/2 P(q + d + l)f dx --t 0 as X --t 00.
o
This completes the proof of (38.6).
Next we consider [Q(X)]1/2 f(x) as x --t 00. Suppose that
Then there exists Xo > 0 such that P(x) 2 fJ2jQ(x) for x 2 X o, and so for
x 2 X o,
J+ J ~tx;
X X
(q d + 1)f2 dx 2 fJ2 q 1 dx
Xo Xo
and this, together with the existence of the limit in (b), implies that a = O. This
contradicts the assumption a < 0, and proves that a = 0 must hold.
REMARK 4: Note that for the original inequality (38.2), the above method of
proof shows that under the given hypotheses on p,q, (a) and (38.5) hold, and
!00
!
00
f2 dx
) 1/2 (
! 00
M(f)2 dx
) 1/2
22 CHAPTER I
Hence a result of the form (38.2) would follow if one could obtain bounds of the
form Ip(O)f(O)f'(O)1 ~ IIfIlIlM(f)II· Unfortunately, as can be seen in [72], even
in the simple case M(f) = -!" such inequalities are both difficult to obtain and
do not lead to sharp values of K(p, q).
REMARK 5: A few specific examples of (38.2) are given in [73, §17], and in
[74, pp. 13-14]. In particular, we state the following two results from [74]: if
f,!" E L 2 (R+), then
where K(Ji) = 4 for -00 < Ji ~ 0, K(Ji) = +00 for Ji > o. If a> 0, f E L 2 (a, 00),
(XT!,), E L 2(a, 00) then
where K(r) = 4 for -00 < r ~ 0, K(r) = [cos [(3 - r)-111"]]-2 for 0 ~ r < 1,
and K(r) = +00 for r ~ 1.
(For (38.7) see also [77].)
W. N. Everitt and M. Giertz [76] also considered inequalities of the form
in the Hilbert function space L 2 ( a, b) for arbitrary open intervals (a, b) with the
differential operator M defined by (38.1).
39. The number C 42 (2, R+) was discovered by J. S. Bradley and W. N. Everitt
[79] in 1974. Namely, C 42 (2, R+) = 2k 11 , where k1 is the smaller of the two
positive roots of the equation
k4 - 2k2 - 4k + 1 = o.
Note that 8.87 < C42 (2, R+) < 8.88.
N. P. Kupcov [80] in 1975 succeeded in reducing the problem of finding C nk (2, R+)
to finding roots of explicit polynomials. In this case inequality
Ilf(k) 112 ~ Mn,kllfll~n-k)/nllf(n) II~/n, where Mn,k = Cnk(2, R+), is equivalent to
where
M;' k = _1 {(~)k/n + (_k )(n-k)/n}.
, 'Yn,k n n- k
An-k+t A n-k+2 An
A n-k+2 A n-k+3 An+1
D('Y) = =0,
An An+1 A n+k-1
where
An-j = f
m=l
c-;;/ exp (iC:7rk ] tir- 1Fm(t) dt) ,
0
J
+00
1- s 2k t
Fm(t) = (s2t1/k _ ~)(s2n _ ts2k + 1) ds,
o
m7rZ
Cm = exp -k- (m = 1, ... ,2k);
40. Some new proofs of results of Kolmogorov- Landau's type are given by 1. J.
Schoenberg [81] and A. S. Cavaretta [82]. Note that Cavaretta gave a surpris-
ingly easy proof of the Kolmogorov inequality.
42. Some related results are given by G. V. Kirsanova [84] and V. K. Dzjadyk
and V. A. Dubovik [85], [86].
Results from [85] and [86] give generalizations of results of A. M. Rodov (see
15.).
45. Let f and f" be continuous on the unit interval I = [0,1] and let IIflll :=
:s :s
ess suplf(t)1 1 and 111"111 A. Then
2+.4 if a :s A :s 8,
(45.2) 11'(1/2)1 < { rnA4
- v2A if A> 8.
46. Some inequalities with bounded derivatives of higher order are given by H.
Kallioniemi in [91]'[92], [93].
(47.1)
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 25
is shown to hold for all f E £P( M, A, u) for which the right-hand side is finite,
where cp = 22 / p for 1 < p ~ 2 and cp = 22 - 2 / p for 2 ~ p < +00. Specializing A
to be d/dx acting on LP[O, +00), the inequality becomes
(47.2)
This is the classical inequality of Hardy, Landau and Littlewood, but with sub-
stantially improved constants.
J. A. Goldstein also proved in [97] the following upper bounds on Cp ( J), where
Cp ( J) denote the smallest constant cp for f E LP( J) with second derivative
f" E LP(J) (in (47.2) of course):
Cp(R+) ~ 2(5/4)2/P -l for 1 < p < 2;
Cp(R) ~ 22 / p - 1 for 1 < p < 2;
Cp(R) ~ 21 - 2/ p for 2 < p < +00.
J
u
C is the space of all continuous 27r-periodic function with the nonn IIIlIe =
max II(x)l.
For the elements I E c(r+1) (r = 0,1, ... ) we have the following sharp in-
equality
51. I. J. Schoenberg [101] proved that if 11111 ~ A and 1If" + III ~ 1 (11·11 denote
the uniform norm on R) then
(52.1)
1 ( n-l)
~ I(x)+{;Fk -b-a
1 Jb I(t)dt
a
M (x - a)n+1 + (b _ x)n+l
n(n+l)! b-a
where Fk is defined by
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 27
53. Denote by X a real Banach space with norm II . II. If A is the generator of
a strongly continuous contraction semi group and x is in the domain of An, some
integer n 2:: 2, then
(53.1)
(53.2)
His elementary proof is based on Gram's inequality (see for example, AI, pp.
45-47). As a corollary the inequality (6.1) is obtained.
In [109], Copson dealt with inequality (38.7) and with an analogous inequality
over (-00, +00).
55. In [57] E. Hille conjectured that for Lp(R), II/(n)lI; ~ 411/I1pll/(2n) lip. Z.
Ditzian [110] showed that
56. Further references of interest are [111], [112], and [113], where relationships
with inequalities of this type are related to approximation problems.
57. Let r be fixed in [0,1] and let u be the solution of the equation r =
4 cos ~u/(3 + cos u), u E [0,1r]. If I is complex-valued with
{
,\ + ,\ coth ~ + 1 tanh~, for 0 :::; A :::; ,\2 coth2~,
11f' ± ,\flll :::; ,\ + 2VA, for ,\2 coth ~ < A.
Sharp estimates are also obtained for 1If'III if Ilflll :::; 1 and 11f" +,\2 fill:::; A.
61. Some integral inequalities are obtained by C. Bennewitz and W. N. Everitt
[118] and P. R. Beesack [119].
62. Some sharp inequalities for the best uniform approximations of periodic
functions are established by V. V. Zuk [120]. For some previous results of the
same author see references from [120].
where Ilgll denotes the upper bound of Igl on R2 and fi 2) the second derivative
in the direction e = (e1' e2).
This is a result from [121] (V. N. Konovalov).
64. R. Redheffer and W. Walter [122] considered an inequality of type (1.1) for
p = +00, but with M~(+oo,I) = max (III-nMo(+oo,I), Mn(+oo,I)) instead of
30 CHAPTER I
Mn( +00, I), where III is the length of interval I. Generalizations for functions
of several variables are also given.
65. Inequalities for fractional derivatives on the half-line are considered in [123]
by V. V. Arestov.
67. B. Neta [125] has determined the numerical values 141 = 0.339246 and
142 = 0.225270, for (39.1). In the same article B. Neta has given lower and
upper bounds for the best possible constants In,k and Mn,k for n ~ 10.
An additive inequality of the form Ilf(n-l)lIp ~ Kllfllp + IIf(n)lIp for f E
Lp(O, l) is considered by V. I. Burenkov [126]. The best possible constant K is
obtained.
68. Here, we shall give several results from [127] of M. K. Kwong and A. Zettl.
Let us consider the inequality (31.1), with conditions given in 31.
This inequality is equivalent to each of the following statements.
1. There exists a constant K, depending only on p, q, r, n, k and such that
for all ..\ > 0 and y E W;'r( J) we have
For any p satisfying 1 ::; p ::; +00 we have K(p, M i ) = K(p, R), i = 1, ... ,7.
Furthermore, the interval [0,1] in the definition of Mi, for i = 1, ... ,5, can be
replaced by any compact interval [a, b].
We also have K(p, Ms) = K(p, R), where Ms = {y E W;(R+): y'(O)y(O) 2:: O}.
Further, let
69. Of course, related weighted inequalities are also of interest (see for example
(6.8)). The following problems are considered:
(69.1) There is a constant K such that for all y in a class D of functions with
domain I,
(69.2) For each c > 0 there is a number K(c) such that for all y E D,
J
I
Nly(j)IP 50 K(c) J
I
WlylP +c J
I
Ply(n)IP.
e
(69.3) There exists ~ 0 (e > 0 for j f= 0), TJ > 0, K > 0, and a set r of positive
numbers such that for all c E rand y ED,
Inequalities of type (69.2) have been given by Evans and Zettl [130] and M.
Kwong and A. Zettl [132],[131], new results for (69.4) have been given by Kwong
and Zettl [127],[134],[133] (see also R. Amos and W. N. Everitt [135] and W.
N. Everitt [136]); for additional references see [137].
Recent work on product type inequalities involving powers of an operator may
be found in M. Kwong and Z. Zettl [134] and [138] and V. Ph6ng [139].
70. Let e > 0; Se a net on R (real line) with e step; Wn(Se,R) - set of all
functions defined on R with locally absolutely continuous derivatives I(n-l)(x),
such that IIllIs. = sup II«x)1 < +00, III(n)IIR = ess supll(n)(x)1 < +00.
xES.
If I E Wn(S", R), then for all m, 0 ~ m < n, we have
72. The result from 59. is extended by the same author in [142].
(73.1)
where ~xn = Xn+l - x n , ~2xn = ~(~xn) and where the norm was that of Ip or
100 , and a is a constant.
In [143] best constants for 12(N) and 12(Z) (where N = {I, 2, ... } and
Z = ( ... , -1,0,1,2, ... ) were determined for (72.1) and estimates were given for
the constant in the Ip case. This result also generalized an earlier result of E. T.
Copson [144].
°
If 1 satisfies I(j) ~ for j
then on [0, T] we have
= 0,1, ... ,n + 1 and 1(j)(0) = °for j = 0, ... ,N - 1,
.- (N-k)!
Cnd +00, [0, TD = (N!)l-"/N"
75. M. Sato and R. Sato [146] gave a simple proof of the Chui-Smith theorem
from 45.
°
Let 1 E C2 (R+), 1(3) exist in R+ and 1 be bounded and non-negative there. If
m(f') ~ 0, M (1(3») < +00 and 1"(0) ~ 0, then M (1(3») ~ and:
(76.1)
(76.2)
These results are due to K. N. Boyadziev [147]. He also proved (76.1) and (76.2)
under slightly different assumptions.
then T" Mol (+00, [0, TD M" (+00, [0, TD ~ P", k = 1,2,3, the upper bounds
P" are exactly described. The inequality is best possible.
A. I. Zvyagintsev and A. Ya. Lepin [149] show that if
then
For such results and for many new results see paper of W. D. Evans and W.
N. Everitt [150].
79. I. J. Schoenberg [151] proved the following result (see also [28]):
Let 9 be k-convex for k = 0, ... ,n + 1 on ( -00,0], and assume that 9 satisfies
Then
(79.1)
80. Z. Ditzian [154],[155] gave some further results about Kolmogorov inequal-
ities. For example, in [155] he proved for a wide variety of Banach spaces of
infinite or biinfinite sequences the analogue of (1.1), i.e. generalizations of (72.1),
given by .
(80.1)
where ~xm:= Xm+1-Xm, ~ixm = ~(~i-lxm) and where c(n,k,B) is the best
constant for E, nand k.
36 CHAPTER I
(r + s + 1)
Ilf (S)11 2,8 -< r 1 / 2(r r_1 /s2+ Ilflll-s/rllf(r)lls/r
l)rs/2r(2r + 1) 2 2,r'
83. Some further remarks about results from 11., 17., 25., and 52. are given
in [158], [159] by J. E. Pecaric and B. Savic. For example, a generalization of
Ostrowski's inequality (11.1) in the case when f satisfies Lipschitz's condition of
order a is given.
84. Let us consider the classical weighted Banach space Xp = LP(J, k) with the
usual norm
(84.3)
Note that Goldstein, Kwong and Zettl did not note that (84.3) is a consequence
of Kurepa's inequality (34.3). They also noted that if A generates a (Co) group
{T(t): - 00 < t < +oo} on (X, 11·11) satisfying N = sup IIT(t)1I < +00, then
fER
(84.4)
85. Let 0 :S a < 1, K(a,R) = 2/(1- a), K(a,R+) = 4/(1- a). Let J = R
or R+. Suppose y E Loo(J), y' exists and is absolutely continuous on compact
subintervals of J, and y"lyla E Loo(J). Then, y' E Loo(J) and
(85.1)
86. V. G. Timofeev [162],[163] proved for I E C(Rd) and fli E Loo(Rd), where
fli = t;=1
(a~.)2 I
•
(86.1)
38 CHAPTER I
where (8/ 8c:)f is a directional derivative and 1\ . 1\ is the Loo( R d ) norm. In fact,
in [163], only d 2:: 3 is dealt with and for d = 2, (86.1) was proved in [162].
Timofeev [163] also proved (86.1) in the case when /1·11 is L 1 (R d ) norm.
87. J. E. Pecaric [164] gave some remarks about results from [145].
J J
00 00
in [166] (the integral on the left may be only conditionally convergent). In fact,
w. D. Evans and W. N. Everitt [166] gave a complete analysis of the integral
inequality (88.1) for f E ~, where ~ = {J: [0, +00) - elf' locally absolutely
continuous on [0, +00) and f,1" + x f E L2 (0, +00) }.
A nonweighted case of (78.1), i.e. the inequality (38.2) is considered in C.
Bennewitz [167].
B. G. Pachpatte [168] proved the following theorem:
Let p, q be real valued continuous functions defined on I = [a, b]; p' exists and
is continuous on I. Let f, g be real valued continuous functions defined on I,
each twice continuously differentiable on I and f( a) = f( b) = g( a) = g( b) = 0.
Then
where L = b - a, An,n-l = (n - 1)!(2n - l)n and the other Anj are given
inductively by An+1,j = Anj + An+1,n, for j = 1, ... ,n - 1.
Some further considerations are given by the same authors in [171]'[172].
Weighted multidimensional inequalities of these types were considered by L.
Caffarelli, R. Kohn and L. Nirenberg [173], C. S. Lin [174], and R. C. Brown
and D. B. Hinton [175].
90. A. 1. Zvyagintsev [176], [177] gave extensions of results from [148] (see 77.).
He also proved inequalities of type (1.1) for n = 4, p = +00, and 1= [0, T]. He
got the following constants in (1.1).
(90.1)
in the case when T4 Mo (+00[0, T])-l M4 (+00, [0, T]) 2:: 48 (17 + 12V2).
91. A generalization of Timofeev's result from 86. for the case of Kolmogorov-
type inequality
where alOej is the derivative in the ej direction, and the nth iterate of the
°
Laplacian D. n f belongs to Loo(Rd), < k,2nj is proved by Z. Ditzian in [179].
This is an improvement of a result of K. N. Boyadzhiev [178]. A generalization
for Banach space of function on Rd is also given in [179].
40 CHAPTER I
93. Various extensions of Fink's result from [145] (see 74.) are given by D. S.
Mitrinovic and J. E. Pecaric [182],[183],[184]. For example, the following result
is valid:
Let i,j, am, bm, an, bn E No = {O, 1,2, ... }, where nand m are positive
numbers such that n ~ m. If f is a two-place completely monotone func-
tion on [0, +00 )2, i.e. let its partial derivatives of all orders exist and satisfy
(_l)n+m Df D;"(f) ~ 0, n,m = 0,1,2, ... , then
1 1 1
( _l)i+i+ m(aH) D~+am D~Hm f(x, y)) m ( _l)i+i D;D~f(x, y)) ,,-m
1
~ (_l)i+j+n(aH)D~+anD~Hnf(x,y))".
94. The following results are obtained by H. Kraljevic and J. E. Pecaric [185]
94.1. Semigroups.
In this section T(t) is a strongly continuous semigroup on X which is uniformly
bounded:
J
t
(94.5)
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 41
Thus
(94.7)
S2 = T(s)x - x - 2"
sAx + "2 11 s
(s - u)2T(u)A3 x duo
o
By solving this system we obtain
Ax = s2T(t)x - t 2T(s)x _ S + t x
(94.8)
ts(s-t) ts
+ 1
o
s
K(t,Sju)T(u)A3 x du,
1 1
8 8
ts t+s
L(t,Sju)du = 6' L(t,Sju)du = -3-'
o o
we obtain (94.2) and (94.3) immediately from (94.8) and (94.9) by the triangle
inequality and using (94.1).
Now, we shall minimize the right-hand sides of the inequaliteis (94.2) and
(94.3) over the domain 0 < t < s. Fix x E D(A3) and put S = at, a > 1, t > O.
We obtain from (94.2) and (94.3)
(94.10)
42 CHAPTER I
where
(94.11)
a(a) = [~~~ ~ ~;) + 1: a] IIxll, b(a) = ~a II A3 x ll,
c(a) = 2 [~(~ ~ ~1 + ~] IIxll, d(a) = M(13+ a) IIA 3 xll.
For fixed a > 1 the minimal values of the right hand side of the first (resp.
second) inequality in (94.10) is attained at t = 2- 1/ 3a(a)1/3b(a)-1/3 (resp.
t = 21/ 3c(a)1/3d(a)-1/3). Taking this minimal value we conclude that the in-
equalities
(94.12)
(94.13)
(94.15)
(94.16)
LEMMA 2. For every x E D(A 3 ) and every t > 0 the following inequalities hold:
M Mt 2
(94.21) IIAxll::::; Tllxll + -6-IIA3xll, and
REMARK: Using formulas (94.7) and (94.8) for any t, s E R (instead of choosing
s = -t) and minimizing over the domain t f= s leads to the same inequalities.
94.3. Cosine Functions.
In this section T(t) (t ~ 0) will denote a uniformly bounded (IIT(t)11 : : ; M)
strongly continuous cosine function of linear operators on X and A its infinites-
imal generator. Now, we have
J(t -
t
(94.27)
LEMMA 3. For every x E D{A3) and for 0 <t <s the following inequalities hold
(94.28)
t4+s4 t2+s2] Mt 2s 2 3
\lAx\l ~ 2 [M t 2s 2(S2 _ t 2) + ~ \lx\l + 36<)\lA x\l, and
(94.29)
2 [t2 + s2 1 ] M(t 2 + s2) 3
\lA xII ~ 24 M t 2s2(S2 _ t 2) + t 2s 2 \Ix II + 30 IIA xII·
Note, that these inequalities can be obtained from (94.2) and (94.3) by substi-
tutions t - t t2, s - t s2, \lAx\l-t !\lAx\l, \lA2x\l - t 112 \1A2 x \l, \lA3 x ll-t l~O IIA3 x \l.
Thus, we get immediately from Theorem 1 and its corollaries:
THEOREM 3. Let A be the infinitesimal generator of a strongly continuous cosine
function T{t) (\IT{t)1I ~ M, t ~ 0) on a Banach space X. For every x E D{A3)
we have
(94.30)
(94.31)
\lA2xll 3 ~ ~M2{1
50
+ a) [M a ~a+- a1) +!]
a
IIx\l·IIA 3xIl 2,
and
46 CHAPTER I
1
(95.3) IIF(x)(b - a)1I ::; 2M + 2 (H(a - x) + H(b - x».
x+h
COROLLARY 1. Let X = Y = R, F(x) = kJ f(t) dt, H(h) = Nh 2 , a < b,
x
h > O. Then we have for a differentiable function f(x) on [a, b + h) such that
If(x)1 :::; M, x E [a, b + h) and l~hf'(x)1 :::; N, x E (a, b), where ~hg(X) =
k(g(x + h) - g(x)),
N
(95.6) (b - a)l~d(x)1 :::; 2M + 2 ((x - a? + (b - x)2).
This is a generalization of Theorem 6 from [1). Namely from (95.6) we get
2M b-a
(95.7) II~hf(x)11 :::; b_ a + -2- N ,
and if b - a 2:: 2JM/N, we have
(95.8)
since the function y = 2Mx- 1 + Ifx has a minimum 2VMN for x = 2JM/N.
But, from (95.6) we can also obtain the following result. Let f(x) be a differ-
entiable function on R such that If(x)1 :::; M and l~hf'(x)1 :::; N, x E R, h > O.
Then (95.5) becomes (i.e. (95.6) for a --t x - H, b --t X + H),
(95.9)
The function y = Mx- 1 + !Nx has a minimum V2MN for x = J2M/N, so for
H 2:: J2M / N we get from (95.9)
(95.10) l~hf(x)1 :::; V2MN.
COROLLARY 2. For X = Rn, Y = R, F(x) = f(x) = f(X1, ... ,X n ), let
(95.11)
n
Ilx - yll = L Ix; - y;i (x,y EX),
;=1
and
11F[:)(h, h)11 = L
;,j
8: •
2
fx' h;h j
J
: :; L Nijlhil·lhjl·
i,j
48 CHAPTER I
ab + cd 5 (a + c)(b + d),
(95.12) and (95.13) give, respectively,
(95.14)
(95.15)
af 1
"(b, - a·)- < 2M + - " N .. (b· - a·)(b· - a·)
n
where N = I: N;j.
i,j
Therefore, if min(b i - ai) ~ 2y'M/n, then
(95.17) t:fi
.=1
S2JMN.
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 49
(95.19)
(95.20)
(95.22)
n of
'"_ <
{4±N
2'
if 0 < N ~ 4
~ OXi - 2..jN, if N > 4.
if
02f
'L.J
" ox.ox· ~A.
i,j , )
50 CHAPTER I
n of M hA
(95.23) , , - <-+-
~ ox' - h 2'
i=l •
n of
(95.24) "~ox· :::; V2MA.
i=l •
It is a generalization of (7.2).
By using (95.23) and (95.24) we can obtain the following generalization of
II: a:;2lxi : :;
(45.2):
(95.25) t
i=l OXi
of (~, ... ) ) :::; {
2 2
2+
V2A,
1, if 0 < A:::; 8,
if A > 8.
we can obtain better bounds. This is a consequence of the first remark af-
ter Theorem 1, and these results can be obtained from previous by substitu-
tions M -+ M/2. For example the following generalizations of results of V. M.
Olovyanisnikov (see 23.) are valid:
nof
(95.17') :::;V2MN, and
Lox.
i=l •
nof
(95.24') :::;VMA,
Lox.
i=l •
where the other conditions for (95.17) and (95.24) are fulfilled, respectively.
( !
00
X- 1 / 2 U dx
) 2:::; 4G ( ! 00
u2 dx +2
(
! !
00
xu 2 dx .
00
xu,2 dx
) 1/2) ,
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 51
99. Zvyaginstev [200] continues his investigation from [176]' [177] of Landau-
Kolmogorov inequalities.
101. If f is only defined on [a, b] then one can get inequalities of the Landau
type 1If'1I2 ~ Kllfllllf"11 under further boundary conditions. For example, in
Gillman, Kaper, and Kwong [202], the boundary condition that f' has a zero in
[a, b]leads to a precise calculation of the best constant.
where p > -1, v is a complex valued function belonging to C 2 ((0, 00)) and such
that x p / 2v and x- p / 2 v" belong to L2(0, 00). The best possible constants cp and
c~ are calculated and the functions for which the equality holds are given.
In studying problems related to the equation cited in the title of the paper, he
has also found the following inequality:
REFERENCES
108. COPSON, E. T., On two integral inequalitie3, Proc. Roy. Soc. Edinburgh
Sect. A 77 (1977), 325-328.
109. , On two inequalitie3 of Brodlie and Everitt, Proc. Roy. Soc.
Edinburgh Sect. A 77 (1977), 329-333.
110. DITZIAN, Z., Note on Hille's question, Aeq. Math. 15 (1977), 143-144.
111. FINK, A. M., Be3t p033ible approximation con3tant3, Trans. Amer. Math.
Soc. 226 (1977), 243-255.
112. CAVARETTA, A. S., A refinement of Kolmogorov's inequality, J. Approx.
Theory 27 (1979), 45-60. (See also Notices Amer. Math. Soc. 25, Nr. 5
(1978), A-530).
113. SCHOENBERG, I. J., "On Nicchelli's Theory of Cardinal C-Splines," New
York, 1976, pp. 251-276.
114. , The Landau problem for motion in a ring and in bounded con-
tinua, Mathematics Research Center Technical Summary Report #1563,
Madison, Wisconsin, and in somewhat abbreviated form in Amer. Math.
Monthly 84 (1977), 1-12.
115. GOODMAN, T. N. T. and S. L. LEE, Landau and Hardy-Littlewood-Polya
inequalities involving derivatives of functions, SEA Bull. Math. 2, (1978),
82-100.
116. SCHOENBERG, I. J., The Landau problem, I: The ca3e of motion3 on
sets, Nederl. Akad. Wet., Proc., Ser. A. 81 (1978), 276-286.
117. SHARMA, A. and J. TZIMBALARIO, Landau-type inequalitie3 for bounded
interva13, Period. Math. Hungar. 9 3 (1978), 175-186.
118. BENNEWITZ, C. and W. N. EVERITT, Some remarb on the Titchmar3h-
Weyl m-coefficient, Proc. of the Pleijel Conf., Uppsala, (1979), 49-108.
119. BEESACK, P. R., A simpler proof of two inequalitie3 of Brodlie and
Everitt, Proc. Roy. Soc. Edinburgh A 84 (1979),259-261.
120. ZUK, V. V., Nekotorye tocnye neravenstva meidu nailuUimi priblizenijami
periodiceskih funkci{ i moduljami nepreryvn03ti vyssih porjadkov, Vestnik
Leningradskogo Univ. No. 19 (1978), 35-42.
121. KONOVALOV, V. N., Tocnye neravenstva dlja normfunkci{ tret'ih ca3tnyh,
vtoryh 3mesannyh iIi kosyh proizvodnyh, Mat. Zametki 23 (1978), 67-78.
122. REDHEFFER, R. and W. WALTER, Inequalities involving derivatives,
Pacif. J. Math. 85, (1979),165- -178.
123. ARESTOV, V. V., Inequalities for fractional derivative3 on the half-line,
Approx. theory, Warszawa 1975, Banach Cent. Pub. 4 (1979), 19-34.
124. MILOVANOVIC, G. V. and 1. Z. MILOVANOVIC, On a generalization
of certain results of A. Ostrowski and A. Lupas, Univ. Beograd. Publ.
Elektrotehn. Fak. Ser. Mat. Fiz. No. 634-677 (1979), 62-69.
125. NETA, B., On Determination of Best-Possible Constant3 in Integral In-
equalities Involving Derivatives, Math. of Comptuation 35 (1980), 1191-
1193.
LANDAU-KOLMOGOROV AND RELATED INEQUALITIES 61
J J
211" 211"
J J
b 2 b
J J
b 2 b
(2.1) f'(x)2 dx 2 (b: a) f(x)2 dx.
a a
where f and f' are continuous functions, f(a) = f(b) and where p is a positive
continuous function on (a, b).
In papers [10]'[11] W. Stekloffproved that (2.1) is valid if either f(a) = f(b) =
b
o or I f( x) dx = 0 is valid. A further generalization of (2.1) is given in Stekloff's
a
paper [12].
4. Inequality (1.3) can be also found in the book [16) of J. Hadamard from 1910,
but in that year Ja. D. Tamarkine [17) proved the following result:
If a continuous function f has two continuous derivatives in (a, b) and if f( a) =
b
feb), J f(x) dx = 0, then
a
J ~ (b: a) J
b 4 b
J a) Jf(x)2 dx.
b 4 b
(4.2) rex? dx ~ (b ~
a a
For some related results see papers of M. Picone [19) and G. Cimmino [20).
5. The form of the inequalitiy (1.2) was slightly changed by E. E. Levi [21)
in order to accomodate other boundary conditions. For example, let f have a
bounded derivative on (a, b) and let f( a) = f( b) = 0. If A and B are disjoint
measurable sets such that A U B = (a, b) then
J
a
b
f(x)2 dx S (b ~ a)
2
J
A
f'(x? dx + Ilflloo(b - a) J
B
1f'(x)1 dx,
and
WIRTINGER INEQUALITY AND RELATED RESULTS 69
If(x)f'(x)1 dx ~ (b - ~~2 +
" A B
6. A. Pleijel [22] proved that if f is periodic of period 27r with f" E L2(a,b)
then
2,..
2,. 2,. (2,..)2
ff'(x)2dx
27r ;,..
f f"(x)2 dx
J
0
f'(x? dx ~ 27r J
0
f(x)2 dx - J
0
f(x) dx
J
2,..
The right hand inequality implies (1.3) but has stronger hypothesis. On the
other hand it is a better inequality as it stands. Janet [23] had proved this part
earlier as well as other results in [24].
As a follow-up of Janet's results, Cimmino [25] considered quotients of the
form
b
f f(n)(x)2 dx
(6.1) 1= .::,"-----
b
(p < n)
f f(p)(x)2 dx
"
under the conditions that f
have continuous derivatives of order n - 1 and f
2n-2P
have an n-fold zero at a and b. He showed that the minimum of I is ( :::! )
where vn,p is the least positive zero of the Wronskian of n independent solutions
of the differential equation
Janet in [23], [24] continued the study of finding the explicit minimum of I. The
papers [23], [24] also contain the following geometric result. Let
The two branches of this curve descend from (0,11"2) to (1,0) and then ascend to
(+00, +00). The result is that if f and f' are continuous on (1,1), then the point
(
! (1(0)2 + f(1)2)
1 '
i
1
f'(x)2 dX)
J f(x)2 dx J f(x)2 dx
o 0
lies on or above C.
Other results related to the quotient I are in M. Janet [28],[29],[3o], G. Cim-
mino [31] and the doctoral dissertation [32] of Tcheng Tchou-Yun.
(ii) (Th. 254) If u > 4, yeO) = 0, y(l) = 1, and y' is L2, then
1 ( y2) d x > -
uy ,2 - - 2-
/
x2 - 1- 2a'
o
/1( uy
,k yk)
- xk
1
dx ~ (k _ 1)(1 _ >.)'
o
where>. is the (unique) root of
u(k _1)>.k-1(>. - 1) + 1 = 0
where
1 (2k. 11"
C = 2k _ 1 -;- sm 2k
)2k
There is equality only for certain hyperelliptic curves.
(v) (Th. 262) Ify(O) = y(I) = 0 and y' is L2, then
/
1 y2
x(I- x) dx <
1/1 ,2
2" y dx,
o 0
unless y = x/(ax + b), where a and b are positive, in which case there is
equality. More generally, if m > k > 1, r = T - 1, and y' is positive and
Lk, then
where
K _ 1 ( rr(m/r) )
-m-r-I r(I/r)r«m-I)/r) ,
unless y = x/(ax r + b)l/r.
Note that Theorems 257 and 258 and Scheeffer's and Wirtinger's inequalities,
respectively, and that for (vi) is given the reference Bliss [34].
8. The Bohr inequality from 1935 (see [35]) for almost periodic functions is
connected to the previous results. Namely, if 1 has no spectrum in [-.x, .x], then
10. Generalizations of results from 7., i.e. of result from Chapter VII of [33) is
given by V. 1. Levin [41). His paper contains the proof and some applications of
three general inequalities:
J J
1 1
J ~J
1 1
and
J J
00 00
(10.3) h(x)w(x)k dx ~ w'(x)k dx,
o 0
WIRTINGER INEQUALITY AND RELATED RESULTS 73
where fjJ E fjJj, i.e. a) fjJ(O) = 0 b) fjJ'(l) = 0, c) fjJ" exists almost everywhere
in (0,1), fjJ" 1= 0, fjJ" ~ 0, and d*) (fjJI+e)' E Lk(O, 1), where c > 0 is arbitrarily
small. If fjJ satisfies instead of d*) the condition d) fjJ' E L k(O,l), we say that
fjJ E fjJ I. The sign of equality can occur in (10.1) only if fjJ E fjJ I, and then it occurs
for y = CfjJ(x) (C ?: 0 if k i:- 2q); if fjJ E fjJj \ fjJI we have in (10.1) strict inequality
with the best constant.
In (10.2) z(x) is the integral of z' (z(O) = 0), z(l) = 0, z 1= 0 and z' E L2 q (0, 1);
g(x) is given by (lOA) with k = 2q, where fjJ E fjJiI, i.e. fjJ satisfies a), b') fjJ(l) = 0,
c) and d*) with k = 2q. Accordingly, fjJ E fjJII, if fjJ satisfies d) with k = 2q instead
of d*). The sign of equality can occur in (10.2) only if fjJ E fjJII, and then it occurs
for z = CfjJ(x); if fjJ E fjJh \ fjJII, we have in (10.2) strict inequality with the best
constant.
Finally, in (10.3) w(x) is the integral of w' (w'(O) = 0), w' 1= 0, w ?: 0 if k i:- 2q
and w' E Lk(O, 00); hex) is given by (lOA) (f -+ h), where fjJ E fjJjII' i.e. fjJ
satisfies a), c') fjJ" exists almost everywhere in (0,00), fjJ" 1= 0, fjJ" ~ 0, and d'*)
e
(fjJI+e)' E Lk(O,e), (fjJl-e)' E Lk(e, oo) with a certain 0 < < 00, where c > 0
is arbitrarily small. We say that fjJ E fjJIII if fjJ satisfies d') fjJ E Lk(O, 00) instead
of d'*). The equality can occur on (10.3) only if fjJ E fjJIII and then it occurs for
w =" CfjJ(x)j if fjJ E fjJ~II \ fjJIII, we have in (10.3) strict inequality with the best
constant.
(11.1 )
and
74 CHAPTER II
(11.2)
(12.1) (~ a ! )
IJ(t)IPdt
lJp
~ ~H (~, q ~ 1)
(a)
aq- l ! 1f'(tWdt
lJq
where
(12.2) J
o
IJ(t) - ~(m
2 P -1 411' P
0
1f'(t)IP dt.
a
Now if one takes p = 2 and J J( t)dt = 0, we get
o
(12.3)
which improves (1.3). This inequality for a = 211' appears in Benson [45]. Of
course a scale change is not important in obtaining such results. For other results
related to (12.1) see E. Schmidt [46], Bellman [47] and B. Sz. Nagy [48].
WIRTINGER INEQUALITY AND RELATED RESULTS 75
13. Northcott's result (8.3) was generalized by R. Bellman [49] who showed that
7r
7r 7r
where k, n are integers and an are the numbers that appear on the right hand
side of (8.3).
For k = n = 1 the inequality (13.1) is weaker than (1.2).
14. Wirtinger type inequalities have a close connection with differential equa-
tions. Beesack [50] established the following one.
THEOREM 1. Let the differential equation
(14.1) Y" + py = 0
be disconjugate (see Ch. 6) on (-a, a), i.e. let a solution Yl exist such that
a
Yl(X) > 0 on (-a, a). Further assume J p(x)dx 2 o. If f(±a) = 0, f' E L2, and
-a
a
J f(x)p(x)dx =0, then
-a
a a
Equality holds if and only if f( x) = AYl (x) where A = 0 if either Yl (a) -j. 0 or
Yl( -a) -j. O.
a
Beesack also considered inequalities where J f'(x)2 dx on the left in (14.2) is
-a
a
replaced by J f" (x)2 dx.
-a
Other higher derivative versions are given by W. J. Kim [51] and Z. Nehari
[52]. Kim's result reads as follows. 1
/
b
f
(m) 2 (b-a) IT_ (2k+1) 2/ (a_x)2m(b_x)2m
(X) > -2-
2m m-l
f(x) dx .
b 2
a k-O a
or
THEOREM 3. Let f' E L2k on [-7r, 7r] with f(7r) = f( -7r) and J f(x)2k-l dx = 0
then
J J
or or
See [53] for the cases of equality. Beesack also replaced the differential equation
(14.1) by
where r, r', and s are continuous on (a, b) with r( x) > 0 and s( x) ~ 0 where r
may have a zero or a discontinuity at some single point.
Further let p = 2k and q = 2;~1' Suppose there is a solution y(x) of (14.3)
on (a,b) which is absolutely continuous and y(x) < 0 on (a,x) and y(x) = 0 on
(x,b). Suppose y also satisfies
near aj and
J
x
near b.
b b
If f is absolutely continuous with Jr(t)f'(t)P dt < 00 and J s(t)f(t)P-l dt < 00
a a
then
J J
b b
b
with equality if and only if f(x) = cy(x) where c = 0 unless J r(t)y'(x)P < 00.
II
15. Various generalizations of (1.2) along the lines of Beesack are given by W.
J. Coles in [54], [55]. One of them is given here. Let m be an integer and set
m
n = 2m. Let k i be numbers that are 0 and 1 so that I: ki is even. Define
i=O
i
hi = I: ki and
j=O
has a solution y(x) such that (-1)mp(x)y(x) ~ 0, with strict inequality some-
where, and satisfying the boundary conditions
J J
b b
For cases of equality, see Coles' papers. He also discusses inequalities of the form
L(-1t+
i=1
i J
II
b
pi(x)f(i)(x)2dx ~ o.
JslulP :s Jrlu'IP
b b
In (16.1) we suppose that r, s are continuous and positive on ( a, b), where -00 :::;
a < b :::; 00. The Euler differential equation corresponding to the minimum
problem (16.1) is
(16.2)
if we suppose y > 0, y' > 0 or y > 0, y' < 0 respectively. In (16.1) the admissible
x
functions u are those of class AC( a, b) which satisfy u( x) = f u' dt or
a
J
b
Suppose the first (second) of differential equations (16.2) has a solution y such
x b
that y >0 and y' >0 (y' < 0) on (a,b) with y(x) = fy' dt (y(x) =- fy' dt)
a x
while as x -t a+ (x -t b-) we have
b
where p > 1 and q = p/(p -1). Let u be any function such that J rlu'IP dx < 00
=! =1
a
and u(x) u' dt (U(X) u' dt). Then the inquality (16.1) holds. Moreover
b
equality holds in (16.1) if and only if u = cy where c= 0 unless J rly'IP dx < 00
a
and r yly'IP-l(X) = 0(1) as x - t a+ (as x - t b-).
By taking y(x) = x(k-l)/p and rex) = )..x p- k with)"
= (p/lk -l1)P we get the
case p > 1 of the well-known Hardy's inequality. Many other special cases of
the inequality (16.1) are given in [56] of which we list only the following here: If
p > 1, u E AC[O, 7r /2) with u(O) = 0, then
(16.3)
o o
WIRTINGER INEQUALITY AND RELATED RESULTS 79
and equality holds only if u = cy( x) where y( x) is the unique solution of the
y
equation x = ~psin(71"/p) J(1 - t P)-1/ P dt 0 ::; y ::; 1. (cf. E. Schmidt [44] and
o
[33, Th. 256], i.e. (iv) from 7.)
(17.1)
where 81,82, ... ,8 n are certain points in the interval [a, bJ. Then the estimate
is valid on [a, b], where the numbers C 1 , C2 , • •• are defined by the expansion
L Ck tk
00
THEOREM 2. Let x(t) satisfy conditions (17.1), where we have one of the chains
of inequalities
(17.4)
M. Hukuhara [59] indicates that M. Tumura [50] has found inequalities in the
form
(17.5)
and
IIlk) 1100 ~ k(b - a t- k IIf(n) llooln( n - k )!, k = 1, ... ,n - 1,
for f E en[a, b] having n zeros in [a, b] including zeros at a and b.
These results are also obtained by G. A. Bessmertnyh and A. Ju. Levin [61].
A proof of (17.4) is also given in Levin [62].
18. J. Brink [63] considered generalizations of these results replacing the infinity
norm by any of the usual p-norms. In particular he has considered the inequality
(1.1) with the hypotheses that f has n zeros on the interval [a, b]. If 1 < p ~ 00
then he proves that one may assume that there are n zeros at the endpoints. He
therefore considered the problem: Find the best possible constants K(n, k,p, q)
such that
(18.1)
with
kk(n _ k)n-k
K(n, k,oo, 00) = I( -k)1.
n.n
so that these numbers are known explicitly. Furthermore, the constants may be
of interest to study in themselves. For example Brink has shown that:
(a) K(n, k,', q) is increasing;
(b) K(n, k,p,') is decreasing; and
(c) K(n,k,·,q) is log convex.
WIRTINGER INEQUALITY AND RELATED RESULTS 81
(19.5)
82 CHAPTER II
For 1 < p, q one may take all the ai at the ends of the intervals. See the discussion
in Chapter VI on disconjugacy for applications of these inequalities. Again the
properties a), b), and c) of 18. hold. We particularly want to mention the
following special cases:
20. H. Federer, W. Fleming and R. Rishel [67], [68],[69] proved the inequality:
(20.1)
WIRTINGER INEQUALITY AND RELATED RESULTS 83
where m > 1, IDul is the length of the gradient Du of u, for every u belonging
to a broad class of functions which vanish at infinity. The inequality (20.1) is
sharp.
Note that the constant in the right-hand side of (20.1) is the isoperimetric
constant.
In connection with this result are papers of M. Miranda [70] [71],[72]. Sobolev's
inequalities are considered in his book [73].
21. A slightly different weighted version of (1.2) is given by Troesch [74]. Let h
be a positive function with -h convex and h' piecewise continuous on [0,1] and
h'(O) ~ O. If f' is piecewise continuous with f(O) = 0, then
1
J h(x)f'(x? dx
(21.1 ) ~o~__~~_____ > __
71"2
1 1 - 4
J hex) dx J f(x)2 dx
o 0
22. Inequalities which involve linear combination of values of f that are some-
what like (1.2) are given by J. B. Diaz and F. T. Metcalf [76].
THEOREM 1. If f is continuously differentiable on (a, b) and suppose felt) =
f(t2) for a ~ tl ~ t2 ~ b, then
(22.1)
j
a
[J(I) - J(t,)[' dl ~ :' max (I, - a)', (b - I,)', (I, ; t, r) j f'(x)' dx.
a
If moreover f(a) = feb) then the right hand side of (22.1) may be replaced by
:2 J
b
b
If further, all of the above hypotheses and (b - a )f( tl? ~ 2f( it) J f( t)dt holds,
a
then
J :2 J
b b
23. In another direction one can replace integrals by sums of integrals as in the
following result, obtained in various ways by I. Halperin and H. Pitt [77], W.
Miiller [78], L. Nirenberg [79], and R. Redheffer [80]. If f" is continuous on [a, b]
then for every c; > 0
J J J
b b b
where K(c;) = ~ + (b3a )2 and P = 1, Q = 12. This is best possible in the sense
that if P < 1 or Q < 12 then (23.1) is not correct for all f, and c; > o.
A. M. Pfeffer [81] generalized (23.1) to
J J J
b b b
(r!J ",-1
k
(1-~), if J is a positive integer,
24. H. D. Block [82] has formed a class of integral inequalities which contain
inequalities of Wirtinger's type. Here, we shall give some special cases of Block's
general results.
Let y be a real function, defined and continuous on a x b and having a :s :s
derivative which is sectionally continuous. Then for every c i- 0
ly(t)1 2 :S cothc~b-a) J b
((y')2 +c2y2)dx.
a
WIRTINGER INEQUALITY AND RELATED RESULTS 85
If yea) = 0, then
and (when c -+ 0)
b
25. R. Redheffer [83] gave several interesting related results. For example he
gave some comments about (21.1), an inequality of Block type (see 24.), etc.
Here we shall give his inequality of Wirtinger type:
Let v and w be absolutely continuous and let v > 0, v' > 0, w > 0, w' ~ o.
Then u( a+ ) = implies°
J
b
u' w dx
r'
~ - J b
w' u 2 dx
v
a a
J
b
26. Some related results are also given in papers by Rybarski [84] and Krzywicki
and Rybarski [85],[86]. For example, the following result is proved in [86]:
Let a real function J(t), defined on the interval [-1,1] and absolutely continous
there, satisfy the boundary conditions J( -1) = J(I) = o.
If a ~ -1 then
J J
1 1
where p = (1 - t 2)-1/2.
There is equality in (26.1) only if either a > 0 and f = const (1 - t 2)(I+a)/2
or a ~ 0 and f = 0 besides the trivial case when the integrals in (26.1) become
infinite.
We also have, for a > -2,
J J
1 1
j = 1,2,
He gave a method for discovering the best possible constants K under the as-
sumptions that f(n-1) is absolutely continuous and f has n zeros at a. For
sufficiently smooth weights v and w the extremal solves a boundary value prob-
lem for a differential equation. For n = 1 and v == w == 1 he obtained the best
constants.
The cases p = q = r = 2, n = 1; p = 4, q = 0, r = 2, n = 1 for v == w == 1 are
considered by O. Arama and D. Ripianu [93] under the hypothesis that feb) = 0
and they do not obtain the best possible constants.
The inequality generalizes both Wirtinger's inequality (1.2) and Opial's in-
equality which is considered in the next Chapter.
WIRTINGER INEQUALITY AND RELATED RESULTS 87
n-l n
(29.1) '
~ " ( Xk- - Xk-l )2 ;::: . 2
4sm ( )
7r " 2
~Xk
'
2 2n-l
k=l k=l
with equality if and only if
. (k-l)7r
xk=Sln , k=I, ... ,n.
2n-l
n
If instead XnH = Xl and L:: Xk = 0 then
k=l
n n
(29.2) "
~ ' ( Xk - Xk-l ) ;:::
2 4sm
' 2 :;;
7r ' " 2
~Xk
k=l k=l
with equality if and only if
2k7r 2k7r
xk=A cos--+B sm--, k=I, ... ,n.
n n
n n
( . 2 27r . 2 7r ) 2
L(xi - XiH)2 ;::: sm -:;;: - sm :;; t;(Xi + Xi+m)
i=l
+4sm -. 2 7r Ln
x·2
n ;=1 •
i = 1, ... ,no
30. In the papers [100) of Mitrinovic and Vasic, and [101) of Janet the history
of inequality (1.2) and its relatives are given. The name of Wirtinger's inequality
is not justified. In the paper [100) the topics are exposed chronologically as in
this Chapter, where the text has been updated and enlarged.
31. Extensions of 7(i) are given by R. Redheffer [102). Some results concerning
Wirtinger's inequality can be found in papers of U. Richard [103) and A. Lupa§
[104].
32. Generalizations of results from 14. and 16. are given by the same author in
[105].
(34.1)
This was shown by L. Nirenberg [109]' and with the constant c = 4/7r by H.
Fujita [110]. G. Rosen [111] gave the best possible constant
(34.2) 4/~
c -_ 3" 7r -_ O. 078 ...
36. I. J. Schoenberg [113) proved Theorem 1 from 17. but with less restrictive
conditions for functions, i.e. x(t) E Cn-1(I), x(n-l)(t) satisfies a Lipshitz con-
dition and X(II)(t) (v = O,l, ... ,n -1) vanishes at some points of I. He also
WIRTINGER INEQUALITY AND RELATED RESULTS 89
noted that his result for entire functions from 1936 [114] is a consequence of
these results.
37. A related result for functions of several result is given -by V. G. Hryptun
[115]. One-dimensional case of his result is proved by S. Mandelbrojt [116] in
1940.
40. G. A._ Sadrin [122] studied the best constants in some inequalities of the
form
J J
11" 11"
(40.1) pu2 dx ~c (pu; + qu 2 ) dx (p(x) > 0, p(x) > 0, q(x) > 0).
o 0
41. N. Anderson, A. M. Arthurs and R. R. Hall [123] proved the following result:
if
(41.2)
(43.1)
45. Let u be any real (or complex) valued function, defined on the whole m-
dimensional Euclidean space Rm, sufficiently smooth and decaying fast enough
at infinity. Moreover let p be any number such that: 1 < p < m. Then
(45.1 ) J
(
lul q dx
) l/q
::; C J
(
IDul P dx
) lip
( 45.2)
C= 7I"-1/2 m -l/p (p - 1 ) l-l/p ( r(l + m/2)r(m) ) lim
m- p r(m/p)r(l + m - m/p)
The equality sign holds in (45.1) if u has the form
l-m/p
( 45.3) u(x) = ( a + blxIP/(P-l) ) ,
(45.4)
46. Wirtinger (or Sobolev) type inequalities are also considered by N. C. Meyers
[130].
47. Sobolev type inequalities are also considered in papers of G. Rosen [131]'
F. B. Weissler [132]' T. Aubin [133] and A. Alvino [134].
J J
b b
positive half line and let J x f (x) 2 dx and J x f" (x) 2 dx be convergent. Then
o 0
00
J J
00 00
where C = 1/ (2(2v - 1)) is the best posible constant, v being the solution be-
tween 1/2 and 1 of
J
x/2
J J
211" 211"
J11'1 ~ JIfI
211" 211"
52. Best constants in integral inequalities of the form (1.2) for functions with n
zeros are considered by Ju. A. Melencova [146].
JIl'l JIfIV
b b
v dt ~ K(v)· (4/(b - a)t . dt
a a
valid for admissible functions u, i.e. functions u such that u(n-l) E AC[a, b],
u(n) E L2(a, b) and which satisfy appropriate boundary conditions. Three main
methods for obtaining such inequalities are given. This paper contains many
examples of new Wirtinger type inequalities, as well as earlier results due to P.
R. Beesack [50]' W. J. Coles [54], K. Fan, O. Taussky and J. Todd [94]' and W.
Leighton [150].
< 7r 4 / 3 + 5 X 10-4 •
(55.1) J
Rn
lu(xW In lu(x)ldJ-ln(x) ~ ~IIVuI12 + lI ul1 2+ IIull2In lIull,
II v ll 2 = J
Rn
Iv(x)12 dpn(x).
b b
If limvlhl P ~ limvlhl P then the two limit tenns in (56.1) may be dropped.
t_a t-b
Conditions for equality are given in [161].
In the case P = 1 the constant ak is best possible (that is, equality can be achieved
for a suitable J). The case P = 00 is (8.3) while the case P = 2r (r a positive
integer) is (13.1).
Ditzian makes an extension from Lp to more general Banach spaces and also
considered the discrete analogues with differences in place of derivatives.
59. Let 1;, i = 1,2, ... , n, have absolutely continuous (k - l)th derivatives on
[a, b]. If in addition
60. Several related results are given in papers [173], [174], [175] and [176] of B.
G. Pachpatte. The first two papers gave generalizations of the weighted version
of Wirtinger's inequality given by J. Traple [177]:
Let p be a real-valued nonnegative continuous function defined on [0, b]. If
f, 9 are real-valued absolutely continuous functions defined on [0, b] with f(O) =
feb) = 0, g(O) = g(b) = 0, then
(60.1) J
o
b
p(s)lf(s)llg(s)lds ::; ~ (
J
0
b
pes) ds
)
J
0
b
(1f'(sW + 19'(s)12) ds.
61. D. S. Mitrinovic and J. E. Pecaric [180] showed that the following result of
N. Ozeki [181] is a generalization of (29.1):
96 CHAPTER II
62. B. Florkiewitz [182]' using a method from [158] and [161] examine certain
integral inequalities of Block type.
(64.1) 11
f!
f2 dxdy ~ ell
f!
l\7fl2 dxdy,
(64.2) 11
f!
f dxdy = O.
WIRTINGER INEQUALITY AND RELATED RESULTS 97
65. There is a close connection between certain eigenvalue problems and Wirtinger
type inequalities. The extrema of inequalities in 18. are solutions to (nonlinear)
differential equations with the values of the constants being eigenvalues. Only
in the case that p = q = 2 are these differential equations linear. So the Raleigh
quotient
b
f (f1(X))2
dx
I=.c.a _ _ _ __
b
f f(x)2 dx
a
The minimum of J is '\0, the smallest (positive) eigenvalue of the boundary value
problem
I = JJ +
r[(8 2f
8X2
82f)2
8y2 dxdy
= 11
R
H f(x,y? dxdy
R
D= JJ [(~~)' + (~)'l
R
dxdy
with the idea of determining their minimum value. He mentions Janet [28] but
not the work of Poincare [14] or Levi [15].
D. M. Mangeron [199] generalized the inequalities of Schwarz (see 3.) by
finding
an f )2
J ... Jp(Xl,"" Xn) ( aXl··· aXn dXl ... dXn
InJn r ..
A .
n = J q(Xl, ... , Xn)f(Xl,"" Xn)2 dXl .. , dx n '
In the case n = 2 he considered instead the quotient K = min (if)1/2 and found
that K < ~ ~ where R is a simply connected domain with perimeter L and area
A. That is is best possible was shown by P6lya [200]. If R is convex the constant
7r /2 can be replaced by 7r /4 according to Makai [201]. See also P6lya and Szego
[202] and P6lya [203] and the papers [201] and [202] for references.
In Boll. Un. Mat. Ital. 8 (1929), 113, 164-165, the inequality
was proved when R is bounded and f is zero on the boundary of R and D is the
diameter of R, see [203]. Refinements of this result are given by I. D. Necaev
[204],[205] as well as the number An.
WIRTINGER INEQUALITY AND RELATED RESULTS 99
(65.3)
R
j u 2 dx :s:; q-l
aR
f u 2 dS, .a.u = 0 on Rj
(65.3') f (~:)
aR
2 dS:S:; q-l j(.a.u)2 dx,
R
U = 0 on aR.
(65.5)
j u2 dx :s:; J-L- 1 j U,i U,i dx, j U dx = OJ
R R R
(65.5')
au j
j u2 dx :s:; J-L- 2 j (.a.U)2 dx, an = 0 on aR, U dx = OJ
R R R
(65.5")
j U,iU,i dx:S:; J-L- 1 j(.a.u)2 dx,
au
- =0 onaR.
an
R R
(65.6)
.a.U = 0 on R, j U dx = OJ
R
f f
(65.6')
u 2 dS:S:; C 1 j(.a.u)2 dx,
au
- =OonaR, U dS=O.
an
R R aR
100 CHAPTER II
(65.7)
au
j U2 dx ~ !1- 1 j(Cl.U)2 dx, u = - =OonoR·
an '
R B
f
(65.8)
8R
u 2 dS ~ p-1 j
R
u,; U,i dx, f
8R
u dS= OJ
(65.8')
(65.81/)
j u,; u,; dx
R
f
~ p-1 (~:r
8R
dS, Cl.u = 0 on R.
(65.9)
au
j u,; u,; dx ~ A- 1 j(Cl.u)2 dx, u = - = 0 on oR·
an '
B B
The optimal values are given as eigenvalues (first unless noted) ofthe boundary
value problems (with consistent notation)
66. Variations of Wirtinger type inequalities which put boundary values into
the inequality are given by A. P. Burton and P. Smith [207]. For example one
of their inequalities is
where the interval is [a, b] with x(a) = a, x(b) = (3, and 0 < >. < 7r/(b - a).
67. Pachpatte [208], [209] gives inequalities of the Sobolev type for two func-
tions on R n and other types. In [210], he gives discrete versions of Wirtinger
inequalities.
j 1(,Wdx:5 k j IV,Wdx
n n
holds for some k and all functions for which IV<p12 is integrable, and 0 is a fixed
open bounded set in Rn. The condition is that for every sequence {<Pn} in HJ(O),
there is a subsequence that converges in L2(0).
69. Kreith and Swanson [212] have given various higher order generalizations
of Wirtinger's inequality. These are inequalities of the form
b n b
for f(n) E L2(a,b), and appropriate boundary conditions. For example, (69.1)
holds for all f with n zeros at a and n zeros at b, with p == 0 provided the Pi'S
are continuous and Pn(X) > 0 on (a, b) and b < 7]l(a), the first n - n conjugate
n
point of Ly = 0 on (a, b). Here Ly = L: Pk(X)y(n)(x).
k=O
70. Everitt and Jones [213] completed a study of an inequality showing that by
choices of coefficients the best possible constants can be arbitrarily large.
71. Two additional references are Necaev [214] and Viszus [215].
102 CHAPTER II
REFERENCES
110. FUJITA, H., On the existence and regularity of the steady-state solutions
of the Navier-Sto!ces equation, J. Fac. Sci. Univ. Tokyo Sect. 19 (1961),
59-102.
111. ROSEN, G., Minimum value for c in the Sobolev inequality II cpa II :5 cllV cfo1l 3 ,
SIAM J. Appl. Math. 21, (1971), 30-32.
112. DENKOWSKI, Z., Inequalities of Wirtinger's type and their discrete ana-
logues, Zesz. Nauk. Iniw. JagiellOIlskiego Prace Matem. Z. 15 CCLII
(1971), 27-37.
113. SCHOENBERG, I. J., Norm inequalities for a certain class of COO func-
tions, Isreal J. Math. 10 3 (1971), 364-372.
114. , On the zeros of successive derivatives of integral functions,
Trans. Amer. Math. Soc. (1936), 12-23.
115. HRYPTUN, V. G. On a certain representation of infinitely differentiable
functions, Doklady Akad. Nauk SSSR 199 (1971), 282-284, (Russian).
116. MANDELBROJT, S., Sur les fonctions indefiniment derivables, Acta Math.
72 (1940), 15-29.
117. WONG, P. K., Wirtinger type inequalities and elliptic differential inequal-
ities, Tohoku Math. J. 23 (1971), 117-127.
118. , Integral inequalities of Wirtinger-type and fourth-order ellip-
tic differential inequalities, Pacif. J. Math. 40 (1972), 739- -751.
119. , A Sturmian theorem for first order partial differential equa-
tions, Trans. Amer. Math. Soc. 166 (1972), 125-131.
120. WINTER, M. J. and P.-K.WONG, Comparison and maximum theorems
for systems of quasilinear elliptic differential equations, J. Math. Anal.
Appl. 40, (1972).
121. PEETRE, J., The best constant in some inequalities involving Lq norms
of derivatives, Ricerche di Matematica 21 (1972), 176-183.
122. SADRIN, G. A., Ob ocen!cah sverhu tocnyh !constant v neravenstvah tipa
Puan!care, uc. zap. Mosk. gos. ped. in-ta im. V. I. Lenia 460 (1972),
33-37.
123. ANDERSON, N., A. M. ARTHURS and R. R. HALL, Extremum principle
for a nonlinear problem in magneto- elasticity, Proc. Cambridge Philos.
Soc. 72 (1972), 315- -318.
124. SHISHA, 0., On the discrete version of Wirtinger's inequality, Amer.
Math. Monthly 80 (1973), 755-760.
125. FLORKIEWICZ, B. and A. RYBARSKI, On an integral inequality con-
nected with Hardy's inequality (III), Colloquium Math. 27 (1973), 293-
296.
126. EASWARAN, S., Fundamental inequalities for polyvibrating operators,
Bul. Inst. Politehn. din 18.§i 19 23 (1973), 71-76.
127. Problem No. 919, Nieuw Archief voor Wiskunde 21 3 (1973), 108-109.
WIRTINGER INEQUALITY AND RELATED RESULTS 109
200. POLYA, G., Two more inequalitei.'l between phY.'lical and geometrical quan-
titie.'l, J. Indian Math. Soc. (N.S.) 24 (1960), 413-419.
201. MAKAI, E., On the principal frequency of membrane and the tor.'lional
rigidity of a beam, "Studies in Mathematical Analysis and Related Top-
ics," Stanford, 1962, pp. 227-231.
202. POLYA, G. and G. SZEGO, "Isoperimetric Inequalities in Mathematical
Physics," Princeton, 1951.
203. POLYA, G., Circle, .'lphere, .'lymmetrization and .'lome cla.'l.'lical phY.'lical
problem.'l, in "Modern Mathematics for the Engineer," second series, New
York-Toronto-London, 1961, pp. 420-442, (edited by E. F. Beckenbach).
204. NECAEV, I. D., Nekotorye obobJcenija mnogomernogo neraven.'ltva tipa
Virtingera. I, Uc. zap. Mosk. gos. ped. in-ta im. V. I. Lenina 460
(1972), 96- -101.
205. , Nekotorye obobscenija mnogomernogo neraven.'ltva tipa Virtingera.
II, Uc. zap. Mosk. gos. ped. in-ta im. V. I. Lenina 460 (1972).
206. SIGILITTO, V. G., "Explicit a priori inequalities with applications to
boundary value problems," London/San Francisco/Melbourne, 1977.
207. BURTON, A. P. and P. SMITH, An approach to Friedrick.'l and Poincare
integral inequalitie.'l U.'ling convexity, Rad. Mat. 5 (1989), 107-114.
208. PACHPATTE, B. G., On two inequalitie.'l of the Sobolev type, Chinese J.
Math. 15 (1987), 247-252.
209. , A note on Poincare and Sobolev type integral inequalitie.'l,
Tamkang J. Math. 18 (1987), 1-7.
210. , On Wirtinger like di.'lcrete inequalitie.'l, Tamkang J. Math.
20 (1989), 211-219.
211. FONTES, F. G., On Friedrick.'l inequality and Rellich.'l Theorem, J. Math.
Anal. Appl. 145 (1990), 516- -533.
212. KREITH K. and C. A. SWANSON, Higher order Wirtinger inequalitie.'l,
Proc. Roy. Soc. Edinburgh Sect. A. 85 (1980), 87-110.
213. EVERITT, W. N. and P. S. JONES, On an integral inequality, Proc. Roy.
Soc. London Ser A 357 (1977), 271-288.
214. NECAEV, I. D., On an inequality of Steklov- Wirtinger Type (Russian),
Gos. Ped. Institut Barnaul, 11pp.
215. VISZUS, E., A remark on type of Poincare'.'l inequality, Acta Math. Univ.
Comen. 56-57 (1990), 19-22.
216. SAITAH, S., Hilbert .'lpace.'l admitting reproducing kernel.'l on the real line
and related fundamental inequalitie.'l, Riazi J. Kar. Math. Assoc. 6 (1984),
25-31.
217. , Norm inequalitie.'l derived from the theory of reproducing ker-
nel.'l, General Inequalities 6, Oberwolfach, 1990.
218. PHILLIPS, G. M., Error e.'ltimate.'l for be.'lt polynomial approximation.'l,
Approximation Theory (Proc. Sympos., Lancaster, 1969), London, 1970,
1-6.
CHAPTER III
OPIAL'S INEQUALITY
J J
a a
J iJ
a a
We obtain
J i Jf'(X)2 dx
a~ a~
If(x)f'(x)ldx ::;
o 0
and
J i Jf'(x - a)2dx.
a/2 a/2
If(a - x)f'(a - x)ldx ::;
o 0
Setting a - x = t in the second inequality and adding we obtain (1.1) and the
cases of equality.
We shall give Mallows' proof of Theorem 2'. First it is sufficient to prove
x
the theorem when f ~ 0 and l' ~ 0 on [0, a]. Indeed if hex) == J 11'(t)ldt
o
a a
then If(x)1 ::; Ih(x)1 and J If(x)1'(x)ldx::; J Ih(x)f{(x)ldx. We may assume
o 0
f(x) ~ 0 and 1'(x) ~ o. By the Buniakowski-Schwarz inequality
(2.1) ! a
If(x)f'(x)ldx ::; ~ !
( a
p(x)l- q' dx !
) 2/q' ( a
p(x)If'(xW
) l/q
x
where 1
q
+ ~q = J
1 and equality holds if and only if 1'(x) = c p(t)l-q' dt for some
0
constant c.
116 CHAPTER III
(a! )
p(x)IJ'(x)19 dx
2/9 (
!a p(xl- 9' dx
) 2/9'
The cases of equality are Ip(x)1/9 f'(x)19 = clp(x)-1/91 9' for some constant c.
P. R. Beesack [3] has obtained this result in the case q = 2 but with a proof
that preceded the discovery of Mallow's proof of Opial's inequality. He also
obtained a similar result which is a direct generalization of Theorem 1. G. S.
Yang [7] simplified Beesack's proofs and at the same time gave a generalization
of Theorem 1.
a
THEOREM 2. Let p be a positive function on [O,a] such that J dx/p(x) < 00,
o
and let q be a positive, bounded and nonincreasing function on [0, a]. If f is
absolutely continuous on [0, a] with f(O) = 0, then
J J J
a a a
x
with equality if and only if q(x) = constant and f(x) = cJ dt/p(t) for some
o
constant c.
The introduction of the measure p, which may not be symmetric, means that
a version of Theorem 1 of Section 1 does not follow from Theorem 2. Beesack
found the right generalization.
a
THEOREM 3. Let p be positive and continuous on (O,a) with J dx/p(x) < 00.
o
Let f be absolutely continuous on [0, a] so that
J J
x a
J J
a a
J J
a' a
dt dt
pet) = pet) = K-.
o a'
Jp~!) Jp~!)
x a
J J
a' a
x a
Equality holds if f(x) = A J p(t)l- q' dt on [0, a'] and f(x) = B J p(t)l- Q' dt.
o x
PROOF: Define a' and K- by (2.3). As above we may assume that f(x) ~ 0 and
f'(x) ~ 0 on [O,a'] and we get (2.1). On [a', a] we may assume f(x) ~ 0 and
118 CHAPTER III
! a
I/(x)f'(x)ldx = 1(~')2 = ~ V.
( a
f'(x)dx
) 2
and we again apply Holder's inequality. The details are the same as for Theorem
1.
If p( x) := 1 we get Opial's inequality.
J. M. Holt [8] gave a generalization of Theorem 2 which we do not reproduce
here.
J J
a a
Various authors have considered this inequality. L. K. Hua [9] showed the
validity of (3.1) if q = 1 and p is a positive integer. G. S. Yang [7] gave a proof
for p, q ~ 1 but P. R. Beesack observed that the proof is valid for p ~ 0, q ~ 1.
For q = 1 and p ~ 0 this inequality was generalized by J. Calvert [10] which we
will quote below. J. S. W. Wong [11) gave a short proof of that result.
Just as Opial's Theorem can be obtained from Theorem 2' the above result
also yields the next theorem.
THEOREM 2. If I is absolutely continuous on [0, a] and 1(0) = I(a) = 0, then
for p ~ 0 and q ~ 1,
J J
a a
P. R. Beesack and K. M. Das [12) obtained more results in this genre, obtaining
a a
inequalities of the form J s(x)l/(xW/I'(x)lqdx < k(a,p,q)Jr(x)lf'(x)/p+qdx.
o 0
We quote one.
OPIAL'S INEQUALITY 119
THEOREM 3. tet p, q be real numbers such that pq > 0 and either p + q > 1 or
p+q < o. Suppose that r, s are nonnegative measurable functions on (0, a) such
a
that Jr(x)-p+!-ldx < 00 and that the constant", is finite, where
o
If f is absolutely continuous on [0, aJ with f(O) = o and f' of constant sign, then
J J
a a
X -1
and f(x) = k2 J r(t)p+q-l dt for some constants kl' k2 with kl ~ o.
o
REMARK: As with Mallows' proof, the condition "f' as of one sign" may be
omitted.
4. We have mentioned J. Calvert's paper [10] above and now we present two of
his results.
THEOREM 1. Let f be absolutely continuous on [0, a] with f(O) = O. Let 9 be a
a
continuous complex valued function defined on the range of f and for J 1f'(x)ldx.
o
Suppose that Ig(t)1 ~ 9 (It!) for all t and that 9 is real for t > 0 and is nondecreas-
ing there. Let r be a positive continuous function in Ll-q(a, b) where!. + _ql = 1
- P
x
(p> 1). If P(x) = J g(t)dt for real x> o. Then
o
! a
Ig(x)f'(x)ldx ~F !
(( a
r(x)l-qdx dx
) l/q (
! a
r(x)lf'(x)IPdx
) liP)
,
120 CHAPTER III
a
with equality if f(x) = c f r(t)l-qdt.
o
.a
The same result holds if f(a) = 0 and equality holds if f(x) = c f r(t)l-qdt.
x
! I~/(C;; I
a
dx ~G
(( a
! r(x)l- qdx
) l/q (
!a
r(x)If'(x)IPdx
) liP)
,
x
with equality if f(x) = cfr(t)l-qdt.
o
P. R. Beesack noticed that the conditions for equality in the above two theo-
rems are sufficient but not necessary.
E. K. Godunova and V. 1. Levin [13] assumed slightly more about 9 to get
better results.
THEOREM 3. Let f be absolutely continuous on [O,a], with f(O) = f(a) = O.
a
Let r( x) > 0 and f r( x )dx = 1. If 9 and h are convex increasing functions for
o
x > 0 and g(O) = 0, then
(4.2) ! a
If(x)f'(x)ldx ~C
(
!
a
s(x)If'(x)IP
) 21p
OPIAL'S INEQUALITY 121
with C = ~ (
a
[s(t)t=P
l)¥
P. Maroni [14] gave inequality (4.2) for 1 < p :::; 2 with C replaced by C· 2P~1.
J J
a a
where'\o is the largest positive eigenvalue of the boundary value problem (5.1)-
(5.2).
6. Note that with a = w == 1 and p = 1 this reduces to Opial's inequality. We
have incorporated the review of this papt)r in our quotation of the theorem. See
Math. Reviews 35(1968), 563 by J. V. Ryff.
r'(x) f(x)2 + r(x) J'(x)2 :::; 0, r(x)s'(x) ~ 0, s'(x) =1= 0, s(x) >0
s(x) s'(x)
on [0, a], then
J:,~:))
a
THEOREM 1. Let f(n-1) exist and be absolutely continuous on [0, a] with f(i)(O) =
0, j = 0, ... , n - 1, (n 2: 1). There exists a constant en such that
J J
a a
D. Willett [17] initiated the study of this inequality using it to prove the
existence and uniqueness of solutions of linear differential equations of nth order.
He proved that en ::; !.
K. M. Das [18] improved this to show that en <
2~! (2n~1) 1/2 unless n = 1 when this reduces to Theorem 2' of Section 1.
D. W. Boyd [19] determined the best possible constants in (8.1). For n =
2m + 1, the best possible constant is 2n(~~1)!' where >'0 is the largest positive
eigenvalue of the (m + 1) x (m + 1) matrix
For n = 2m the best possible constant is (>'o2 n (n _1)!)-1 where >'0 is the small-
est positive solution of Det B(>.) = 0 where B(>.) = (b;j(>'» is an m x m matrix
with
10. We may also have discrete versions of Opial's inequality, J. S. W. Wong [11]
proved the following version of inequality (3.1) for q = 1.
OPIAL'S INEQUALITY 123
and
P- 1 q(n + 1)P)
Kn -_ max ( K n- 1 + -
pn- , ~-_--'-- (n = 1,2, ... ).
p+q p+q
where
q . ( pn P- 1 q(n + 1)P)
CO = - - and C n = mm C n - 1 + - - , (n=1,2, ... ).
p+q p+q p+q
Hp,q ~ 1,
if p 2: 0, p + q < 0, tben
n
C1 = 1 and C n = 1 + - p - I:i P - 1 (n = 2,3, ... ).
p + q i=2
11. The papers of D. O. Banks [23]' O. Arama[24] and A. Lup8.§ [25] are also
connected to Opial's inequality.
J J
a a
° J °< x <
x
x a
Let f be any function sucb tbat f(x) = I f'(t)dt and I ,1f'lp+1dx < 00. Tben
o 0
J J
a a
a
Equality holds in (14.1) if and only iff == cy, where c = 0 unless both J ry,PH dx <
o
00 and lim x ..... a y(xHr(x)y'P(x) - (p+ l)-lS(X)Yp(x)} = O. The inequality (14.1)
is sharp if either
J
x
(16.3)
126 CHAPTER III
where G(u) = uQ (W(l/u)), H(u) = F(u)Q (w(a/u)), with equality if and only
ify(x) = Ax, rex) = Bx, w(u) = CF()..u), where A, B, C and)" are constants.
For y(x) = f(x), f'(x) > 0, f(O) = 0, f(a) = b, Ij>(u) = u, F(u) = cp(u) = u 2 ,
and Q( u) = v'f+U, we get from (16.3) the following inequality of G. P6lya [34]
J
a
n ) 1/2
r =
(
~x; ,
K = ( + )-1 ( (p + q - 1) )P RP(p+q-n-A)«q(P+q_l))-l
1 qP q p + q-n-A') 0 •
where
K2
_ ( 27r n / 2
- r(n/2)
) s/p(p-s) (_p_)
(p-s)
p~.
(n+)..)
W:N
.
OPIAL'S INEQUALITY 127
J i J11!,(x)11 dx.
a a
IU(x),!,(x))ldx:S 2
o 0
(19.1 )
x
where g2(X) = (p + 1) J IflPIf'ldx - If(x)iP+l (~ 0) for 0 :S x :S a. If either
o
a a
p < -1 and both J IflPIf'ldx < 00, and J If'lp+ldx < 00, or -1 < p < 0 and
o 0
a
J IflPIf'ldx < 00, the reverse inequality holds.
o
For p > 0, equality holds in (19.1) if and only if f(x) = cx for some constant
c; for p < -1 equaliy never holds; for -1 < p < 0 equality holds if and only if
f(x) = cx for some constant c f:. O.
(20.1)
a a
x
valid for integrals f(x) = J f'dt having f(a) = O. Here, -00 :S a < b :S
a
+00, p ~ 0, q ~ 1 (or p < 0, q < 0 if f' does not change sign), and R(x) =
X )P(l-q)/q
r(x)(w(x))(p+q)(l-q)/q, Sex) = s(x)(w(X))q-l ( [w-1dt where w is a
x
positive, measurable function on (a, b) such that J w- 1 dt < 00 for x E (a, b)
a
128 CHAPTER III
while r > 0, s 2:: 0 are the coefficients in a boundary value problem involving the
differential equation
(20.2)
i
J
b
~ RIf'lkdx
a
x
valid for integrals f(x) = J f'dt. In (20.3),
a
(!W-'dt)
ll!.=..il
W(x) ~ (w(xJr' · ,
~
R(x) = r(x)(w(x)) q ,
and
where w is as in (20.1), and r > 0, s 2:: 0 are now the coefficients in a boundary
value problem involving the differential equation
in the latter case. As before, it is assumed that the boundary value problem has
a
x
a solution y(x) = j y' dt with y' > °on (a, b) (and also y(b) < 00 if k/q < 0).
21. In 1976, G. I. Rozanova gave two new papers with realted results ([39],[40]).
For example, the following results are proved in [40]:
Let y and r be absolutely continuous functions which have n absolutely con-
tinuous derivatives (n ~ 1), with y(il(O) = r(il(O) = 0, i = 0,1, ... , m - 1,
1 :::; m :::; n, y(nl(a) = r(nl(a) = 0, y(n+ll(x) and r(j)(x) (j = 0,1, ... , n + 1) do
not change sign for x E [0, a].
Let sex) > 0, and let ¢>(u) be a positive convex increasing function, < q < 1,
p> -q. Then
°
J a
where
g(x) = Ir(ml(x)1 (Ja (t - x)m-ls(t))
Ir(t)1
l':Q dt) l-q
J
a
(p + q) Ja
r(x)q(q-l)s(x)qlf(x)IPIf'(x)lqdx
o
There is equality if and only if f' does not change sign, s is a positive constant,
x
q = 1 and f(x) = A J r(t)-qdt.
o
a
THEOREM 2. Let rex) be positive on [O,a] with J r(x)-qdx < 00. Let sex) > 0,
o
a
be nonincreasing and both J r(x)q(q-l)s(x)qlf(x)IPlf'(x)lqdx < 00 and
o
a
J r(x)q(p+q-l)s(x)qlf'(x)IP+qdx < 00. H f is absolutely continuous and f(O) =0
o
then for all p < -q, q ~ 1
J a
q r(x)q(q+p-l)s(x)qlf'(x)IP+qdx
o
~ a p+q p J a
r(x)q(q-l)s(x)qlf(x)IPlf'(xWdx
([ r(x)-q) 0
J
a
r(x)-qh(x) d
+p x )p+q x,
o ([ r-q(t)dt
where
unless 1 == o.
f
a
f
e
s(x) = p(t)dt
x
=f
x
f
fJ
0/
If(x)I>'If'(x)ldx < _f3_
- A+1
~p
f If'(x)I~+ldx
0
- _a_
A+1
~O/
f
0
If'(x)I>.+ldx.
This inequality is best possible and includes Opial's inequality and its general-
izations due to S. L. Wang [46] and Z. J. Liang [47].
132 CHAPTER III
26. Y. Hong, H. Yang and D. Du [48] obtained an inequality for convex increas-
ing functions. As a corollary they obtained a result from [8].
27. Results from [12] are extended to a Fourier type integral operator in a paper
[49] of M. Dighe and V. M. Bhise.
(29.1) J a
lJ<k)(xWIJ<n)(x)lqdx ~ M(k)a(n-k)p J a
If(n)(xw+qdx,
o 0
P(I-)')
where M(k) = )..q).q ( (n:k~(l~)') ) ((n - k)!)-P and)" = l/(p+q).
For k = 0 we have Das' result, but the reverse implication can be easily proved.
Agarwal and Thandapani used (29.1) in obtaining some results for Gronwall-
type inequalities.
i (i I/(S,t)ll/,,(S,t)ldt) d.
iU
then
IU
Then
II(S,tWIf,,c,,tll"dt) ds
iU
Then
o
nondecreasing and convex. The short proof uses Jensen's inequality.
35. C. T. Lin and G. S. Yang [59] and B. G. Pachpatte [60] gave Opial type
inequalities with two functions. For example, the following result is given in [59]:
J
(1
Tben
j (i
o 0
q(s,t)lf(s,t)g(s,t)lm
X (lf12(S,t)g(s,tW + If(s,t)g12(S,tW)dt)ds
:::;2(mn+n) ( ~
b)2m+n-l
K
Ja Jb p(s,t)q(s,t) (lfI2(S,t)1 2(m+n)
o 0
a b
X = - and Y = -.
2 2
For f = g, 2m + n =a and n = (3, we have a result of C. T. Lin from [64].
: :; L M(k)qka(n-k)p JIf(n)(x)IP+qdx.
n-l a
k=O 0
136 CHAPTER III
This result of G. S. Yang [65] is a simple consequence of a result from 29. and
the arithmetic-geometric mean inequality.
A related result is given by B. G. Pachpatte [66].
Recently, L. Ju-Da [67] proved the following similar result:
n
Let Pi (i = 0,1, ... , n) be nonnegative real numbers and satisfy P = L: Pi >
;=0
Pn > 0, P > 1, f(x) be of class en on 0 S x S a and satisfy f(O) = f'(O) = ... =
f(n-l)(o) = o. Suppose that h(x), r(x) are positive functions on 0 S x S a, such
that
x
where Q =
n-l
11
i=O
«n - i - l)!)P;.
37. We shall say that function f : [0, a] ---t R belongs to the class U( v, K) if it
can be represented in the following form.
x
If the function x 1-+ <f;(x 1lq ) is concave, then the reverse inequality in (37.1) is
valid.
This is due to D. S. Mitrinovic and J. E. Pecaric [69].
The following result is a special case of the above:
Let <f;, q and p be defined as in the above result. If f(n-l) E AC[O, a], with
either f(k)(O) = a for k = 0,1, ... , n -lor f(k)(a) = a for such k, then
J J
a a
40. Recent papers on Opial's inequality include those of C. T. Lin [73], [74], [75],
L. M. Shieh [76] and Pachpatte [77]-[82]. Lin's papers give alternate proofs to
the results of Pachpatte in [77] and [78], which are on discrete versions of Opial's
inequality. Shieh [76] gives a weighted version.
n-l
: :; L Mkrk(b -
k=O
a)(n-k)p J
a
b
w(x)lf(n)(x)IP+qdx
1
a=--.
(p+ q)
REFERENCES
1. OPIAL, Z., Sur une inegaliU, Ann. Polon. Math. 8 (1960), 29-32.
2. OLECH, C., A 8imple proof of a certain re8ult of Z. Opial, Ann. Polon.
Math. 8 (1960), 61-63.
3. BEESACK, P. R. On an integral inequality of Z. Opial, Trans. Amer.
Math. Soc. 104 (1962),470-475.
4. LEVINSON, N., On an inequality of Opial and Bee8ack, Proc. Amer.
Math. Soc. 15 (1964), 565-566.
5. MALLOWS, C. L., An even simpler proof of Opial's inequality, Proc.
Amer. Math. Soc. 16 (1965), 173.
6. PEDERSEN, R. N., On an inequality of Opial, Beesack and Levinson,
Proc. Amer. Math. Soc. 16 (1965), 174.
7. YANG, G.-S., On a certain result of Z. Opial, Proc. Japan Acad. 42
(1966), 78-83.
8. HOLT, J. M., Integral inequalities related to non-08cillation theorems for
differential equations, SIAM J. 13 (1965), 767.
9. HUA, L.-K., On an inequlaity of Opial, Sci. Sinica 14 (1965), 789-790.
OPIAL'S INEQUALITY 139
1. Some preliminaries.
THEOREM 1. If p > 1, an ~ 0, and An = al + a2 + ... + an, then
unless all the a are zero. The constant (pS-)P is the best possible.
The corresponding theorem for integrals is
x
THEOREM 2. Ifp> 1, f(x) ~ 0, and F(x) =
.
J f(t)dt,
0
then
00 00
then
(2.3)
unless f == o. If
(2.4) fr(x) =
x
then
(2.5)
J J
x 00
then
(2.6)
unless f == 0.
The constant is the best possible. It is also easy to verify that, when
p = 1, the two sides of (2.6) are equal.
(iv) (Th. 331, E. T. Copson [16], Hardy [18]) If p > 1, then
(2.7)
then
(2.8) ~.A
00
A:
(A)P
<
(
P~l
)P ~.Ana~,
00
(vi) (Th. 337) If 0 < p < 1, f(x) ~ 0, J fPdx < 00, F(x) = J f(t)dt, then
o x
(2.10)
(2.11)
(xi) (Ths. 350,351 and 352, Hardy [19]) If p > 1, K(x) > 0, and
1 K(x)xS-1dx = ¢(s),
then
1 (1
dx K(XY)f(y)dy)P < ¢P (~) 1 x p - 2 f P dx,
1 (1
xp- 2 K(XY)f(Y)dY) P < ¢P (;,) 1 fP dx.
J Fpf dx ~ ~~ ( J
P
dx
)Pf/P
(xii) (Th. 271, Hardy and Littlewood [20]) If p > 1 and GU) is the geometric
mean of f over (0, x), then
where e is the best possible constant, and where the conditions of (v) are satisfied.
3. The inequalities (1.1) and (1.3) are the special cases cjJ = xt (0 <t < 1) ,
cjJ = log x, of (Knopp [23], [13, Th. 365]):
(3.1)
where an 2:: 0 (not all an = 0), 0 < q < 1, q > 0, 0 < p ~ 1, and ,X is defined by
(q + l),Xl-m
O<'x < 1, u= .
(q,X + 1)
5. In the following results under the (weighted) geometric mean of the nonneg-
ative function I(x) we define the function
where the weight-function p(x) satisfies the following conditions: p'(t) 2:: 0,
x
p(O) = 0, so that p(x) = J p'(t)dt. Furthermore we suppose that p' is the integral
o
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 149
of pIt, p(X) --+ 00 for x --+ 00 and p(x)logp'(x) --+ 0 for x --+ O. If g(x) ~ 0 is the
integral of g' (x) and p( x) log g( x) --+ 0 for x --+ 0, V. Levin [32] proved that
J J
00 00
J J
00 00
J J
00 00
J J
00 00
J JeX~
00 00
Let f(x) ~ 0 for x ~ 0 and set F(x) = Jf dx (>.. > -1), F(x) = Jf dt
o x
(>.. < -1).
If q ~ p ~ 1 and >.. -! -1, then
(7.1) ( ! 00
x-l-qA(F~X)r dx
) llq ( 00
5,B(P,q,>..)! x-l-PAfP(x)dx
) lip
The case q > p > 1, >.. = _p-l is due to Hardy and Littlewood [20), who
conjectured the best (least) value of the constant B(p,q,>..) in this case; this
conjecture was proved by Bliss [21) (see (2.13)). The case>.. > -1 is included
in the following more general result involving the Riemann-Liouville integral of
order r (see 2.2)).
If either q ~ p ~ 1 and r > p-l - q-l, or q > p > 1 and r = p-l - q-l; and if
>.. > -1, then
x-l-PAfP(x)dx
) lip
8. For some related results see papers [36) ofN. Levinson and [37) ofR. P. Boas.
Boas gave generalizations of some results of A. A. Konyuskov [38).
N. Levinson [36) proved the following generalization of Hardy's inequality:
Let p > 1, f(x) ~ 0, rex) > 0 for x> 0, and let r be an absolutely continuous
function. If for some >.. > 0 and for almost all x
p- 1 xr'(x) 1
-p- + rex) ~"X'
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 151
and if
J
x
J J
+00 +00
H(x)Pdx ~ AP f(x)Pdx.
o 0
9. The following result was attributed by V. I. Levin and S. B. Steckin [39, Th.
D83] to V. V. Stepanov:
00
J JfP
+00 +00
J J
00 00
J J
b b
for all al > 0, ... , an > O. N. G. De Bruijn [41] established the asymptotic
behavior of An if n - 00.
He showed that
(10.2) 2~2e
An = e - (logn)2 +0
(1)
(logn)3 .
H. S. Wilf [42] gives a similar result for Hardy's inequality in the case p = 2,
i.e. for
(10.3)
He shows that
(10.4) 16~2
An = 4 - (logn)2 +0
(log log
(logn)3
n) .
11. Hardy's inequality was also considered in F. A. Sysoeva [43], and J. Kadlec
and A. Kufner [44].
n 00
Then
00 00
where Fn (a, p) denotes the quasi-arithmetic mean of a with weight p, with respect
n
to F. If further C = lim ~ L: bk , then the constant in (12.1) is best possible.
n-+oo k=l
The following results are special cases of this theorem:
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 153
(12.2)
(12.3)
(12.4)
(12.5)
00
L
n=l (
n: 1
)
(
II a~n(k1r/n)
n
k=l
) tan(:"-j2n) 00
(12.7)
Since e- 1 < (n!)l/nj(n + 1), (12.7) implies Carleman's inequality. The in-
equality (12.7) is due to B. Akerberg [46].
Note that (12.1) is also a generalization of some results from [29]-[31]. Various
integral inequalities are also obtained in [45].
Many discrete and integral inequalities of Carleman type are obtained in Go-
dunova's paper [47]. For example, the following theorem is valid:
THEOREM. If
(12.8) I
o
'l1(x)exp (~ ]
0
IOgf(t)dt) dx 5, I
0
s(x)f(x)dx
13. D. W. Boyd [48] used Hardy's inequality to prove the following result:
Let f be a nonnegative measurable function, and let a 2': 0, ).. > 0, q 2': 0, p
be real numbers with p + q > 1 and q).. < 1. Then
2': °
(13.1)
00
°
14. Let K(t) 2': on Vi = {t = (tl, ... ,tn)IO < ti < 00, i = 1, ... ,n}, with
J K (t )dVi = 1, and let Vx and Vy be similarly defined. Further, let <p( u) be
v,
a nonnegative convex function for u 2': 0, f(y) 2': 0, Y E Vy , f ¢ 0, and let
<p (f(x)) I(xl ... x n ) be integrable on Vx . Then
I(xl ... xn)-l<p ((Xl ... xn)-l I K (~:, ... , ~:) f(Yl, ... 'Yn)dV y) dVx
v'" v.
:s I(xl ... ;n)-l<P(f(x))dVx .
v'"
Hardy's inequality is a simple consequence of this result of E. K. Godunova
[49].
°
15. R. P. Boas [50] considered the two cases r
replaced by C l f(t) 2': if r = 0), namely
= p and r = °
of (1.6) (with f(t)
00 00
I FJdx :s pP I pdx,
o 0
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 155
for p > 1, and noted that these inequalities suggested there may be others of the
form
J J J
00 00 00
x 00
Cl = [-p ]
p-l
P - 1, C2 = pP -1.
(15.2)
1 (l
(15.3)
In [50], it is noted that the constants in (15.2), (15.3) are not best possible. The
proofs of (15.2), (15.3) are obtained from some general inequalities concerning
integral operators and their convolution. These results are proved by using the
156 CHAPTER IV
LEMMA (Boas [50]). If r.p is convex and continuous, f is nonnegative and mea-
surable, A is non decreasing and bounded with L = A( 00) - A(O), then
so
JJ
00 00
JJ
00 00
= L- 1 Clr.p(f(t» dt dA(U)
o 0
J
00
1 (i t
The inequality (15.4) reduces to
dt) , dx ::; a- J
00
if p > 1 or p < o. If 0 < p < 1, these inequalities are reversed. In the first of these,
let J(t) = tl-ag(t) and a = (k -1)/p, and in the second, let J(t) = tHPg(t) and
(3 = (1 - k)/p. This gives
(15.5)
(15.6)
iJ k < 1, p> 1 or k > 1, p < o. (The cases p > 1 give inequality (1) as stated.)
Similarly, with -0 < p < 1, we obtain
16. G. Talenti [52], [53], G. Tomaselli [54], B. Muckenhoupt [55] have investi-
gated the problem of finding necessary and sufficient conditions of measures J1.
and 1/ such that for a fixed p ~ 1 we have
(16.1 )
B = ~~~ (
!
00
uPdx
) IIp (
! r
v- p ' dx
) IIp'
< 00,
and this is the condition obtained earlier by Talenti and Tomaselli. For this
x 00
case an analogous result in which Jf dt is replaced by Jf dt was also given by
o x
Muckenhoupt [55].
17. M. Izumi and S. I. Izumi [56] proved several results similar to Hardy's
inequality. For example the following results are valid:
(i) Let p > 1 and s < -1 and let f be nonnegative and integrable on (0,11").
If x 8 f(x)P is integrable, then
(17.1)
x
where G(x) = J rl f(t)dt.
xl2
(ii) Let p > 1 and s =/: 1 and let f be nonnegative, nonconstant and integrable
on (0,00). If x 8 f(x)P is integrable on (0,00), then
J J
00 00
18. In [57, pp. 273-275], G. H. Hardy proved that the arithmetic mean of (an)
in (1.1) can be replaced by a more general mean which contains the Euler mean,
Cesaro mean or Holder mean as particular cases. Another more general case has
been studied in G. M. Petersen [58] and G. S. Davies and G. M. Petersen [59]
where the following has been proved:
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 159
for all k ~ m ~ n.
Suppose there exists a sequence f(k), f(k) /' 00, such that
for all k ~ m ~ n
and
L
00
(18.1)
The case Cm,k = 11m, k ~ m, Cm,k = 0, k > m, and f(k) = k satisfies the
condition ofthe theorem and (18.1) reduces to (1.1).
M. Izumi, S. 1. Izumi and G. M. Petersen [60] investigated various generaliza-
tions of Hardy's inequality and of the above Theorem 1.
L. Leindler [61] proved among others the following similar result:
THEOREM 2. Let an ~ 0 and An > 0 (n = 1,2, ... ) be given. Then, using the
notation
n n
Am,n= Lai and ~m,n = L Ai (1 ~ m ~ n ~ 00),
i=m i=m
we have
and
160 CHAPTER IV
19. Let p > 1, r =1= 1, and f(x) be nonnegative and Lebesgue integrable on [0, b)
or on [a, 00) according as r > 1 or r < 1, where a > 0, b> O. If F(x) is defined
b =
as in (iii) of 2, then if J x-r(xf)P dx < 00 in (18.1) and J x-r(xf)P dx < 00 in
o a
(18.2),
(19.1)
b b
20. Let {An} be an arbitrary fixed sequence of real numbers. The aim of the
paper [66) of B. Florkiewicz and A. Rybarski is to estimate the optimal constant
IJ in the inequality
(20.1)
which should hold for every sequence {an} of real numbers. They proved the
following results:
(i) If a sequence {An} is nondecreasing, then IJ ::::; 4.
=
(ii) Let ~n = Al + ... + An. If ~n --t +00 for n --t 00 and :E ~;;1 = +00, then
n=1
IJ ~ 4.
21. C. Bennett [68, Th. 6.3] proved the following results: If a > 0, 1 ::; q < 00,
and J is nonnegative on [0,1], then
J1[
o
(I-logt)'" Jt]q
0
J ds dp,(t) ::; a- q J1
0
[t(I-logt)l+'" J(t)]q dp,(t),
where dp, = dtl [t(I -logt)]. See also Bennett and K. Rudnick [51] for some
related inequalities.
22. Some inequalities related to Hardy's inequality are obtained by L. Gluk [69],
T. M. Flett [70] and K. F. Andersen [71].
J J
x x
J
00
Ja FP~-ccp
00
JFP~-ccp ~ (c ~ 1) Jp~p-ccp
00 00
dx P dx.
a a
00
JFP~-ccp (1 ~ c) JfP~P-ccp
00 00
dx ~ P dx.
a a
b
(iv) If 0 < b ~ 00, 0 < p ~ 1, c < 1, and J FP~-ccp dx converges, then
o
We note that if we take cp(t) == 1 in these inequalities then (i), (iv) with b = 00
reduce to cases of (2.6), as do (ii), (iii) on letting a --t 0+.
In [72) Copson also considered the case c = 1 and obtained the following
results: let cp, f satisfy the preceding hypotheses, and ~ be defined as before. Set
J J
x 00
JFP~-l'P J
b b
JFP~-l'P J
00 00
24. In [73]' H. P. Heinig proved the following two inequalities. Let p > 0 and
00
(24.1)
00
Similarly, if p >0 and 2p - 1 > A- sp, while J tA-SPlf(t)IPdt < 00, then
o
(24.2)
J
00
We illustrate by proving (24.2). We have e = exp [_p2[ yp-l log y dy], so the
change of variable t = xy in (24.2) reduces it to
164 CHAPTER IV
where A = pi [(2 + S)p -.A - 1]. By Jensen's inequality the left side of this
! {p i
inequality does not exceed
J
00
By setting f = log Igl in this result, it follows that: if J ef dx < 00, and
o
J e-stf(s)ds
00
(24.4)
°
25. Let f be continuous and nondecreasing on [0,00) with g(O) = 0, g(x) >
for x > and g( 00) = 00. Let p 2:: 1, k :f 1, and let f be nonnegative and
°
°
Lebesgue-Stieltjes integrable with respect to 9 on [0, b] or on [a, 00) according as
k> 1 or k < 1, where a > b> 0. Suppose
J J
x 00
Then
b b
with both inequalities reversed when 0 < p ~ 1. Equality holds in either case
when p = 1 or f == 0, and the constants on the right sides are best possible when
the left sides are unchanged.
This result is due to C. O. Imoru [74]. Shum's results from 19. follow by
taking g( x) == x in the above results.
1
is valid for all f(x) E L2(-1, 1), where F(x) = - J f(t)dt.
x
This inequality is due to D. C. Peaslee and W. A. Coppel [76].
They used (27.1) in a proof of the following inequality of E. Bombieri [77J:
1
28. Results from 16. were further extended by J. S. Bradley [79J to the case
of mixed norms (two parameters p,q) and to values of p,q < 1 by H. Heinig
[80J, and Beesack and Heinig [81J. We present here a statement of the results in
[79],[81].
In all cases u, v, f denote nonnegative, extended real- valued measurable func-
x
tions on R+ = (0,00), and f is called admissible if the integral Jf dt (or
o
166 CHAPTER IV
00
Jf dt in parts (b) below) exists (finite) for 0 < x < +00. In all cases p',p
x
denote conjugate exponents. In case k = +00 we interpret any integral norm
J Iglkdx llk = essup Ig(x )1, and adopt the usual convention O· (+00) = o.
A xEA
r
THEOREM 1. (Bradley [79]) (a) Suppose that 1 ::; p, q ::; +00, and J v- P' dt
o
exists (finite) for 0 < r < +00. If there exists a constant C> 0 such that
(28.1 )
uq dx
) llq (
! r
v- p' dt
) lip'
= B < 00.
Moreover, in case p ::; q above, then (28.2) is also sufficient for (28.1) to hold for
all admissible f.
r
(b) If 1 ::; p, q ::; +00 and 0 ::; J v- p ' dt < 00 for 0 < r < 00, then the existence
o
of a constant C > 0 such that
(28.3) {
!
00 [ 00
U(X)! f dt
1 q
dx
} llq
::; C
{
! 00
[v(x)f(x)JP dx
} lip
u q
dx
) l/q (
!00
v- p' dt
) l/p'
=B < 00
when p ::; q, (28.4) is also sufficient for (28.3), to hold for all admissible f.
Finally, in all cases, if C denotes the best ( = least) constant in (28.1) or
(28.3) then B ::; C ::; Bpl/q(p,)l/p' for 1 < p ::; q < 00, while C = B if p = 1 or
q = +00.
r
THEOREM 2. (Beesack and Heinig [81]) (a) Suppose p, q < 0 and 0 < J v- p' dt <
o
00 holds for all r > O. If there exists C > 0 such that
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 167
(28.6)
r
Moreover, in case q ~ p < 0, and 0 < J V-pI
dt < 00 for all r > 0, and K1(r)
o
not only satisfies (28.6) but is non decreasing on R+, then (28.5) holds for some
C> 0 and all admissible I.
00
(b) Ifp,q < 0,0 < J V-pI dt < 00 for all r > 0 and
r
(28.7) { !
OO[ 00
U(X)! I dt
]q}l/q
~C !
{OO
[v(x)I(x)JP dx
}l/P
(28.8)
00
Moreover, in case q ~ p < 0, and 0 < J V-pI dt < 00 for all r > 0, and J1(r) is
r
nonincreasing on R+ and satisfies (28.8), then (28.7) holds for some C > 0 and
all admissible I.
As in Theorem 1, the best (= largest) constant C in (28.5) or (28.7) satisfies
B ~. C ~ Ipll/q(p,)l/P' B if q ~ p < o.
00
THEOREM 3. (Beesack and Heinig [81]) (a) If 0 < p, q < 1 and 0 < Ju q dx,
00 r
J V-pI dt < 00 for all r > 0, and if (28.5) holds for all admissible I, then (28.7)
r
J u q dx, J V-pI
00 00
x ) q } l/q { lip
~
00 ( 00 }
{ oq l Ifdt
I x - dx Ipi I Ix(O+l)P-lfP(x)dx
laqllalql p
o 0 0
q } l/q { lip
~ Ipi I
00 ( 00 ) 00 }
{ x Oq - l I f dt dx . Ix(O+l)P-l fp(x) dx
I laqllal qlp
o x 0
29. A. Kufner and H. Triebel [S2, §2] also dealt with weighted norm inequalities
of the form
b b
30. L. Y. Chan [S3] gave some special inequalities with mixed weights involving
Stieltjes integrals. We shall state these results briefly. Let 9 be continuous and
nondecreasing on [0,00) with g(O) = 0, g( +00) = +00, g(x) i= 0 if x i= 0,
g(l) = 1, g(x) i= 1 if x i= 1; let f be nonnegative and measurable on [0,00) and
let F;(x) = J f dg for i = 1,2,3,4 where El = [x, 00) (1 ~ x < 00), E2 = [l,x]
Ei
(1 < x < 00), E3 = [O,x] (0 < x ~ 1), E4 = [x, 1] (0 ~ x < 1). For real p,q i= 0,
r i= 1, set A = (Iq/(r - l)I)(P-l)qlp (Ip/(r -1)1), S = [(q - r + l)p/q] = 1. Then
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 169
for 1 ~ p ~ q < 00, or -00 < q ~ p < 0 where i = 1 if (r - l)/q < 0 and i =2
if(r -l)/q > o. Also, for the same p,q,r,
jX-I
I
(J I
fdt/lOgX)P dx" (p~lr jX'-'f'(X)dX,
I
J J
o
I
X-I
(
z
l)P
f dt/l1og xl dx ~ (p ~ 1) P J
0
I
x p - I fP(x)dx,
all for p > 1. Some more general results involving boundary terms were also
given in [84].
31. The unnatural hypotheses concerning the existence of the integrals FP rjJ -c dx J
in (i) and (iii) from 23. can be replaced by the assumption that the larger integral
J fPrjJP-c dx converges. See also Beesack [85].
We shall give such a theorem for the case p < O.
Let p < 0, and let rjJ, f be positive measurable functions on R+ such that the
integrals
J J J
z z ~
<flex) = <p dt, F(x) = f<p dt (if c < 1), or F(x) = f<p dt (if c > 1)
o 0 z
exist for x > o. If 0 < b < 00, then for c > 1 we have
170 CHAPTER IV
J
00
jPiJ?P-cr.p dx.
a a
32. Shum's results from 19. are extended in Mitrinovic, Beesack and Vasic [143]:
Jf
x
Jf
00
or F(x) = dt if(k-l)p<O.
x
b
Then: (a) ifp < 0 or p > 1 and J t p - k jP dt < 00, we have for (k -1)p > 0,
o
J I J -I
b b
00
J I rJ -I
00 00
b
(b) if 0 < p < 1 and JC k FP dt < 00 we have for (k - l)p > 0,
o
J I ~ rJ -I
b b
J I ~ liP J -I
00 00
Moreover in all cases inequality can hold only if f == 0 (or never in case p < 0).
The constant Ipl(k - 1)IP is best possible in each case.
The following result was also proved in [143]:
If k > 0, and p > 1 or p < 0, then
(32.6) dx :::; k- P J
00
-00
ekpxgP(x) dx;
J
00
If 0 < p < 1, then the inequalities (32.6), (32.7) hold with:::; replaced by ::::: if
k > 0 or k < 0 respectively. (If p = 1, both (32.6) and (32.7) hold, with equality.)
Equality holds in (32.6) or (32.7) only for 9 == 0 (or never if p < 0) and the
constants k- P or (-k)-P are best possible.
The inequality (32.6) is a generalization of (9.1).
33 . .Let the function space Lp and the norm II . lip (1 :::; p < (0) be as usual. Let
K(t, s) denote a nonnegative, measurable function of t, s > 0 and let K be the
operator defined by
J
00
J
00
o
172 CHAPTER IV
J
t
R2 R2
35. Let f: 1-+ R+ be a convex function, Xi E I (i = 1,2, ... ), C = (Cl> cz, ... )
a positive sequence, and let for every n ~ 1, q(n) = (q~n), ... , q~n») be a positive
(35.1) (k ~ 1),
n=k
then
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 173
(35.2)
If J is a concave function and if in (35.1) the reverse inequality holds, then the
reverse inequality in (35.2) holds also.
This result is proved by P. M. Vasic and J. E. Pecaric [97]. The following
result is also proved in [97]:
Let Pi ;::: 1 (i = 1,2, ... ), Xi > 0 (i = 1,2, ... ) and let J be a positive function
such that x-I J( x) decreases for x > O. Suppose that c = (CI, C2, ••• ) is a positive
sequence, such that
Then
(35.4)
If x-I J(x) increases and if (35.3) is reversed, then (35.4) is reversed also.
36. Some inequalities for sequence spaces are established by P. Chandra and R.
N. Mohapatra [98]. A proof uses results from [59].
I (s!X) ]
0 0 0
sex - t)J(t) dt) P dx'; (:~)' J(f(t)), dt
38. Further generalizations of a result from 25. are obtained by C. O. Imoru
[100]-[102].
174 CHAPTER IV
(40.1)
Theorems 2 and 3 from [105] are similar theorems for upper triangular and
nontriangular matrices (a mn ). Theorems 1 and 2 include discrete Copson's gen-
eralizations of Hardy's inequalities.
(Note that (40.1) is related to (33.2).)
Generalizations of results from [72] and [44] and some analogous results are
obtained in [106].
Hardy's inequality in Orlicz-type sequence spaces for operators related to gen-
eralized Hausdorff matrices is obtained in [107]. It is a generalization of results
from A. Jakimovski, B, E. Rhoades and J. Tzimbalario [111] and D. Borwein
[112].
J. A. Cochran and C. S.Lee [113] proved the following results:
(i) If A and p are real numbers with p > 0, and f(t) is measurable and non-
negative on (0,00), then
(40.2) ~ e(A+1)/p f
00
t A f(t) dt.
o
(ii) If A and p are constants with A ~ 0 and p ~ 1, and 0 < Xn ~ 1, then
(40.3)
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 175
°
ibn} satisfy an = bn = for n ~ N, then the best constant AN is smaller than
the best possible constant A for all sequences and AN converges to A. This book
discusses the computation of AN and its rate of convergence to A. In most cases
AN is an eigenvalue of a Hermitian matrix.
45. R. Brown and D. Hinton [158] give Hardy inequalities of the form
46. As this book goes to press, the book [159] by B. Opic and A. Kufner has
appeared. It is devoted to finding conditions under which there is a constant C
such that
J }J
o
00 bt
at
g(h)dh
8
:! ~ K J :!,
00
0
Ig(t)1 9
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 177
6
bl=.l.
for 1 :S () :S 00, 0 < a < b < 00, and K = (
[s' 9 ds )
51. An additional reference is the paper of Carlson [173] and the alternate proofs
of his result given in Ahiezer [174].
REFERENCES
8. KNOPP, K., tiber Reihen mit p03itiven Gliedern, J. Lond. Math. Soc. 3
(1928), 205-211.
9. CARLEMAN, T., Sur le3 fonction3 qua3i-analytique3, in "Conferences
faites au cinquieme congres des mathematiciens scandinaves," Helsing-
fores, 1923, pp. 181-196.
10. POLYA, G., Proof of an inequality, Proc. London Math. Soc. 24 2
(1926), Records of Proc. 1vii.
11. VALIRON, G., "Lectures on the General Theory of Integral FUnctions,"
Toulouse, 1923.
12. OSTROWSKI, A. M., tiber qua3i-analyti3che Funktionen und Be3timmtheit
a3ymptoti3cher Entwicklungen, Acta Math. 53 (1929), 181-266.
13. HARDY, G. H., J. E. LITTLEWOOD and G. POLYA, "Inequalities,"
Cambridge, 1934.
14. KNOPP, K., tiber Reihen mit p03itiven Gliedern (2te Mitteilung), J. Lond.
Math. Soc. 5 (1930), 13-21.
15. HARDY, G. H., Note3 on 30me point3 in the integral calculu3 (LXIV),
Messenger of Math. 57 (1928), 12-16.
16. COPSON, E. T., Note on 3erie3 of p03itive term3, J. Lond. Math. Soc. 2
(1927),9-12.
17. , Note on 3erie3 of p03itive term3, J. Lond. Math. Soc. 3
(1928), 49-51.
18. HARDY, G. H., Remark3 on three recent note3 in the Journal, J. Lond.
Math. Soc. 3 (1928), 166- -169.
19. , The con3tant3 of certain inequalitie3, J. Lond. Math. Soc. 8
(1933), 114-119.
20. HARDY, G. H. and J. E. LITTLEWOOD, Note3 on the theory of 3eri3
(XII): On certain inequalitie3 connected with the calculU3 of variation3, J.
Lond. Math. Soc. 5 (1930), 283-290.
21. BLISS, G. A., An integral inequality, J. Lond. Math. Soc. 5 (1930),
40-46.
22. SUNOUCHI, G. and N. TAKAGI, A generalization of the Carleman in-
equality theorem, Proceedings Phys.-Math. Soc. Japan 16 (1934), 164-
166.
23. KNOPP, K.,Neuere Siitze: Reihen mit p03itiven Gliedern, Math. Zeitschr.
30 (1929), 387-413.
24. MULHOLLAND, H. P., On the generali3ation of Hardy'3 inequality, J.
Lond. Math. Soc. 7 (1934), 208- -214.
25. , Concerning the generalization of the Young-Hau3dorff theo-
rem and the Hardy-Littlewood theorem3 on Fourier con3tani3, Proc. Lond.
Math. Soc. 35 (1934), 257-293.
HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 179
136. NEHARI, Z., The Schwarzian derivative and schlicht functions, Bull.
Amer. Math. Soc. 55 (1949), 545- -551.
137. FLANDERS, H., Remarks on Almgren's interior regularity theorems, Illi-
nois J. Math. 13 (1969), 707- -716.
138. JANET, M., Sur la methode de Legendre-Jacobi-Clebsch et quelquesunes
de ses applications, Bull. Sci. Math. 53 (1929), 144-160.
139. TROESCH, B. A., An isoperimetric sloshing problem, Comm. Pure Appl.
Math. 18 (1965), 319-338.
140. BEESACK, P. R., Integral inequalities involving a function and its deriva-
tive, Amer. Math. Monthly 78 (1971), 705-741.
141. FEINBERG, J. M., Some Wirtinger-like inequalities, SIAM J. Math.
Anal. 10 (1979), 1258-1271.
142. FLORKIEWICZ, B., Integral inequalities of Hardy type, Dissertation.
143. BEESACK, P. R., D. S. MITRINOVIC and P. M. VASIC, Integral in-
equalities, (The manuscript not achieved and not published - we used this
manuscript in writing sections 15,16,21,23-25,28-32).
144. PACHPATTE, B. G., On some extensions of Levinson's generalizations
of Hardy's inequality, Soochow J. Math. 132 (1987), 203-210.
145. OSTROGORSKI, T., Analogues of Hardy's inequality in Rn, Studia Math.
88 3 (1988), 209-219.
146. SINN AMON, G., A note on the Stieltjes transformation, Proc. Roy. Soc.
Edinburgh Sect. A 110 1- -2 (1988), 73-78.
147. BENNETT, G., Some elementary inequalities II, Quart J. Math. Oxford
Ser. (2) 39 156 (1988), 385-400.
148. PACHPATTE, B. G. and E. R. LOVE, On some new inequalities related
to Hardy's integral inequality, J. Math. Anal. Appl. 149 (1990), 17-25.
149. MOHAPATRA, R. N. and D. C. RUSSELL, Integral inequalities related
to Hardy's inequality, Aeq. Math. 28 (1985), 199-207.
150. MOHAPATRA, R. N. and K. VAJRAVELU, Integral inequalities resem-
bling Copson's inequality, J. Aust. Math. Soc. Ser. A. 48 (1990),124-132.
151. MARTIN-REYES, F. J. and E. SAWYER, Weighted inequalities for Riemann-
Liouville fractional integrals of order one and greater, Proc. Amer. Math.
Soc. 106 (1989), 727- -733.
152. PITTENGER, A. P., Note on a square function inequality, Ann. Prob. 7
(1979), 907-908.
153. WILF, H. S., "Finite sections of some classical inequalities," Berlin, Hei-
delberg, New York, 1970.
154. BATUEV, E. N. and V. D. STEPANOV, Weighted inequalities of Hardy
type (Russian), Sibirsk. Mat. Z. 30 (1989), 13-22.
155. SINNAMON, G., A weighted gradient inequality, Proc. Roy. Soc. Edin-
burgh Sect. A. 111 (1989), 329-335.
186 CHAPTER IV
1. The inequality
(1.1 )
(1.2) LL
00 00
amb n <
(
1T
) (
L
00
aP
) lip (
Lb
00
P'
) lip'
JJ
00 00
f(x)g(y) dxdy
x+y
o 0
(1.3)
unless f == 0 or 9 == O.
187
188 CHAPTER V
These theorems, as well as the following are known as Hilbert's theorems (see
[4, p. 234), Theorems 315, 316 and 323).
L a~ L b~
00 ) IIp ( 00 ) IIp'
(1.4) ( ) (
(p)
• 7r 1r
sm m=O n=O
00 00 \l/p' IIp
"\;"' "\;"' /\ m V n
(2.1 ) L..J L..J A M amb n
m=l n=l m + n
00 00 n
3. Suppose that an ~ 0, and that A(x) = L: anx n , A*(x) = L: a'!.~ . Then
n=O n=O
J J(e-"
1 i
(3.3) J 1
(3.4)
oo
(3.5) L Loo 110g(mln) I 2~ ~ aman
(
maxmn aman ~ L...J L...J maxmn •
m=l n=l (,) m=l n=l ' )
The last two inequalities are consequence of the following Hardy's inequality
for quadratic forms ([16], [4, p. 257]):
00
J
00
Then
~ ~ ~ 00
!00
fP(x) dx
) IIp (
! 00
gP' (x) dx
) lip'
,
JJ
00 00
fP dx
) IIp (
!
00
gP' dy
) IIp'
(4.1)
HILBERT'S AND RELATED INEQUALITIES 191
is convergent, then
(4.4)
(4.6) ~AnBnn-'(IOgn)-" ~ K (
!
00
a'(x) dx
) l/p (
!00
b'(x) dx
) l/q
°
When l/ = and so q = p', the best possible value of K is 7r I sine 7r I p).
The special case of (4.5) is
(4.7) ff
2 2
amb n
(mn (log(mn))l-")
~K (f a~)l/P (f
2 n 2
b'h)l/q,
n
and when l/ = 0, K = 7r1 sin(7r/p), and this is the smallest K for which the
inequality holds.
192 CHAPTER V
H. P. Mulholland [18] gave some similar results for power series (in form related
to (4.4)). These results are generalizations of results of G. H. Hardy and J. E.
Littlewood [19].
where
8. v. Levin [23] proved that in (1.4) for p = q = 2 the factor (m+n+ 1)-1 can
be replaced by the larger one
(m + n) loge m + n) + (m + n + 2) loge m + n + 2) - 2( m + n + 1) loge m + n + 1).
9. By the Hilbert-Riesz inequality is meant the following inequality: let p > 1,
q > 1, p-1 + q-1 > 1, .x = 2 - p-1 - q-1 = p,-l + q,-l (0 < .x < 1), f(x) ~ 0,
g(y) ~ 0, and all integrals are finite, then
00 00
(9.1) JJ Ix+ylA
-00
f(x)g(y) dxdy
£00 £
-(X)
(9.3)
00 00
J J Ix + yl"
-00 -00
f(x)g(x) dxdy
where
L(p,q) = ~ sin >.; sin~ sin ~r(l- >.)r (;,) r (;,)
-00 -00
J J
00 00
f(x)g(y)h(x + y) dxd y ,
xayb(x + y)c
o
J
0
x
u(x) = x- c f(y)g(x - y) dy ,
ya(x _ y)b
o
with the usual conventions if p, q, r, or r' is infinite. Thus, for example, F = IIfllp
in all cases, so that if p = 00, F = ess sup If(x )1. Let p, q, r, a, b, c satisfy
(10.1') I~KFGH.
This result was proved in 1938 by H. R. Pitt [26], and is the integral analogue
of a corresponding theorem for infinite series involving Cauchy products due to
Hardy and Littlewood [27]. (The latter paper involves parameters a, {3, 'Y related
to a, b, c by a = a + p-t, {3 = b + q-l, 'Y = c + r- 1.)
In [26] Pitt also states the following result as a corollary of the above (by
expressing the new I as a sum of six integrals over (-00,0] or [0,00)):
HILBERT'S AND RELATED INEQUALITIES 195
1= JJ
00 00
f(x)g(y)h(x + y) dxd y ,
Ixlalylblx + ylC
J
-00 -00
00
11. Let A*(x) be defined as in 3. and let B*(x) be defined similarly. The
following generalization of (3.1) and (3.2) is given by Y. C. Chow [28]:
(11.1)
(11.2)
where PI > 1, P2 > 1, ... P1 I + p:;1 + ... + p-;;,I = 1, and the sequences are
nonnegative.
196 CHAPTER V
(12.1)
max
Xi
LL ~iXj.
n
lIZ
n
+J
/ n
LX~ = 7r{1 + O(log-l n)}.
1
14. S. B. Steckin [32] proved that the constant in (9.2) is the best possible
constant for a discrete analogue of (9.1), i.e.
(14.1)
(14.2)
Then
(14.3)
Levin's result was also considered in F. F. Bonsall [33]. He used the following
elementary application of Holder's inequality.
HILBERT'S AND RELATED INEQUALITIES 197
(14.4)
Then
n
Hence, if It, ... , fn, U1, ... , Un, ware nonnegative measurable functions and k is
a positive constant such that
then
(14.6')
where
JJ
(Xl (Xl
JJ
(Xl (Xl
= + u du (y = xu)
o 0
Hence
1:5. { ( t) }11q, {
7rCSC ;, 7rcosec
( t) }11P' 1I/llpllqllq.
:'
By elementary calculus, the minimum value of this upper bound occurs for t =
A-I, and this gives the constant factor preceding II/lIplIgllq the value
( 7r csc >.;, ) 11q, ( 7r csc A~' )l/p
l
=
(
7r CSC A~'
)A ,since A~' + A~' = 1, so that 7r -
A;' = ;;,.
We can obtain a discrete analogue from the integral one by taking I(x) = am
for m - 1 :5. x < m and g(y) = bn for n - 1 :5. y < n (m, n = 1,2, ... ), or by
taking dJ.L to be a sum of point masses.
Taking n = 3 in (14.4) and proceeding in the same way as in the proof of
Levin's inequality, Bonsall [33, Th. 8] proved that
(14.8) J J J (xI(x)g(y)h(z)
+y+ dxdydz:5. kll/ll Z)2A p1 ·lIgllp2 ·lI h ll p3
o 0 0
with
Levin's inequality is the special case K(x, y) == (X+y)-l of the following theorem.
HILBERT'S AND RELATED INEQUALITIES 199
THEOREM 1. (Bonsall [33, Th. 2]). Let p > 1, q > 1, >. = (p')-1 + (q')-I, let f,
g be nonnegative and measurable on [0,00), and let K( x, y) be nonnegative and
homogeneous of degree -1, with
J
00
u- 1 / Ap' K(l, u) du = k.
o
Then
JJ
00 00
JJ
00 00
where
II = IIfll~llglI~,
12 = II
o 0
fP(x) (;) -tip' K(x, y) dxdy
J
00
Ilgll~ J
00
J
00
t 1
13 = IlglI~ uqr- K(l,u)du.
o
JJ
00 00
(14.10) { II 00 ( 00
K(x, y)f(x) dx
) P
dy
} lip
:::; kllfll p,
00 ( 00 ) q } llq
(14.11) { / / K(x, y)g(y)dy dx :::; kllgll q.
JJ JJ
00 00 00 00
n=l
nb! = B2 <
00, then
L
00 00
J J
00 00
then
° °
JJ
00 00
16. F. C. Hsiang [35] proved that, ifao,al, ... ,aN and bo,b1, ... ,bN are se-
quences of nonnegative real numbers,
18. N. G. De Bruijn and H. S. Wilf [39] showed that the best possible constant
CNin
N N N
n < C '"' a 2
L...JL...J m + n - NL...J n
'"' '"' ama
n=l m=l n=l
202 CHAPTER V
is given by
This is an extension of the result given in [30], [31] and [40] (R. A. Fairthorne
and J. C. P. Miller).
JJ (Yl
V" Vp
u - , ... ,Yn)
Xl
-
Xn
(Yl ... Yn ) ~ ( X1, ... ,Xn )-~
Xl··· Xn
20. Further related papers are R. Redheffer [42], V. P. ll'in [43], H. Davenport
and H. Halberstam [44], K. R. Matthews [45], F. A. Sysoeva [46],[47],[48], P.
L. Walker [49], A. Erdelyi [50], F. Feher [51] and A. Kufner [52]. For example,
Walker [49] proved that the constant in (9.2) is asymptotically best possible.
21. The following result was proved by P. M. Vasic and J. E. Pecaric [53]:
Let f: [kl, k2] x [£1,£2] --+ R+ be a real function, and let a = (aI, a2, ... ),
b = (b l , b2 , ••• ) and C = (Cl, C2, ••• ) be three positive sequences. Suppose that for
every n ~ 1, q(n) = (q~n), ... , q~n») is a positive n-tuple such that qin) = 1. t
k=l
(a) If H( s, t) = f (K-l( s), L -1 (t») is a convex function and if dk are positive
numbers such that
L
00
(21.1) cnqin) :5 d k (k ~ 1)
n=k
then
where K n (a, w) and Ln (b, w) are quasiarithmetic mean of a with weights w with
respect to the functions K and L respectively.
(b) If H( s, t) is concave and if in (21.1) the reverse inequality holds, then (21.2)
is reversed.
22. Let S be a complex inner product space with inner product {".} and let
A be a real, nonnegative constant. If X n , Yn E S for m, n E Z with X 2 =
00 00
00
(22.1)
{Xm' Yn} < 71"XY
m,n=-oo
m - n +A - Isin 7r AI'
and equality holds if and only if either X = 0 or for some complex constant 8,
00
Yn = 8 l: xm/(m - n + A) for all nEZ, where X < 00.
m=-oo
This is the main result of the paper [54] of R. M. Redheffer and P. Volkmann,
and is an extension of Schur's inquality [2], which is just (22.1) with S = C and
(xm' Yn) = xnYn'
23. The Hilbert-Riesz inequality (9.1) can be given in equivalent form (see [4,
Th. 382])
JJ~;X2g;,}
00 00
r (I-A)
K(p q) = 7r A - 1 / 2 2.
, r(l-~)
24. D. S. Mitrinovic and J. E. Pecaric [57] gave several remarks about results
from [15],[16] and [28].
1° First they proved that (11.1) can be proved by using Hilbert's inequality
204 CHAPTER V
1 1
00 00
1
00
and so
1 1~~ 1 1
1 1 00 00
1
o
1
A(x)B(x) dx = 11
00
dt
00
e-tua(u) du 1
00
e-tvb(v) dv
= 11
0 0 0
00 00
a(u)b(v) dudv
u+v
o 0
00 ) IIp ( 00 ) IIp'
,; ffin(i)
( ) (
/aP(x)dx /1I"(X)dX
(24.1) LL amb n
m +n +1
::; ( 7f'
sin (~:)
) (1 0
00
ZP-2 aP (z) dX) IIp
00 ) IIp'
X (
/ zp'-2bP' (z) dz
HILBERT'S AND RELATED INEQUALITIES 205
PROOF: We have
~ (! 'I, (!
r
z'-'A'(z) dZ) z"-'B"(z) dZ) ,/,'
,; r mG)(r--'a'(Z)
r dz
X (f00
J
00
Kl(x,y) = Ko(xt)Ko(yt) dt
o
(so that Kl is symmetric and homogeneous of degree -1), and that
J
00
where
00
J -_jK1(w,1)d
..;w w.
o
PROOF: We have, in fact, putting t = mw and using the symmetry and hom-
geneity of K 1 ,
00 00 ~ 00 00
J J
00 00 00 00
00 00
where ¢a(t) = ~ anKo(nt), ¢b(t) = ~ bnKo(nt), and this does not exceed
1 1
ff
1 1
log
m-n
(~) ambn ::; 7r (f f
1 1
ama n ) 1/2
m+n
(f f 1 1
bmb n ) 1/2,
m+n
where the coefficient on the left-hand side is to be interpreted as lin when m = nj
(25.3)
where
This follows from (1.2) and from the following general results:
THEOREM 1. Let a mn ;::: 0 and for every p > 1 and nonnegative X m , Yn the
following inequality is valid
00 00 ( 00 ) IIp ( 00 ) IIp'
(25.5) ~ ~ amnXmYn ::; K(p) ~ X~ ~ y~'
If (25.1) and (25.2) are valid, then
(25.6)
~~a~"xmY" S K' (G. G-~ +.)) -') (~x~) 'I. (~Y!) 'I.
First, we shall prove the following
208 CHAPTER V
LEMMA. Let
Then
(25.9)
(25.10)
(25.11)
HILBERT'S AND RELATED INEQUALITIES 209
and
k2 > 0, kl + k2 = -1 (1-
2 P
+ -1q + >. ) ;::: -21 (1P- + -q1 + -p'1 + -q'1 ) = l.
We also have 0 < lip! < 1, i.e. 0 < ~ - + >. i < 2>'. Indeed, the first inequality
is valid since >. >
- 2 - 1 q > 1.
p - 1 p i.e. 1 >
q - l, 11 q. Similarly we can obtain the
second inequality.
In the case when
(25.12)
is valid, we have a result from 14. But, in this case we can also obtain an integral
analogue, i.e. the following theorem is valid:
210 CHAPTER V
THEOREM 2. Let k(x, y) ~ 0 and for every P > 1 and nonnegative f(x), g(y)
the following inequality is valid
!!
00 00
! 00
fP(x) dx
) lip (
!
00
gP' (y) dy
) lip'
JJ
00 00
k\x,y)f(x)g(y) dxdy
o
! !
0
00 ) lip ( 00 ) l/q
::; J<A(>..q') ( fP(x) dx gq(y) dy
Since (1.3) is valid, Theorem 2 gives the integral inequality of V. Levin from
[24J (see 9.).
Using Theorem 3 from 1. we get that the factor (m + n)-A in (1.4) can
be replaced by (m + n - I)-A. Similarly we can also given generalizations of
Theorems 341 and 342 from [lJ (see (i) and (ii) from 2.).
Of course, we can similarly obtain multi dimentional generalizations of Theo-
rems 1 and 2.
THEOREM 3. Let a mn ... s ~ 0 and for every PI > 1, ... ,Pr > 1, such that
r
I: pi = 1 and nonnegative x m , Yn,.'"
l Zs the following inequality is valid
i=l
l/Pl ( / ) 1/P2
~ K(P1, ... ,Pr) (/ fPl(X) dx ) gP2(y) dy
l/pr
... (/ hPr(z) dz )
t(~)
i=l q.
~1, (n - 1),x = t (1~) ,
i=l q.
then
REFERENCES
INEQUALITIES OF LYAPUNOV
AND OF DE LA VALLEE POUSSIN
1. Let p be a real continuous function of the real variable x on [a, b]. If the
°
differential equation
y" + p(x)y =
has a nontrivial solution y that vanishes at two points of [a, b], then, according
to A. M. Lyapunov [1], p is subject to the inequality
f
b
This inequality is sharp in the sense that the constant 4 cannot be replaced by
a larger number.
REMARK 1: This inequality may be applied in two ways. If one considers the
eigenvalue problem
(1.3) A> 4 i
(b - a) IP(X)ldX) -,
This furnishes a lower bound. If more is known about p, then this inequality can
be improved. This topic is disussed below in Section 9.
For the second interpretation, we consider a generalization to a scalar nth
order linear operator of the form
216
INEQUALITIES OF LYAPUNOV AND OF DE LA VALLEE POUSSIN 217
(1.5)
2. In [3] de la Vallee Poussin (see also [44]) proved that (see (1.4» Lx = 0 is
disconjugate on the interval [a, b] if
(2.1)
(2.3)
Note that (2kk [k;-l]! a]!)-l :s: t if, so that the coefficients of each power of
h in inequality (2.1) are smaller than the corresponding coefficients in inequality
(2.2).
It is easy to convince oneself that conditions (2.2) and (2.3) are independent
(for n > 2).
Another refinement of (2.1) (and of (2.2» is obtained by G. A. Bessmertnyh
and A. Yu. Levin [7]:
n
(2.4) L Bk,nPk hk < 1,
k=l
where
in particular,
(3.2)
(ii) If, in addition, Pl(t) == ... == Pm(t) == 0 for some m, 1 :S m < n - 1, then
(3.1) can be relaxed to
(3.3)
1 ) n-l
exp ( 211PoII1 L
(C
2 1
m:
) ( k +km ) k-m hkllpklh :S 1;
k=m
(iii) Po, ... ,Pn-l satisfy
(3.5)
where
(3.6)
(3.7)
220 CHAPTER VI
and pci and Po are defined by pci(x) = max (O,p(x)), Po(x) = max (0, -Po(x)).
(4.1)
([ ~],
2
[~])-lJbl t )I(t-a)n-l(b-t)n-l dt
. 2 rn (b-a)n-l
a
b b
1 J n-l (2 n --1) J
1 (t - a)k-l(b - t)k-l
+2 n- lrt(t)ldt+ ~ (k-1)! Irk(t)1 (b-a)k-l dt
a k-2 a
~ 1,
then
(4.2)
(5.1) Ln,p[Y] = 0,
we denote by Y n the set of the solutions of the equation (5.1), that is the set of
functions y: [a, a + h] - t R with the properties
(i) y has a derivative of order (n - 1) which is absolutely continuous on
[a, a+ h],
(ii) y verifies (5.1) almost everywhere on [a, a + h].
DEFINITION 1: The differential operator Ln,p has the interpolation property
In [a, a + h) iffor each
or
(5.2)
For some related results see also papers [11],[12] of O. Arama and D. Ripianu,
[13],[14] of D. Ripianu, and [15] of B. Szmanda. For example, for 17 = 3, p = 2,
O. Arama and D. Ripianu [12] showed that if
(5.3)
222 CHAPTER VI
(5.4)
in order that the differential operator L 4 ,2 to have the property H 4 [a, a + h).
D. Bobrowski [17] obtained the following inequality, as a sufficient condition
for the property H 4 [a, a + h) of the operator L 4 ,00,
(5.6)
then the operator L 2,00 has the interpolation property 12[a, a + h).
2)If
(5.8)
(5.9)
INEQUALITIES OF LYAPUNOV AND OF DE LA VALLEE POUSSIN 223
(6.1)
where
L Jh(t)lvn-k+l(t) dt:::; 1.
n b
(6.3)
k=la
(7.1 )
are very important in application to disconjugacy criteria for linear differential
equations. This inequality (see Chapter II. on Wirtinger's inequality) is consid-
ered in papers [33] of J. Brink and [34],[35] of A. M. Fink. Here, we shall give
disconjugacy tests from [35].
Consider differential equations of the form
(7.2)
with integrable coefficients on [a, b]. If a non-trivial solution of (7.2) has n zeros
on [a, b], then there are sequences ai and bi such that
(7.3) a :::; ao :::; ... :::; an-l = bn- 1 :::; bn- 2 :::; ••• :::; bo :::; b
and
224 CHAPTER VI
(7.4) Y(i)(a I o )
-- y(i)(b1 -0,
o ) - ;- 0
fi- , ••• ,
n -
1 •
Write e = an-l and note that a < e < b. We want to derive necessary conditions
for such a solution to exist so that the denial of these conditions give disconjagacy
of the equatio{n(. : : thiS) ;'t;po(se bwe tak)e1;p =} ao and b = bo. Furthermore, let
p=oo.
THEOREM 1. Equation (7.2) is disconjugate on [a, b] if one of the following in-
equalities holds:
k=l
Cancelling lIy(n) lit one obtains
n
(7.5) 1 < L lIakllpiH(k, oo,Pi)(e - a)k--h,
k=l
INEQUALITIES OF LYAPUNOV AND OF DE LA VALLEE POUSSIN 225
k=l
If r' :5 p,
n-l
lIy(n-l)lIp < l.)e - a)lIakllrlly(n-k)llr f
k=l
:5 (e - a)1+~-tllalllrlly(n-l)lIp
+L lIakllrH(k _l,r'p)lIy(n-l)lIp(e - a)k+~-t
k=2
and thus
(7.6)
226 CHAPTER VI
To prove (iv) take max over [a, e] and use Holder's inequality on the right. To
c
prove (v) and (vi) integrate (7.6) from a to e and replace J It - al!aky(n-k) I by
a
(e - a)lIaklipi lIy(n-k) lip'. for (v) and by IlakllooliCt - a)lIp'lIy(n-k)llp for (vi).
All of the above pro~fs derive an expression of the fo;m
n
(7.7) 1 < L IlakllpkBk(e - ayk
k=l
on [a, e] and a similar expression on [a, b]. The numbers Bk and Sk are inde-
pendent of a and e, and are the same for the expression on [e, b]. An attempt
to combine these was made in Zaiceva [36], but the last inequality was wrong.
Her method was successfully completed by Hartman [8] who was inspired by the
paper of Nehari [37]. His ideas with slight modification work here also.
THEOREM 2. Suppose one has inequalities of the form (7.7) on [a, e] and [e, b].
Let m = minsk> 0, then
t lIakllpkBkTm-~
k
Now using lIakll;. + lIakll;. ::; 2~lIakllp. and the geometric mean arithmetic
mean inequality, we get the theorem.
From the last theorem and the proofs of Theorem lone can read off more
disconjugacy tests. Some of these constants in the case of p-norms are better
than those of Willett [9] or Martelli [37].
Generally, for k near n these are small, while Willett's are small for k small.
For other theorems that give disconjugacy see Willett [38] and Kim [39] and
[40]. One of Kim's theorems from [39] we may quote here. The differential
equation (7.2) is disconjugate on (-c, c) if
A < (
n ! 7l'n
vp(x) dx
)' u rex) ~ 5p'(x)' - 4p(x)p"(x) " 0,
and
(8.2)
P. H. Hartman and A. Winter [43] improved this to
(8.3)
The best possible inequality of this type was obtained by Z. Opial [10] who
proved that
(8.4)
where equality can only hold when 9 == O.
The proof is elementary but long, so we give the proof of a somewhat weaker
inequality (also due to Opial)
(8.5) 11"2 $ 1I"IIgilooh + Il/lIooh 2 •
Let x solve (8.1) and multiply by x and integrate from 0 to h. We get
J J J
h h h
J J J
h h h
J J
h h
(8.8) Jh
o
Ix(t)1 Ix'(t)1 dt $
(
J 0
h
x'(t)2 dt J
0
h
x{t)2 dt
) 1/2
~J
h
$ x'(t)2 dt.
o
INEQUALITIES OF LYAPUNOV AND OF DE LA VALLEE POUSSIN 229
J J J
h h h
(8.10)
and a fortiori
J JIg(t)1
h h
(8.12)
(b - a)h
IIplh 2: (b - a)h 2 + 2(b - a)2 cot g 2 > 4.
if p is monotone then 4 may be replaced by 7.88 ... and by 9.397 ... if p is convex
([50]). See D. C. Barnes [51] for more results of this kind and a bibliography of
related results.
A very interesting result is obtained by A. Galbraith [52]. He has shown that
if a and b are successive zeroes of the differential equation
(9.1) y" + py = 0
J
b
(b - a) p( x) dx 2:: rr2.
a
J
b
and that these are the best possible bounds. The constant ~A~ = 9.478132 ...
and rr2 = 9.869604 ... so that we have a delicate test for the spacing of zeroes for
linear p. Furthermore, when the restriction that p 2:: 0 is removed, rr2 is still the
upper bound for he shows that the function for fixed a and b is an even function
of the slope of p, that is decreasing for positive slopes.
He also considered inequalities when p is concave. A. Elbert, in a series
of papers [54]-[58] considered Lyapunov inequalities of various kinds for (9.1),
especially when p is concave. For example, he showed [55] that
J
b
(9.2)
a
implies that every solution of (9.1) has n zeros on [a, b]. The coefficient of n 2rr2
cannot be replaced by a smaller number.
INEQUALITIES OF LYAPUNOV AND OF DE LA VALLEE POUSSIN 231
(12.2) (,X2
o
where ,X2, K and E are defined by k = 4(,X2 + 1)K2/h2
11'/2 11'/2
K= j(1_,X2sin2f/»-1/2df/>, E= j(1_,X2sin2f/»1/2df/>.
o 0
232 CHAPTER VI
The constant in the right hand side of (12.2) is the best possible.
For some applications in ordinary differential equations, see Borg [67].
Also connected to Borg's inequalities are the papers of H. O. Cordes [68] and
H. J. Cohen [69].
For other applications, as well as generalizations of (1.6) to complex differential
equations and univalent functions, see Z. Nehari [70] and P. R. Beesack [71].
13. Lyapunov inequalities for nonlinear equations can be found in A. Elbert
[72] and S. B. Eliason [73}, [74]. Elbert considers
y"(t) + p(t)y(q(t)) = 0
while Eliason [73], [74] considers
with yea) = y(b) = 0, where for each I' > 0, there is a v > 0 so that yJl+1 fey)
is increasing on (0,1'] and lim yJl+1 f(y) = o. If (13.1) has a solution y with
y~o+
14. Disconjugacy and Lyapunov inequalities are related by the boundary condi-
tions yea) = y(b) = O. If one considers the boundary conditions yea) = y'(b) = 0,
then one says that y" + py = 0 is disfocal on [a, b] if no non-trivial solution stat-
isfies these boundary conditions at c and d with a ~ c ~ d ~ b. St. Mary [75]
observed that if
/ r- 1(x )dx / p+(x )dx > 1, where p+(x) = max (0, p(x)) .
a a
b
if Jp ~ 0 on [a, b] and p is nonincreasing, while
t
(b - a) J b
p(x)dx ~
11"2
"4
a
JJ
b t
B. J. Harris [77] has extended these results. A simple version of his result is an
interated versions of (14.2), to wit, that if a non-trivial solution exists then
JJ JJ
b t b t
p+(x) p+(r)drdsdxdt ~ 1.
a a x a
S
1~ L Ckllpkll1(b - a)6-k
k=O
S
·
{:or t h e disconJugacy f (6)
0 y = "L..J Pi (x) Y(i) on [b]·
a, , C,0 = 3280S'
1 C1 = 2S+34y'iii"
9112S0 ,
i=O
C2 = 3!0' C3 = 310' C4 =l and Cs = ~.
16. A simple proof of (1.1) can be found in Leighton [SO], and some details on
(1.1) can be found in Krein [SI].
18. Beesack [83] has proved some higher order versions of (1.6). For example,
if j<n) and f(n) / f are continuous on [a, b], f(a) = feb) = 0, and f has n zeros
on [a, b] counting those at a and b, then
In case f(2m) and f(2m) / f are continuous on [a, b] and f has an m-fold zero at a
and at b, then
(18.2) J
b
and
(18.3)
b
Both (18.1) and (18.2) give (1.1) as special cases and (18.3) gives (1.1)'. The
inequality (18.2) implies that if y(2m)(x) + p(x)y(x) = 0 has a solution with and
m-fold zero at a and at b, then
J
b
(2m - 1)!22m
p(x)dx> (b _ a)2m-l .
a
with
x(0)=x(N+1)=0
and p(k) nonnegative on the set {2,3, ... ,N}. More general limiting conditions
are also considered in [34].
INEQUALITIES OF LYAPUNOV AND OF DE LA VALLEE POUSSIN 235
REFERENCES
32. CASADEI, G., SuI teorema di unicita di De La Vallee POU3sin per equzioni
differenziali del terzo ordine, Rend. Sem. Mat. Univ. di Padova.
33. BruNK, J., Inequalities involving IIflip and IIf(n)lIq for f with n-zeros,
Pacific J. Math. 422 (1972), 289-311.
34. FINK, A. M., Conjugate inequalities for functions and their derivatives,
SIAM J. Math. Anal. 5 3 (1974), 399-411.
35. , Differential inequalities and dis conjugacy, J. Math. Anal.
Appl. 49 (1975), 758-772.
36. ZAICEVA, G. S., A multipoint boundary value problem, Soviet. Math.
Dokl. 8 (1967), 1183-1185.
37. MARTELLI, M., SuI criterio di univitd di de la Vallee Poussin, Atti Ac-
cad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 45 8 (1968), 7-12.
38. WILLETT, D., Oscillation on finite or infinite intervals of second order
linear differential equations, Canad. Math. Bull. 144 (1971), 539-550.
39. KIM, W. J., Disconjugacy and nonoscillation criteria for linear differen-
tial equations, J. Diff. Eq. 8 (1970), 163-172.
40. , On the zeros of solutions of y(n) + py = 0, J. Math. Anal.
Appl. 25 (1969), 189-208.
41. MAKAI, E., Uber Eigenwertabschatzungen bei gewissen homogenen lin-
earen Differentialgleichungen zweiter Ordnung, Composito Math. 6 (1939),
368-374.
42. , Uber die Nullstellen von Funktionen, die Losung Sturm-
Liouville'scher Differentialgleichungen sind, Comment. Math. Helv. 16
(1943-4), 153-199.
43. HARTMAN, P. H., and A. WINTNER, On an oscillation criterion of de
la Vallee POU3sin, Quart. Appl. Math. 13 (1955), 330-332.
44. SANSONE, G., "Equazioni differenziali nel campo reale," Vol. I, 2nd ed.,
Bologna, 1948.
45. FINK, A. M. and D. S. ST. MARY, On an inequality of Nehari, Notices
Amer. Math. Soc. 16 (1969), 91.
46. LEVIN, A. Yu, On linear second order differential equations, Dokl. Akad.
Nauk SSSR 153 (1963),1257-1260, (Soviet. Math. 4 (1963),1814-1817).
47. HOCHSTADT, H., On an inequality of Lyapunov, Proc. Amer. Math.
Soc. 22 (1969), 282-284.
T
48. FINK, A. M., The functional T J R and the zeroes of a second order linear
o
differential equation, J. Math. Pures et Appl. 45 (1966), 387-394.
49. LEIGHTON, W., On the zeroes of solutions of a second-order linear dif-
ferential equation, J. Math. Pures et Appl. 44 (1965), 296-310.
50. BANKS, D., Bounds for the eigenvalues of some vibrating systems, Pacific
J. Math. 10 (1960), 439-474. •
238 CHAPTER VI
51. BARNES, D. C., Bounds Jor the eigenvalues oj some vibrating systems,
Pacific J. Math. 29 (1969), 43- -61.
52. GALBRAITH, A., On the zeros oj solutions oj ordinary differential equa-
°
tions oj the second order, Proc. Amer. Math. Soc. 17 (1966), 333-337.
53. FINK, A. M. On the zeroes oj y" + py = with linear, convex and concave
p, J. Math. Pures et App!. 46 (1967), 1-10.
54. ELBERT, A.., On the zeros oj the solution., oj the differential equation
y" + q( x)y = 0, where [q( x)]" i., concave, Studia Sci. Math. Hungarica 2
(1967), 293-298.
55. , On a Junctional connected with the zero., oj the .,olution.,
oj the differential equation y" + q(t)y = 0, where q(t) i., a nonnegative,
monotonic, concave Junction, Ann. Univ. Sci. Budapest. Ro!. Eot. nom.
sect. Math. 11 (1968), 129-136.
56. , On the zero., oj solutions oj ordinary second order differential
equations, Publ. Math. (Debrecen) 15 (1968), 13-17.
57. , On the solutions oj the differential equation y" + q( x)y = 0,
where [q(x)]" is concave, 1. Acta Math. Acad. Sci. Hung. 201-2 (1969),
1-11.
58. , On the solution., oj the differential equation y" + q( x)y = 0,
where [q( x)]" i., concave, II, Acta Math. Acad. Sci. Hung. 4 (1969),
257-266.
59. SCHWARZ, B. On the extrema oj the frequencies oj nonhomogeneous
strings with equimeasurable density, J. Math. and Mech. 10 (1961), 401-
422.
60. BEESACK, P. R. and B. SCHWARZ, On the zeros oj solutions oj second-
order linear differential equations, Canad. J. of Math. 8 (1956), 504-515.
T/2
61. ELIASON, S. B., The integral T J p(t)dt and the boundary value prob-
-T/2
lem x" + p(t)x = 0, x( -T /2) = x(T/2) = 0, J. Diff. Eq. 4 (1968), 646-
-659.
T/2
62. ST. MARY, D. F. and S. B. ELIASON, Upper bounds oJT J p(t)dt and
-T/2
the differential equation x" + p(t)x = 0, J. Diff. Eq. 6 (1969), 154-160.
63. LEVIN, V., On some integral inequalities involving periodic Junctions, J.
London Math. Soc. 10 (1935),45-48.
64. SATO, T., Seizi Senkei Bibunhoteisiki no Zero-ten no Bunpu ni tuite,
Kansu Hoteisiki 22 (1940), 39-43.
65. TUMURA, M., Kokai ZyobibunhOteisiki ni tuite, Kansu Hoteisiki 30 (1941),
20-35.
INEQUALITIES OF LYAPUNOV AND OF DE LA VALLEE POUSSIN 239
66. BORG, G., Uber die Stabilitiit geisser Klassen von linearen DilJerential-
gleichungen, Ark. Mat. Ast. Fys. 31 A (1944),No. 1, 31 pp.
67. , On a LiapounolJ crieterion of stability, Amer.J. Math. 71
(1949), 67-70.
68. CORDES, H. 0., An inequality of G. Borg, Amer. Math. Monthly 63
(1956), 27-29.
69. COHEN, H. J., Problem E 1205, Amer. Math. Monthly 63 (1956), 582-
583.
70. NEHARI, Z., Univalent functions and linear dilJerential equations, in
"Lectures on Functions of a Complex Variable," Ann Arbor, 1955, pp.
148-151.
71. BEESACK, P. R., Nonoscillation and disconjugacy in the complex do-
main, Trans. Amer. Math. Soc. 81 (1956), 211-242.
72. ELBERT, A., Oboscenie odnogo neravenstva Ljapunova dlja dilJerencial'nyh
uravnei{ s zapazdyvajuscim argumentom, Ann. Univ. Sci. Budapest. Rol.
Eot. nom. sect. Mat. 12 (1969), 107-111.
73. ELIASON, S. B., A Lypanov inequality for a certain second order non-
linear dilJerential equation, J. Lond. Math. Soc. 2 2 (1970), 461-466.
74. , Lypanov inequalities and bounds on solutions of certain sec-
ond order equations, Canad. Math. Bull. 17 (1974), 499-504.
75. ST. MARY, D. F., Some oscillation and comparison theorems for (r(t)y')' +
p(t)y = 0, J. Differential Eq. 5 (1969),314-323.
76. KWONG, M. K., On Lyapunov's inequality for disfocality, J. Math. Anal.
Appl. 83 (1981), 486-494.
77. HARRIS, B. J., On an inequality of Lyapunov for disfocality, J. Math.
Anal. Appl. 146 (1990), 495- -500.
78. AGARWAL, R. P. and G. V. MILOVANOVIC, On an inequality of Bogar
and Gustafson, J. Math. Anal. Appl. 146 (1990), 207-216.
79. BOGAR, G. A. and G. B. GUSTAFSON, ElJective estimates of invertibil-
ity intervals for linear multipoint boundary value problems, J. Differential
Eq. 29 (1978), 180-204.
80. LEIGHTON, W., On Liapunov's inequality, Proc. Amer. Math. Soc. 33
(1972), 627-628.
81. KREIN, M. G., Ob odnom predpoloienii A. M. Ljapunova, Funk. Anal.
Priloz. 7 (1973), 45-54.
82. ELBERT, A., Nekotorye zadaci kacestvenno{ teorii dilJerencial'nyh urav-
neni{ vtorogo porjadka, Univ. Druzbi narodov im. P. Lumumby, Moskva,
1971.
83. BEESACK, P. R., On the Green's function of an N -point boundary value
problem, Pacific J. Math. 12 (1962), 801-812.
84. CHENG, S. S., A discrete analogue of the inequality of Lyapunov, Hokkaido
Math. J. 12 1 (1983), 105-112.
CHAPTER VII
J
a+h
(1.1) (f"(x))2 dx ~ 2!3 [f(a + h) - 2f(a) + f(a - h)]2 ,
a-h
with equality if and only if f is given by
C1 {(h - a + x)3 + 6h 2(a - x)} + C2x + C3 (x E [a - h, aD,
{
f(x)= C1(h+a-x)3+C2x+C3 (x E [a, a + h]),
where C1, C2, C3 are arbitrary real constants.
The mentioned Zmorovic's result is an improvement of the inequality
J
a+h
(1.2) (f"(x))2 dx ~ 2~3 [I(a + h) - 2f(a) + f(a - h)]2 ,
a-h
(1.3) J
-1
1
where
h
PROOF: Put 'Y(t) = (g(a - t) l~r + g(a + t) l~r ) l-;r. Then (2.3) becomes
Since
a+h h
putting
PI = g(a - t), P2 = g(a + t), ZI = f"(a - t), Z2 = f"(a + t),
and using the inequality (see [4],[5])
r-I
(2.6)
IZI + ... + znlr:s tp:~r ( )
(PIlzd r + ... + Pnlznn
(ZjEC, Pi>O(i=l, ... ,n), r>l),
we have
j g(x)If"(x)lr dx ~ j
a+h h
1f"(a~t)+f"(a+?lr r - I dt
a-h 0 (g(a-t)t=r+g(a+t)t=r)
h
I I
PI Zl r - l = ... = Pn IZn Ir - l and ZkZj ~ 0 (k,j = 1, ... , n),
!P
Icp(x)w(x)1 dx ~ !
( P
Icp(x)IP dx !
) l/p ( P
Iw(x)I' dx
) 1/,
if and only if Icp(xW = Clw(x)I', where C is a real constant (see [6, p. 54]), we
conclude that equality in (2.2) holds if and only if
J
a+h r-l
J
a+h
g(X)«Ii (f"(X)) dx,
a-h
where «Ii(t) = IW (r > 1), we conclude that «Ii is a convex function. We are led
to consider now general versions of Theorem 1.
First, we give the following definition.
DEFINITION: A continuous function «Ii: R -+ R+ belongs to the class M if there
is a convex function F: R -+ R+ and real numbers ,X and m so that for each
x E R the inequalities
are valid.
A convex function is continuous and a Jensen convex function as defined in
[5].
On of the possible generalizations of Theorem 1 is:
J
a+h
g(X)«Ii(f"(X))
a-h
D f( x) = h d~~) , ~ f( x) = f (x + ~) - f (x - ~) ,
J.Lf(x) = ~ (f (x +~) +f (x - ~)) ,
(3.1) 11'1>11. ~ ~ Uh.Z'p( x )1'1>(x )1' dx )". 11'1>11. ~ 1I<!i1l._, (r > 1),
O"n(r) = ( 2h
1 Jh (h - t) ~ (p(a -
0 r-1 t)t=r
1 + p(a + t)t=r1) dt) r;1
,
(3.2)
where
(3.3) (n is even),
n-1 D
-2- 2i - 1
= J.L~ - L (2i -
• =1
1)!
(n is odd) .
(3.5)
THEOREM 4. Let f E Cn[a - h, a + h], p E CIa, b], p(x) > 0 (x E [a, b]). Tilen
a ln- 1 ) r~l
(3.8) f(n)(t) = A ( It ~(t) sgn(t - a) (A E R),
wben n is even.
Theorem 5 follows from Theorem 4.
ZMOROVIC'S AND RELATED INEQUALITIES 247
11) -. If(a + h) -
.-1
REMARK 1: For n = 2 and r = 2k, this resut reduces to the Theorem 264 from
[2], i.e. to Theorem 2 from l.
REMARK 2: If fECI [a, b], according to Theorem 5 (for n = 1), we have
J
b
The aim is to find the best upper bound K~ = K~(d,g,p) in the inequality
n
We define w(x) = w(x: d) = Il (x - Xj) and
j=O
1 n (Xi _ t)+-l
(4.1) Qn(t) = Qn(t: d) = ( -1)'
n .
I:
i=O
'( .) ,
w x,
t E [a,b],
where
(X _ t)+-l = { (x - t)n-l, a:::; t :::; X :::; b
0, a:::; X < t :::; b.
The case 9 = 1 and p = 2 is the easiest.
248 CHAPTER VII
THEOREM 1. Iff E c(n) [a, b) and a < Xl < ... < Xn-l < b, then
(4.2)
where
If
n (Xi _ t)~n-l
( 4.4) f(t) = f*(t) = ct; W'(Xi) + Pn-l(t),
P n - l being an arbitrary polynomial of the degree n - 1 and C a real constant,
then equality holds in (4.2).
PROOF: If f E c(n)[a, b], then
In other words
b
f( X ) = Pn - l () (n)( )(x - t)+.-l d
X + If t (n _ I)! t,
a
I
b
I Q~(t)
b
dt = [a,xI,'" ,xn-r,b;f*)
a
_ (_l)n
- (2n - I)!
n-l
~ t;
n (Xi - xd~n-l _ _ 1_ K d 2
W'(Xi)' W'(Xk) - b - a [ n( )) .
ZMOROVIC'S AND RELATED INEQUALITIES 249
According to (4.5)
= Kn(d)· IIf(n)lb,
Taking into account that (x - a)(b- x) :::; (b7)2, x E [a, bl, we obtain If(x)1 :::;
(b-,:t IIf"lb x E [a, bl. Therefore, with the notation IIfiloo = max If(x)1 we
4v3 xE]a,b]
have prove the following
THEOREM 3. Iff E C(2)[a, b], f(a) = f(b) = 0, then
n
where ~ n f(a) = L: (_l)n-k (;)f (a + ~(b - a)). From Theorem lone finds the
k=O
following proposition
THEOREM 4. f E C(n)[a, b], then
where
1/2
1 n (_1)n- k k 2n - 1 )
I:
(
C* - - - 2
n - nn-l (n + k)!(n - k)!
k=l
Moreover,
Let p > 1 and QnCt) = Qn(t: d) be defined as in (1). Taking into account that
Qn ?:: 0 (see [10)) denote
C4.6)
,*
J\.n = KnCd, g,p) =
* (! b ) (p-l)/p
[Qn(t)]p/(p-1)
[g(t)p/(p-1) dt ,
-
where g E C[a, b] is a positive function. By means of Holder's inequality from
(4.5) we obtain
THEOREM 5. If f E c(n)[a, bl and (d): a < Xl < ... < Xk-l < b is an arbitrary
system of knots, then
(4.8) I[a,x,b;fll~
P-1)~ ·11f"llp,
( 2p-1
with equality for
a~t~x
f(t) = { 1
(
2 )
b-a
P':l (t-a)p-l
~ I
-(t-a)(b-a)2(~=I)'
2
a<t<a+b
- - 2
{
f(t)=C 1
00, p) = t (2~-=-11)
£::.l
This result is K( n, 1, P (See 18. of Ch. II).
252 CHAPTER VII
J
t
J J
t 81
J J J
t 81 Sn-l
where Wi( t) are positive functions of class C n - I on the closed interval [a, b]. Then
{ud ~ is called Extended Complete Tchebycheff System or an ECT-system.
A function f defined on the open interval (a, b) is convex with respect to {ud~
if
Uo(to) uo( t l ) uo(tn+l )
UI (to) UI (t1) UI(t n+l )
U ( UO,UI, ... ,un,!)_ ~O
to,t}, ... ,tn,tn+1 -
un(to) Un(tl) un(tn+t)
f(to) f(td f(tn+d
for all choice of {ti}~H satisfying a < to < tl < ... < t n +1 < b. In symbols we
write
f E C(UO,UI, ... ,U n ).
Note also that [12, pp. 382 and 395] the function
f(t) = fn(tj x)
0, a ~ t ~ x,
is contained in n;=oC(UO,UI, ... ,Uk). For n = 0, fn(tjx) = wo(t) for x ~ t ~ 0
and equals zero otherwise. (Notice that fn(tj a) = un(t).)
Let Dj, j = 0,1, ... , n, denote the first-order differential operator
d (f(t) )
(Djf)(t) = dt Wj(t) .
ZMOROVIC'S AND RELATED INEQUALITIES 253
Suppose that the function I satisfies the condition that DnDn-1 ... Dol is con-
tinuous on [a, b], then (see[12, pp. 388-389])
(5.1) I(t) =
i=O a
b
where 11/11 = sUPa991/(t)1 and Co = J In(b; x )dx.
a
We use the notation
THEOREM 1. Let I: [a, b] -+ R be a real function such that Dn ... Dol is con-
tinuous on [a, b] and a < t1 < ... < tn < b, then
(5.3) iU(UO,U1,
a, t
... ,un,/)i<K
t b - g,n liD
1, ... , n,
n ... D/II
0 g,p,
where
(5.4)
254 CHAPTER VII
where
H
n+l n
I(x) = I*(x) = C~)-l)kFkgk(x) + Laiui(X),
k=1 i=O
where C, ao, ... , an are arbitrary real constants, then equality holds in (5.5).
THEOREM 3. Let I: [a, b) -+ R be a real function such that Dn ... Dol is con-
tinuous on [a, b) and
lu (uo,Ut,
a, t
... ,Uk")1
t b -< K
1, •.. , k,
g,n,k liDn ···D0 III g,p,
where
(5.7) Kg,.,. ~ (i (Q.,.(z)' /g(z).I.) dZ) 'I. , p-' + q-' ~ 1, p E (1, +00),
Qn,k(X) = U (uo, Ut,···, Uk, In).
a, tt, ... , tk, b
ZMOROVIC'S AND RELATED INEQUALITIES 255
THEOREM 4. Let f: [a, b] -+ R be a real function such that Dn ... Dof is con-
tinuous on [a, b].
(i) If a = to < tl < ... < tn+! = b, then
J J
n t Sn-l
(ii) Let a = to < tl < ... < tk < tk+l = b. If (5.6) holds, then
J J
k t Sn-l
where
and
256 CHAPTER VII
then
t
(6.1)
ZMOROVIC'S AND RELATED INEQUALITIES 257
where [Xo, ... , xnlJ is the n-th order divided difference of f at the points Xo, ... , Xn
n
and x is the center of mass of these points x = n~1 ;E Xi. For further general-
.=0
izations see E. Neumann [19].
A multivariable version of this is also true, E. Neuman and J. Pecaric [15].
They proved that for Xi E Rn,
(6.2) 1
f(x) ::; vol(a)
J 1 n
f(x )dx ::; n + 1 ~ f(Xi)
t7 -
if f is convex, x is as above and a is the convex hull of the points {x 0, ••. , X n}.
Equality holds if and only if f is a polynomial of degree 1. The differential version
of this is the following. Let DO be a differential operator of order n (Ial = n).
Suppose DO f is convex on a, then
REFERENCES
1. Let al, a2, ... be a sequence of nonnegative real numbers not all equal to zero,
+00
and let I: k2a~ < +00. Then
k=l
(1.1 )
2. Two related results are given in the books P6lya and Szego [2, II, pp. 114,218,
and I, pp. 57, 214, respectively] and Hardy, Littlewood and P6lya [3, p. 166]:
10 If a 2:: 0, b 2:: 0, a =1= b, and <P is nonnegative and decreasing, then
unless <p = C, where C > 0, in (O,A), and <p = ° in (A,+oo) (A may be 0).
259
260 CHAPTER VIII
(2.2) (j o
xa+bcfo dX)' > {1- (.: ~! S} j 0
x'"¢ dx j
0
x"¢ dx,
unless cfo = c.
A. M. Fink and Max Jodeit Jr. [5] showed that (2.2) is valid if a, b > -1/2
even if a = 1. Indeed (2.2) is equivalent to
JJ
1 1
(2.3) cfo(x)cfo(y)K(x,y)dxdy ~ 0,
o 0
where
If G(s, t) =
8
1 1
ff
t
K(x, y)dxdy then G(s, t) ~ °on [0,1] X [0, 1] and G(s, t) = °only
if (s, t) = (0,0) or s = 1 or t = 1. We prove this.
With a = a + f3 = b + t, t, t1 = t'\ t2 = tfJ, S1 = sot and S2 = sfJ, 2G = (S1 -
S2)2+(t1-t2)2_(S1t1-S2t1)2. But (S1t2-s2t1) = t1tt2(S2-S1)- 8 2 ;81 (t 1 -t 2)
so
Since S1 - S2 = sot - sfJ and t1 - t2 = tot - tfJ have the same sign each of the
three terms is nonnegative. If neither s nor t are 1 and both are nonzero, one
of the three terms is positive. This property of G is used in the following way.
x
Since r.p is increasing, r.p( x) = r.p( 0) + f d/-l( t) for some nonnegative measure and
o
x
we may write r.p(x) = f duet) where u has a mass of r.p(0) at zero. The inequality
o
(2.3) may be written as
JJ JJ
1 1 x y
1 1
(2.5) j j G(t,s)dO'(t)dO'(s) ~ 0.
0 0 ·
This proves (2.2) and gives the cases of equality. For if equality holds in (2.5)
the measure dO'(t)dO'(s) must be zero except at (0,0), that is !pet) == !p(o).
J. E. Pecaric [6] has observed that the above proof gives
1 1
J
1
when f(O, 0), fleX, y), hex, y) and f12(X, y) ~ on [0,1] x[O, 1], and a" b ~ -1/2.
To see this, we observe that (2.6) is equivalent to
°
1 1
f(x"y) =j jd.\(s,t),
o 0
1 1
(2.8) j j G(s,t)d.\(s,t) ~ 0,
o 0
J J JJ
s t B t
So that dA(S,t) 2: o.
The special case when f(x,y) == c.p(x),¢(y), with c.p and '¢ increasing and non-
negative, yields
J J
1 1
J J
1 1
and
(8.2)
c= (-.!..)8 (~)t
pS qt
(r (~) r(t±t)) 1-8-t,
('\+Il)r(l~~:t)
s=
Il , t=---
,\
Pll + q'\ PIl + q'\
where 8 = PIl + q'\ - Il - ,\ > O. In the interval 0 < k < 1 the function z(k)
is continuous and strictly monotonic. Therefore its continuous inverse function
k = k(z) exists on 0 < z < +00.
Consider the identity
(8.4) J00
f(x)dx
o 0
+J
00
~~
[1- k(yx)) x- q x q f(x)dx,
o
fx) dx
o
J
00
where
II = { [pi (z)Z-Hp~l
00
dz
}-}r
12 = { [[1 -
oo
k(Z)]ql z-I-~ dx
}1r
= (* ~)
--2L--
Now, from (8.4), for y p#+q). we get
(8.7) I
o
f(x) dx :5, y-~ lIP + yf 12Q = (:8) pa (:t) qt Irs lit PPSQqt.
If we change arguments in integrals II and 12, we get
IP =
1
I p'q
P'p>. + pq'J-l
J' 00
k i -.-. (1 - k)~ · dk
o
J
00
and analogously
, qt (1-:-t) r (1-!-t)
r--~--~--~--~
r (2±L)
lq - ----
2 - A + I-'
1-8-t
Now, from (8.7) we obtain (8.1) with (8.2).
We should also prove that the constant (8.2) is the best possible. Indeed
equality in (8.1) with constant (8.2) is valid if
T (T dx)·' (T
inequality
(9.1) x·lf( x )1' $ K x·, If(x )1" x·, If(x )1" dX) ., ,
where the exponents are real numbers subject to certain conditions, was consid-
ered by B. Kjellberg in [17]-[19]. (In fact, by a simple transformation we can
obtain (9.1) from the analogous inequality for a = f3 = 0.)
10. For Carlson's inequality and its generalizations see also Levin and Steckin
[20].
r
11. If f is a nonnegative and nonincreasing real-valued function on (0,+00), if
1 :::; q1 :::; q2 < +00 and 1 :::; p < +00, then
(T (tt f(t))" ~
y; -;'; (T
1
This inequality was attributed to A. P. Calderon by O'Neil [21], and was used
later by Hunt and Weiss to give a short proof of the interpolation theorem of
Marcinkiewicz on quasi-linear operators [22].
where A is defined by
n k
and E Ai = sr1, E Ai = (k - s)rz, 0 < s < k, rz > r1.
i=1 i=8+1
20 IfAII=~+h(v-~),then
(~ a.) ('+'~'+"
where
/1=0 /1=1
where a = ?: aj,
n-1
}=1
1/1 = 2; 7, and
()
C=2~(~~~1
( _1)
~-=-~
~ {r (.!..=.1. a ) n-1
IIr(ai)
} ~
n I/r
"f (7
2: Wi -I n, then
i=1
CARLSON'S AND RELATED INEQUALITIES 269
n
where Wo = ,n/(1 + Wo (1 + ,/(1), Wi = wi(l + v/(1), 1 ::; i ::; n - 1, O"i = L: J.lij,
j=l
and the matrix (J.lij), c,"(1, Vi are as above. Further for equality it is necessary
and sufficient that for all points (x 1 , ... , X n) E nn,
This result is due to B. Klefsjo [25) (we suppose that all integrals are finite).
where
(ap + l)l/ p (bq + l)l/ q
C= .
(l+a+b)
20 If a, b, p, q are defined as in 10 and if 9 is a nonnegative, non increasing
convex function, then (15.1) holds with constant
where K(x" t) 2:: 0, when x E (0, +00), t E (0, +00) and h is a non decreasing
function. H p, q are two real numbers such that p-l + q-l = 1, p > 1, then
(16.2)
+00
f
o
ft(x)h(x)g(x) dx
where
+00 ) -l/g}
X ( / K(x,t)!4(x) dx .
+00
f
o
ft(x)h(x)g(x) dx
= f+00+00f
o 0
ft(x)h(x)K(x, t)dh(t) dx
CARLSON'S AND RELATED INEQUALITIES 271
= !+00 {!+00
dh(t) K(x, t)ft(x)f2(x) dx
}
+00
::::; C /
(+00
dh(t)/ K(x, t)fa(x)dx
) IIp
!
(+00
K(x, t)f4(X) dx
) l/q
::; c
+00
J dh(t)
( p
'xp J
+00
K(x,t)fa(x) dx + ,X~q J
+00)
K(x,t)f4(X) dx
q!+ 0 0 )
o
p!+00
0 0
,XP ,X-q
=C ( fa(X)g(X) dx + f4(X)g(x)dx .
For
we obtain (16.2).
A = (Th(X)9(X) dX) -" ( T /.(x)g(x) dx
1
r1
r'" r' r,
Then,
(16.7)
where
(16.8)
J t: J t: J
+00 +00 +00
(16.9) I(x) dx ~ x 4 /(x) dx ~ x 4 /(x) dx.
t t 0
I(x) dx ~ x b/ lI dx
)lI(t x-btu l(x)1/u dx)U
!
+00
f(z) dz ~ t H • (b: v r! ( +00
z-"Of(z)I'U dx
) u
+ c"
(
!
+00
z" f(z) dx
)
,
whence for
CARLSON'S AND RELATED INEQUALITIES 273
we obtain (16.7).
Forb=Oweget
(16.11)
where
(16.12)
REFERENCES
1. CARLSON, F., Une inegaliU, Ark. Mat. Astr. Fys. 25B 1 (1935), 1-5.
2. POLYA, G. and G. SZEGO, "Aufgaben und Lehrsatze aus der Analysis,
I, II," Berlin, 1925.
3. HARDY, G. H., J. E. LITTLEWOOD and G. POLYA, "Inequalities,"
Cambridge, 1934.
4. GAUSS, C. F., "Werke, Gottingen, 1863-1929."
5. FINK, A. M. and Max JODEIT, Jr., Jensen inequalities for functions with
higher monotonicities, Aeq. Math. J. 40 (1990), 26-43.
6. PECARIC, J. E., Result not published.
7. HARDY, G. H., Note on two inequalities, J. London Math. Soc. 11
(1936), 167-170.
8. GABRIEL, R. M., An extension of an inequality due to Carlson, J. London
Math. Soc. 12 (1937), 130-132.
9. BEURLING, A., Sur les inUgrales de Fourier absolument convergentes
et leur application a une transformation fonctionelle, in "Congres des
mathematiciens scandinaves 1938," Helsingfors, 1939.
10. LEVIN, V. I., Notes on inequalities. I - Some inequalities between series
(Russian), Mat. Sbornik (N.S.) 3 (1938), 341-345.
11. CATON, W. B., A class of inequalities, Duke Math. J. 6 (1940), 442--461.
12. SZ.-NAGY, B., On Carlson's and related inequalities (Hungarian), Mat.
Fiz. Lapok 48 (1941), 162- -175.
13. , tiber Integralungleichungen zwischen einer Funktion und ihrer
Ableitung, Acta Sci. Math. Szeged 10 (1941), 64-74.
274 CHAPTER VIII
(1.1 )
with equality if and only if all the row sums are equal, or all the column sums
are equal, or both.
F. V. Atkinson, G. A. Watterson, and P. A. P. Moran [1] proved the following
result. They used (1.1) to obtain
m n m n
(1.2) LL aijPiqj L arjPr L aisqs 2
i=1 j=1 r=1 s=1
m n
where Pi 2 0, q; 2 0, L: Pi = L: qi = 1.
;=1 ;=1
Putting in (1.2) m = n, aij = aji, and Pi = qi one obtains an inequality
conjectured by S. P. H. Mandel and I. M. Hughes [2] on genetical grounds.
JJJJ
a b a b
(2.1) ab K(x,t)K(x,y)K(s,y)dxdydsdt
o 0 0 0
2 (
/! a b
K(x, y)dx dy
) 3
275
276 CHAPTER IX
(3.1) a n- 1 JJ
o 0
Kn(x,y)dxdy 2: (J J
0 0
K(X,Y)dXdy)n
This inequality was proved in [1] for all n of the form 2 r 3 s , and was conjectured
for other positive integral values of n.
J. F. C. Kingman [3] gave a more direct proof of (1.2) which is based on the
convexity of f(x) = xk for k 2: 1.
A generalization to products of sums of aijk... taken over subsets of the index
set {i,j, k, ... } was also obtained in [3]. For example
mnl nl ml mn
LLL aijkPiqjrk LL aistqsrt LL asjtpsrt LL astkPsqt
;=1 j=1 k=1 s=1 t=1 s=1 t=1 s=1 t=1
z= Pi = i=1
z= qi = i=1
z=
m n l
for Pi 2: 0, qi 2: 0, ri 2: 0, and ri = 1.
i=1
Noting that the notion of a "partial average" is a particular example of a
Radon-Nikodym derivative, J. F. C. Kingman [4] established an inequality in-
volving such derivatives, which is a generalizaion of (1.2).
Using Kingman's method of proof, P. R. Beesack [5] proved the somewhat
weaker inequality than (3.1), namely
(nj
(3.2) !!
a a
Kn(x, y)dxdy 2: a n+ 1 exp 0 0
jIogK(x, y)dx dY )
a2 '
INEQUALITIES INVOLVING KERNELS 277
for n ~ 1 and K(x, y) ~ O. In case K = const > 0, both (3.1) and (3.2) are true
with equality holding, but in general (3.1) is better than (3.2), P. R. Beesack
in [51 showed that the conjecture (3.1) is true for all n ~ 1 and nonnegative
symmetric kernels K. He used a result for symmetric matrices with nonnegative
elements due to H.P. Mulholland and C. A. B. Smith [6]. More generally
"(! i
o 0 0
V(X)V(Y)K(X,Y)dXdY) n
4. P. R. Beesack and J. E. Pecaric [7] extended results from [5] to general mean
values of kernels and also obtained some reverse or complementary inequalities
involving such means. Here, we shall give their results.
Let p, be a nonnegative measure on a a-algebra of subsets of A with p,(A) < 00.
Write An = A X .•• X A (n times) and dp,n for the product measure. p, X •.• x p,
on subsets of An. Suppose K(x, y) is a nonnegative P,2 measurable function and
'I/J is a strictly monotone continuous function. A sequence of generalized kernels
'I/J n K ( x , y) are defined as follows:
If 'I/J is the identity function then 'l/JnK(x, y) = Kn(x, y) is the ordinary n-th
iterated kernel in the theory of integral equations. It is easy to see that
(4.2)
(4.3) 'l/JnHK(x, y) = J
A
'l/JlK(x, z)'l/JnK(z, y)dp,(z).
278 CHAPTER IX
We may also define the n-th order means of K with respect to"p. For n =
1,2, ... they are
In [7], there is a careful discussion on the validity of the results when some
integrals are +00. We refer to that paper for this. Here we will assume all the
integrals are finite.
If X is another continuous and strictly monotonic function with X1K E L 2(A 2)
then one can form iYn,x(K).
THEOREM 1. With the above hypotheses and supposing X( u) > 0 for u > 0, let
</>(u) = X ("p-l(u)) on R+. Then Eorn ~ 1
(4.5) iYn,,,,(K) ~ iYn,x(K)
if either X is increasing and </> is convex and submultiplicative or X is decreasing
and </> is concave and superrnultiplicative.
PROOF: In the first alternative, we have by (4.2) and Jensen's inequality (with
k = 1jJ(K))
•(f "n
*
K dp/ p(A)n+l )
~. ( f k(X;_"X;)dPn+,fp(A)n+l)
~n+l
~ J (~k(Xj-l,Xj)) IJl(At+1dJln+l
</>
An+l
= J IT
An+l 1
Xn K dJl2/Jl(A)n+1.
The above results are extended in the following way. If r is a non-zero real
number let
(4.7)
and
(4.8)
For r = 0 let
and
Note that if 1j;(u) = u T , r i= 0, then K!:,l = 1j;nK. There is no 1j; such that
a~l K = an,,,,(K) but
J IT
log K(xj_l,xj)dl1n+1 = tJ 10gK(xj-l,Xj)dl1n+l
J
An+l 1 1 A n +1
so that
THEOREM 2. Let t/J be continuous and strictly monotone and let cfJ( u) = log t/J-I (u)
for u > o. If either t/J(K) E L 2(A 2) or t/J ~ 0, then
(4.9) an,,,,(K) ~ a~)(K)
if cfJ is convex, t/J-I is submultiplicative, and J log+ K dP2 < 00. If cfJ is concave,
A2
t/J-I is supermultiplicative, and either t/J ~ 0 and J t/J(K)dp2 < 00 or t/J(K) E
A2
L 2 (A 2 ), then inequality in (4.9) is reversed.
The above two theorems have a corollary dealing with the means (4.8).
COROLLARY 1. Let K be measurable and nonnegative, then for -00 <S ~ r <
00
(4.10)
provided a~t)(K) is finite for some t > O.
PROOF: (Sketch only.) For s < r with r =/:. 0, take X(u) = u r , t/J(u) = US so that
cfJ( u) =U r / S is both sub- and supermultiplicative. If s < r < 0 we get the second
alternative of Theorem 1 while if s < 0 < r or 0 < s < r the first alternative is
correct.
If s < 0 = r we apply Theorem 2 with t/J( u) = US so that cfJ( u) = U) log u is
convex and t/J-I is multiplicative. Here J
K t dp2 is finite for some t > 0, and
A2
J log-I Kdp2 is finite.
A2
The last case s = 0 < r follows from Theorem 2 in the same way with cfJ( u) =
(~) logu and t/J(u) = u r •
REMARK 1: The special case of (4.10) for r = 1 and s = 0 appears in Beesack
[5] with the hypothesis that K E L 2 (A 2 ).
A second special case is noted in the following Corollary.
COROLLARY 2. If X ~ 0 and either
a) X is increasing and cfJ = X( t/J-I) is convex and submultiplicative or
b) X is decreasing and cfJ is concave and supermultiplicative, then
(4.11) t/J-I (t/Jn(K(xQ,xn))/p(At- l ) ~ X-I (XnK(XQ,xn)/p(At-I).
In particular for t/J(u) == u, and Kn the usual iterated kernel,
(4.12) XnK(xQ,xn) ~ peAt-Ix [Kn(xQ,xn)/p(A)n-l]
if (b) holds and
(4.13) XnK(XQ,xn) ~ p(At-lx [Kn(XQ,xn)/p(At- l ]
if (a) holds.
INEQUALITIES INVOLVING KERNELS 281
For Volterra type kernels with A = [0, a] and K(x, y) = 0 for x < y the above
can be applied with ~(O) = 0 and increasing on [0,00) and continuous. One can
easily show that the iterated kernels ~nK are also of the Volterra type and that
the representations
J~nK(x, z)~lK(z,
x
~n+lK(x, y) = u)dJ-l(z)
y
( 4.14)
= J~lK(x, z)~nK(z,
x
u)dJ-l(z)
y
J... JII ~
Xn X2 n
(4.17)
282 CHAPTER IX
(4.18)
(4.19)
(4.20)
(4.21 )
(4.22)
provided 0 < m ::; K(x, y) ::; M, 'IjJ(R+) = R, and ¢>' is strictly increasing, while
the opposite of (4.9) and (4.22) give
(4.23)
INEQUALITIES INVOLVING KERNELS 283
provided 0 < m ::; K(x, y) ::; M, ~(R+) = R, and ~' is strictly decreasing.
As an application of Theorems 3 and 4 we again consider the r- means O'~l (K)
defined by (4.8) and (4.9). For r,s f= 0 the functions X, ~, 1> considered in
Corollary 1 are all actually multiplicative.
Let 0 < m::; K(x,y)::; M for (x,y) E A2 and for r,s f= 0, n ~ 1 let
(4.24)
Then
(4.25)
(4.26)
REFERENCES
1. ATKINSON, F. V., G. A. WATTERSON and P. A. P. MORAN, A matrix
inequality Quart. J. Math. Oxford Ser. (2) 11 (1960), 127-140.
2. MANDEL, S. P. H. and I. M. HUGHES, Change in mean viability at a
multiallelic locus in a population under random mating, Nature (London)
182 (1958), 63-64.
3. KINGMAN, J. F. C., On an inequality in partial averages, Quart. J.
Math. Oxford Ser. (2) 12 (1961), 78-80.
284 CHAPTER IX
CONVOLUTION, REARRANGEMENT
AND RELATED INEQUALITIES
(1.1)
with equality only if all the x, or all the y, or all the x but one and all the Y but
one, are zero.
This inequality is due to W. H. Young [1]-[3], and it is very important in the
theory of Fourier series. (Young does not consider the question of equality, Hardy,
Littlewood and P6lya Inequalities, 1934.) The proof uses Holder's inequality.
Here, we shall first give generalizations of Young's inequality given in Hardy,
Littlewood and P6lya's book as theorems 278-284:
°
10 (278.) If.x > 0, J.L > 0, ... ,1/ > .x + J.L + ... + 1/ < 1, and
then
<5 1-..\-1'-···_'"
1 (W) ~ <5_1
1-..\
(X)<5_1
1-1'
(y) ... <5_1
1-1'
(Z),
unless all numbers of one set, or all but one of every set, are zero.
20 (279.) If
en = L ail ai 2 ... ai k ,
i1+i2+···+i n
then
J
00
then
J
(
h'-;-" dx
) l-A-I'
<; \-l
(
i'C' dx
)
J
I-A (
g'C" dx
) 1-1'
unless f or 9 is null.
4° (281.) If>. > 0, f.L > 0, >. + f.L < 1, and
J
x
then
unless f or 9 is null.
5° (282.) If k is an integer and
J... J
00 00
then
6° (283.) If
LXi ~ x,
1 (1f~ex)dxr-'
Xi ~ 0,
then
1'e X)dX';
J... J
7r 7r
then
(1.2)
(1.3)
where (a(j),x) = f
i=1
a~j)xi. The proof of the theorem depends on an earlier
inequality of Brascamp, Lieb and Luttinger [7] (this result will be given later).
288 CHAPTER X
2. Many papers prove convolution inequalities, see for example papers of Sub-
rahmanyam [8] (he finds the best possible constants in some convolution inequal-
ities), Boas and Imoru [9], Beesack and Pecaric [10]. Here we shall give results
of Beesack and Pecaric from [10].
°
Let f and g be nonnegative Lebesgue measurable functions on (0,00) and for
each x 2: consider the measurable nonnegative convolution f * g defined by the
formula
x x
f *g(x) = J f(t)g(x-t)dt = J g(t)f(x-t)dt. If ~: (0,00) -+ (0,00) is continuous,
o 0
onto, and strictly monotonic, we consider bounds for ~fl * ~h * ... * ~fn(x),
where 'IjJ fj == ~(fJ), as well as in inequalities involving generalized means of
~h * ... * ~fn' To this end introduce the definitions
(2.1 )
f3n,p(h, ... ,fn;x) = ~-1{(n-1)!~h * ... *~fn(X)/xn-l},
I
(2.2)
p~l(f" ... , In; x) ~ exp {(n -1) (x - t)"-' ~ log !;(t)dt/x n-' } ,
x
where, for simplicity, we always assume Jlog+ fj(t)dt < 00 for 1 ::; j ::; n (n 2: 2).
o
°
(0,00), and let ¢J, X: R+ -+ R+ be continuous, onto, and strictly monotonic. Let
¢J(u) = X ('IjJ-l(u)) for u > 0. Then, for n 2: 2 and x> we have
(2.3)
(2.4)
provided ¢Jl is convex and 'IjJ-l is submultiplicative, while the opposite inequality
holds if ¢Jl is concave and ~-l supermultiplicative.
CONVOLUTION, REARRANGEMENT AND RELATED INEQUALITIES 289
J... JII
Xn :£'2 n
2
(:=- I)! = Jxo ... JX dXl ... dxn-t.
n
n-l
valid for n 2:: 2, Xn 2:: 0 (where Xo = 0) under our hypotheses. Note that
all integrals exist, finite or infinite, under these hypotheses. The last identity
is clearly true for n = 2 and can be proved by induction. We omit further
details of the proof, only noting that the main step is the application of Jensen's
inequality with D = {(Xl, ... 'Xn-d: 0 ~ Xl ~ X2 ~ ... ~ Xn-l ~ x n}, v =
Lebesgue measure in Rn-l, and q = l/v(D) = (n -1)!x~-n.
COROLLARY. For r f= 0, n 2:: 2, and nonnegative measurable fj (1 ~ j ~ n) on
R+ define the r- means
Then for -00 < S < r < 00 we have for n 2:: 2, X > 0,
(2.6)
This follows by taking x(u) = u r , .,p(u) = US if -00 < S < r < 00 with r,s f= 0
in Theorem 1, etc. In this case the proof (when r = 0 or S = 0) is simpler since
x
we are assuming the finiteness of all the integrals Jlog+ h(t)dt.
o
The special case S = 0, r = 1 of (2.6) was proved in [11, Theorem 2].
From the above results, we can obtain bounds for xit * ... * Xfn in terms of
the ordinary convolution it * ... * fn by taking .,p(u) = u in (2.3). We find that
n l
(
(2.7) xft * .. ·Xfn(x) ~ (nX_1)!X
-
(n -1).
,ft* ... *fn(x))
x n- 1
290 CHAPTER X
PROOF: ([12]) Without loss in generality we can suppose that b = band b1 > 0
(by adding a constant). Now, using the Abel identity we get
and
REMARK: For a version of Theorem 1 where a and b are in Rn but the sums are
from i = 1 to k, see 13.
The case of m vectors was considered in [13]:
THEOREM 2. Let aj = (aj;, ... ,ajn) (j = 1, ... ,m) be nonnegative vectors.
Then
(3.2)
CONVOLUTION, REARRANGEMENT AND RELATED INEQUALITIES 291
PROOF: ([12]) For m = 2, (3.2) follows from (3.1). Suppose that (3.2) is valid
for m - 1. Then, if we suppose that am = am, we have
If F(x, y) is a negative set function (the reverse sign in the above definition
inequality is valid), then the reverse inequality in (3.3) is valid.
(ii) Let F(X1'"'' x m ) be a positive set function for each pair of variables, i.e.
F is I-superadditivefunction, on IX1 x··· X Ix m • Ifxj E I;j (j = 1, ... ,m), then
n n
(3.4) LF(X1i, ... ,X m i) ~ LF(X1i, ... ,X m i).
i=l i=l
292 CHAPTER X
If F is a postive set function for every pair of variables Xi, Xj when i,j > k
or i,j ~ k and also a negative set function when i ~ k and j > k, then (3.4)
becomes
n
(3.5) L F(Xli, ... ,Xki, Xk+l,i,···, Xmi)
i=1
n
~ L F(~li'···' ~i' Xk+1,i, ... , Xmi).
;=1
REMARK: Let t/J(a, b) be a function of two variables. Let us define the matrix
t/J(x,y) where X,Y are two vectors, by
t/J(x, y) = [
t/J( Xl.:' Yl) t/J( Xl! Yn) 1
t/J(XmYl) t/J(Xn" Yn)
For a positive function t/J we say that it is totally positive of order 2 (TP2) if
every sub determinant of the matrix t/J( x, y) is nonnegative. It is known that the
function t/J is T P2 iff F( a, b) = log t/J( a, b) is a positive set function. In this case
(3.4) becomes
n n
(3.6) II t/J(Xli, ... ,Xmi) ~ II t/J(Xli, . .. ,Xmi);
~l i=l
where t/J is positive function which is T P2 as a function of every pair of variables.
EXAMPLES: 1° The function t/J(xt, ... ,x m ) = ~in Xi, Xi > 0, is TP2 and a
l<.<m
positive set function in every pair of variables. S; the following inequalities are
valid
n n n
(3.7) II min(xi' 'fI) ~ II min(xi' Yi) ~ II min(xi' Yi) ([19]);
~l ~1 ~l
n n
(3.8) II
i=1 1<min
"<m x""
_1_
l'
<
-
II lim x""
i=1 1< "<m
_1_
l'
([20]);
n n
(3.9) '"" min X""
~l<"<m J' <
- '"" min x""
~1<"<m JI
([20]).
i=1 _1_ ;=1 _J_
n n n
(3.12) 2:: J(~;tii) (»5 2:: J(XiYi) (»5 2:: J(XiYi)
i=1 i=1 i=1
([16]);
n n n
(3.13) 2::JWi! x i) 5 2::J(yi! x i) 5 ~JWi!~i) ([16],[21],[22]).
i=1 (» i=1 (» i=1
n n n
2::(1 + 17k)~k 5 II (1 + YkYk 5 II (1 + Yk)Xk ([16]).
k=1 k=1 k=1
n
II [(Y2i_1Y2i)m + al(Y2i_1Y2i)m-1 + ... + am]
i=1
n m n m
(3.15)
i=1 j=l i=l j=l
L ai b1r(i) ~ L aib.,.(i) ,
i=1 i=l
where 7r(i) and u(i) are two permutations of the set {I, ... , n}.
Z. Dar6czy [27] proved the following two results with several functions:
n
THEOREM 4. The function F(x) = :z= 9i(Xi) (Xi E (a, b), i = 1,2, ... ,n)satisfies
i=1
the inequality
for all choices of Xi E (a, b) iff the functions t I-t 9k(t)-9k+1 (t) (k = 1,2, ... ,n-1)
are non decreasing on the interval (a, b).
THEOREM 5. Let 9j(t), j = 2,3, ... , 2n, be real-valued functions defined on
interval (0,00). For any two n-tuples X and Y (Xi, Yi E (0, +00), i = 1, ... , n) we
have
n n n n
(3.17) LLgi+k(XiYk) ~ LL9i+k(~iU)
i=1 k=l ;=1 k=l
CONVOLUTION, REARRANGEMENT AND RELATED INEQUALITIES 295
if the functions 9j (t) - 9j+ 1 (t) (j = 2, 3, ... , 2n - 1) are non decreasing on interval
(0, +00).
Theorem 5 is, in fact, a generalization of an inequality of F. Wiener: If C2 ;:::
2: C2n ;::: 0, and x and y are two nonnegative n-tuple Then
Ca ;::: •••
n n n n
(3.18) L L CHkXiYk ~ L L CH Hii1!k·
;=1 k=l i=l k=l
n n n
(3.19) L akbn-k+1 ~ L akbuk = L akbk,
k=l k=1 k=l
where a denotes any permutation of the natural numbers 1,2, ... ,n.
Further, let be
then
4. Let a = (a_ n , ... , ao, ... , an). Define the following rearrangements of a: The
sequence a+ is defined by: at 2: at 2: a::!:1 2: a::!:2 ;::: ... j the sequence +a is
defined by: +ao 2: +a-I 2: +al 2: +a-2 ;::: +a2 ;::: ... , and the sequence a* is
defined by: aD 2: ai 2: a: 1 ;::: a:2 ;::: .... The set a* is said to be symmetrically
decreasing.
The following theorem of R. M. Gabriel [29] is a generalization of results from
[4],[30] (see Hardy, Littlewood, and P6lya, Inequalities, pp. 265-272).
296 CHAPTER X
for a < u, v < b, -00 ~ a < b ~ +00, and suppose that the function G defined
by
G(u,v,w) = F(u,v) + F(u,w) - F(v,w) (a < u,v,w < b),
is nonincreasing with respect to u and w for u ~ mine v, w).
If x is an n-tuple with elements from (a, b), then
n
(4.1) L F(Xi' Xi+!)
;=1
where
Ey = {x: If(x)1 > y}.
Furthermore, since m(j, y) is a non-increasing function of y we let 1* (x) be
that function which is inverse to m(j, y) and continuous to the left.
The following results are valid: (a) the functions f and 1* are "equimeasur-
able" (see [4, p. 277] and [33]), so if f 2:: 0 they have equal integrals over (0,1)
1 1
and J F(j*) dx = J F(j) dx, for any measurable F for which the integrals exist
o 0
[4, p. 277].
(b) A monotonic rearrangement of a positive concave function is also a con-
cave function.
An integral analogue of (3.4) is given by G. G. Lorentz [14].
THEOREM 1. The necessary and sufficient conditions that a continuous function
<fo(X,Ul, ... ,U n ) defined for 0 < x < 1, Uk 2:: 0, k = 1,2, ... ,n, satisfies the
inequality
J J<fo(x,J;(x),···,f~(x))
1 1
J
6
for every 0 < x < 1, Uk 2:: 0, k = 1, ... , n, h > 0,0 < S < x, S < 1- x, and i "# j.
If <fo is continuous the second partial derivatives in all variables,the conditions
(5.2) and (5.3) are equivalent to
(5.2a)
(5.3a)
m(f,y) is finite for all positive y. We may define an even function It(x) by
agreeing that
(1
I t "2 m (f, y) )_
- y,
and that It ( - x) = It (x); or, that is the same thing, that It is even and It ( x) =
J*(2x) for positive x. Then It is the symmetric decreasing rearrangement of I.
The following result is valid ([33], [4, pp. 279- -287]):
THEOREM 2. If I, g, h are nonnegative functions defined on R, then
(5.4) JJ
+=+=
-ex:> -00
I(x)g(y)h( -x - y) dxdy::; JJ
+=+=
-00 -00
It(x)gt(y)ht( -x - y) dxdy.
We may plainly suppose that none of j, g, h is null. We may also replace -x-y
by ±x ± y without changing the significance of the inequality.
The following generalization of this result is given in [34]:
THEOREM 3. The following inequality is valid
Jii
Rn .=1
!Fi(Xi)Hi(Xi - xi+dl dx ::; JIT
Rn .=1
F/(Xi)Ht(Xi - Xi-1) dx,
where all the functions are nonnegative and defined on R, and Xn +1 = Xl.
In the same paper the authors suppose the validity of one more general result,
which is proved in [7], namely the inequality
J J
b b
JG (1f*'1) J
b b
PROOF: As in [37], [39] let the multiplicity n(y) of f at the level y be the number
of roots Xk = Xk(y), k = 1, ... ,n(y), of the equation y = f(x), in [O,b]. The
basic relation connecting the derivatives of f is obtained in [37]:
n
1f*'(x)I- 1 = L 1f'(Xk)I- 1 .
k=l
Using x* as independent variable for the rearranged function f*, we have as
in [39]
1f*'(x*)1 = Idy/dx*l,
hence from the basic relation
n n
= H (~If'(Xk)l-l) dy,
300 CHAPTER X
where H(x) = xG(l/x). It is obvious that the function H(x )/x is nonincreasing,
so the following inequality is valid (see [4, p. 83])
H (I» ~ LH(x).
Thus we obtain
and let
f*(x) = 11- I (X).
The basic relation for f E PC I is derived by integration over the level surface
f = f* in the domain D involved. If dn denotes the inward normal differential,
and \1 f the gradient, then 1\1fldn = df. Since
I1(Z) = J J
f?z
dV =
f?z
dndS,
we find
dl1 = L JdndS= L JI~I dS.
CONVOLUTION, REARRANGEMENT AND RELATED INEQUALITIES 301
The summation runs over all components of the level surface f = f*. However,
df*
dp. = -1f*'(x)1
while Idfl = Idf* I. Comparing, we obtain the m-dimensional basic relation (see
[37],[39]):
1
1f*'(x)1 =
dS J IVfl·
L I=!"
The m-dimensional analogue of the multiplicity function is now the level surface
f
area
S=L dS.
I=!"
THEOREM 4. Let G( x) be non decreasing for x > o. Then
(6.4) J (If*'1)
G dV ::; J (IVfl)
G dV.
PROOF: The integral on the left has the differential
::; 2: f (I Vfl-
H 1) dSdf
1=/*
= 2: f (IVfl) dSdf
I=!"
= 2: f G(IVfl) dSdn
I=!"
= 2: f G(IVFI) dV.
1=/*
The result now follows, as in the one dimensional case, by integration over the
domain D.
The following result, related to Theorem 2, was also proved in [37]:
302 CHAPTER X
THEOREM 5. Let n(y) be the multiplicity of f at the level y, and f E C1[0, b).
Then the following inequalities hold for the indicated ranges of p:
(a) for p ~ 1
(6.5)
b 1 jb 1 jn(f(x))IPI
(6.7) j 1f'(x)l'pl dx '.5. 1f*'(x)I'PI dx '.5. 1f'(x)l'pl dx.
o 0
Equality holds on the right in (a) and (c), and on the left in (b), if and only if
the values of If'(Xk)1 are independent of k, k = 1,2, ... , n for almost all Xk.
Multidimensional generalizations for spherical symmetric equimeasurable de-
creasing rearrangement are also given.
The following generalizations were given in [39):
THEOREM 6. Let f be differentiable almost everywhere in [0, b) and let G(y) be
a function convex for y ~ O. Then with n finite a.e.
(6.8)
O(x) = sup
0<6<z
(~JZ
x-
J(t) dt) .
v
- 6
j
0 0 0
s (O(x)) dx ~ j s {~ ] ret) dt} dx.
a
If
~
J is defined on [a, b] we can in the definition of 0 replace ~ 8 < x by
8 < x. This function is the well-known Hardy-Littlewood maximal function.
°
Generalizations of Theorem 1 for functions of two variables are given in [41].
The following results for maximal function are also valid:
J J
b b
J J
b b
These results are given, for example, in [42], and it is noted that (i) is due to
Hardy and Littlewood, (ii) is due to Marcinkiewicz and Zygmund, and (iii) is due
304 CHAPTER X
to Babenko [43]. Note that in [42] there are 77 references on these results. Chen
in [41] also gave many new results. For example, he proved a general theorem
which includes both (iv) and (v).
A generalization of (iii) to Cr-valued functions is given in [44] where the fol-
lowing theorem is proved: For f = (ft, 12, ... ) a sequence of functions on Rn
form the sequence B = (B 1 , B2 , ••• ), where Bk is the classical maximal function of
fk. Writing IIf(x)llr for the Cr-norm of the sequence (ft(x),h(x), ... ), then
1IIB(x)lI~
R"
dx ~ Ap,r
R"
1IIf(x)lI~ dx
(1+
00
00
IIlp-1 U( x) d)
(III+lx-xIll x
(~1 (V(
III
I
x
))-I/(P-l) d )
x
p-l <B
-,
where B does not depend on I, and XI denotes the center of the interval I.
Suppose also that any of the following conditions is valid:
(a) There exists A > 0 such that U(y) ~ AU(x) and V(y) < AV(x) for
x ~ y ~ 2x and x > o.
(b) There exists A > 0 such that U(y) > AU(x) and V(y) > AV(x) for
x ~ y ~ 2x and x > o.
(c) There exists A > 0 and d > 1 such that U(x) ~ AV(y) for x/d ~ y ~ dx
and x> o.
(d) f(x) is a monotonic function on [0,+00).
Then there exists a constant C, independent of f, such that
1 1
+00 +00
B(x)PU(x) dx ~C If(x)IPV(x)dx.
-00 0
Some results on maximal function are also given in [53]. For example the
following result is valid:
Let f be a decreasing function such that l' E £P(O, +00) (p > 1), and let the
x
function F(x) = ~ J f(t) dt be defined for x ~ o. Then
o
1!;!2-I/P
x
s
wbere tbe p-norm is defined by Ilg(s)ll~ = f 11(t)IP dt.
o
lex) =
1r
-~
7r
J I(t)
2tant(t-x)
dt
-1r
J J
1r 1r
J J
1r 1r
J J
1r 1r
For some generalizations see [42]. Now, the symbol J{ will be used for the
constant
1 -2 3-2 + 5- 2 7- 2 +
J{ = - - ... = 0.7424537 ...
1- 2 + 3- 2 + 5- 2 + 7- 2 + .. .
(iv) (Kolmogorov, Davis) [55]. When IE L the conjugate function 1 satisfies
the inequality
J
21r
ym{x: lj(x)1 > y} :S J{-1 II(x)1 dx, y > 0,
o
306 CHAPTER X
Jlj(x)1 J
2~ 2~
JJ
t t
Jlj(x)1 J
2~ 2~
dx ~K j**(t) dt,
o 0
fU(t)
sup < Kess sup IJ(x)l,
O<t<2~ 1 + log(271" It) - O<x<2~
(H J)(x) = .!.
71"
J
+00
J(t) dx,
t-x
x E ( -00, +00 ).
-00
Hilbert transforms and conjugate functions of even and odd functions are con-
sidered in [58].
CONVOLUTION, REARRANGEMENT AND RELATED INEQUALITIES 307
9. Let f be real-valued, Lebesgue measurable, and defined in Rd. For 0 < p < 00
we write
Ia(f)(x) = f
Rd
f(y)lx - yla-d dy.
~;((f)(x) = sup,-d
r>O
f If(x + y)1 dy.
Iyl<r
We will denote various constants, independent of f, by A.
The Hardy-Littlewood-Wiener maximal theorem (see also [59,1.1]) is
p> 1,
If f is supported by a finite ball B, then [59, I.5.2]
f
B
M(f) dx :S A f (1 +
B
Ifllog+ If I) dx.
The following theorem is due to Hardy and Littlewood [60] for d = 1, and to
Sobolev [61] in the general case (a simple proof is given in [59, V.1.2]):
(i) Let 0 < a < d, 1 < p < q < 00, and l/q = l/p - a/d. Then
IIIa(f)IIQ :S Allfll p ·
If f is supported by a ball B, and l/q = 1 - aid, then Ia(f) E U(B) if
J Ifllog+ If I dx < 00.
B
Now, we shall give two results from [62]:
(ii) Let f ~ 0 be measurable on Rd. Then
IIIao(f)lIr :S Allfll~-oIIIa(f)II:'
with 0 < a < d, 0 < () < 1, 1 < p < 00, p < q < 00, and 1/, =
(1 - ())/p + ()/q.
(iii) Let f be as in (ii). Then
IIIao(P)lIr :S Allfll~-oIIIa(f)II:,
with 0 < a < d, 0 < () < 1, 0 <p< 00, 0 < q :S 00, () < t < () + (1 - ())p,
and 1/, = (t - ())/p + ()/q.
308 CHAPTER X
10. Monotonic rearrangements can be extended to the case when the elements
aj of n-tuple a are from any partial ordered set (see [63]-[65] and [15]). So,
a generalization of Theorem 3 from 3. is given in [15], as well as an integral
analogue.
71' ) l/r
Jr(f) = (
2~!. If(OW dO
Suppose that Co, C2, C_lI C2, C-2,. .. is a sequence of complex numbers which
tend to zero, and let Co ~ ci ~ c~l ~ c; ~ c~2 ~ ••• be the sequence
Ico I, IC11, ie-11, ie21, IC-21,· .. arranged in nonincreasing order.
Hardy and Littlewood [66],[67] proved the following result:
Suppose that 1 < r ~ 2 and that L: c n en8i is the Fourier series of a function
f E Lr(-1I",1I"). Then L:c~en8i is the Fourier series of the function ft (see 5.)
from Lr( -11",11") and the following inequality is valid
J
00
h**(x) ~ xf**(x)g**(x) + f8(t)g*(t) dt.
x
CONVOLUTION, REARRANGEMENT AND RELATED INEQUALITIES 309
13. P. W.Day [15] has noticed that if a, bE R+ with aj, bj ~ 0 for i = 1, ... , n,
then for k = 1, ... , n
k k k
L ajQj ~ L ajb j ~ L buajQj.
j=1 j=1 j=1
N. Komaroff [79] used these inequalities to prove that if aj > 0, then for
k = 1, ... ,n
k k
L !!,jbj ~ L !!,jQj, and
j=1 j=1
k k
L!!,jbj ~ Laibj.
j=1 j=1
14. S. Saithoh [80] shows that if q is a positive integer and F j E L2(R+), then
with equality if and only if every Fj(t) = Cje-tu, for some fixed u independent
2q
of j. Here II * means the iterated convolution product. A related result is given
j=1
in Saithoh [81].
REFERENCES
dx
(2.1) dt = F(t, x), x(a) = Xo,
dx
(2.2) dt =F(t,x)±c, x(a)=xo±c,
dx
(2.3) dt =F(t,x)+cn' x(a)=xo+cn.
then also
If all of the inequalities in (2.4) and (2.5) are reversed then the result also holds.
PROOF: We prove the case with <. Suppose that for some t E (a, b1 ) we have
yet) :::; x(t). Let r = inf{t E (a, bJ) I yet) :::; x(t)}. Then a < r < b1 and by
continuity y(r) = x(r) while x(t) < yet) on [a,r). However x'(r) < F(r,x(r)) =
F(r,y(r)) = y'(r). Since x(r) = y(r) this implies that x(t) < yet) on (r - c,r)
which is contrary to the definition of r. The proof for> is analogous.
316 CHAPTER XI
or
S_={(t,r)la~t~bl' ro~r~rl(t)}CD ifG~O.
If 0 ~ (±F(t, r)) ~ (±G(t, r)) for (t, r) E S±, then all solutions r of the initial
value problem
dr
(2.9) dt = F(t, r), rea) = ro
exist on [a, bl ). Moreover
ro ~ ret) ~ rl(t) on [a,bl ) ifG ~ 0, or
rl(t) ~ ret) ~ ro on [a, bl ) if G ~ o.
P. R. Beesack [5, pp. 40-48] used the previous result in a proof of the following
result.
INEQUALITIES OF CAPLYGIN TYPE 317
u, ~ .up { U EJ Iro + i U ds El } .
(2.11)
if a ;::: 0 (a ~ 0). Moreover, if a ;::: 0 and k and 9 have the same sign (a ~ 0 and
k and 9 have opposite signs), then b1 ;::: b2 and we also have
(2.12)
D+(t)
o = 1·Imsup ¢>( t) - ¢>( to)
t-tt t - to
is the right upper derivative, and the right lower (D+ as liminf). The left upper
D- and left lower D_ derivatives are defined analogously. We have D+(to) ~
D+¢>(t o) and D_¢>(to) ~ D-¢>(t o) and ¢>+(to) exists if and only if D+¢>(t o) and
D+¢>(to) are finite and equal, and analogously for ¢>'-(to).
Proofs are similar to the previous ones so we shall give only a proof of the
following theorem:
318 CHAPTER XI
THEOREM 1. Let the function <ft(t,u) be continuous for t E [O,T] and lui < 8,
and let the continuous function v satisfy
Then
Then
Then 4> is dominated by the solution of the scalar linear differential equation
y' = p( t)y + q( t) that agrees with 4> at to. That is
The following Theorem is also proved in Mamedov,Asirov and Atdaev [12, pp.
12-13]:
THEOREM 3. Let 4>(t,u) be a continuous function on [O,T) x [-0",0"). Suppose
VI(t) is a solution of the differential inequality
(3.9)
Suppose that this is not true, i.e. that for some value T, w( T) < VI (T). Let TO
be the lower bound of numbers s for which we have w(t) < VI(t) for s ::; t ::; T.
Then W(TO) = VI(TO) and w(t) < VI(t) for TO < t < T. Therefore on the segment
[TO, T) we get
'ljJ (t,w(t)) = 4> (t, VI(t)).
Using inequality (3.8) we get
J
T
+J
T
and from (3.11) and (3.12) we get vI(r) ::; w(r), which is in contradiction with our
supposition. This proves the inequality (3.10). But now because (3.10) implies
that a solution of inequality (3.11) is also a solution of inequality (3.9), we see
that that result follows from (3.10).
du
(4.1) dt = ~(t,u)
defines a maximal E. solution of problem (4.1 )-( 4.2). It is not the case that w( t)
is a maximal or minimal solution of (4.1)-(4.2). Indeed, the maximal E. solution
of the equation
which passes through the point (0,0) is the function w(t) = 0, but the maximal
solution is u(t) = t 2 , and the minimal solution is y(t) = _t 2 •
THEOREM 1. If the differentiable function v(t) satisfies
tben
(4.7) v(O) < Uo, D+ vet) < I/> (t, vet)) on (0, TJ,
tben
REMARK 1: In all cases wet) can be obtained as sup{v(t)} over all functions vet)
which satisfy the corresponding theorem, i.e. satisfy the inequalities (4.4), (4.6)
and (4.7), respectively.
5. It is clear that differential inequalities are very important for numerical anal-
ysis, especially for numerical solution of differential equations. Of course, here
it is of interest to discuss comparison theorems with the unique solution of a
differential equation as was proved by Caplygin [6] (see also Berezin and Zitkov
[16, pp. 261-264] or Rabczuk [17, pp. 11-16]:
THEOREM 1. Let functions f(t, y) and F(t, y) be continuous in a domain
In fact, if at any point tl inequality (5.4) is strict then it is strict on [tl' tl + a].
In practice, when solving the equation y' = J( t, y) we look for functions F and
¢> such that F(t,y) ~ J(t,y) ~ ¢>(t,y), and that the equations X' = F(t, X) and
x, = ¢>( t, x) are easily solved. For then we have x :::; y :::; X.
EXAMPLE: Let us consider the initial value problem
and
(5.6) X ~ y ~ x.
(5.7) X = Yo + Y and x = Yo - Y,
t
where Y = J eL(t-s)IJ(s,yo)lds and L is the Lipschitz constant for the function
to
J. Indeed
Xl(t) < X2(t) < ... < xn(t) < ... < yet) < ... < Xn(t)
< Xn-l(t) < Xl (t),
in which we only solve linear differential equations.
INEQUALITIES OF CAPLYGIN TYPE 323
We now exposit the previous remark and give Caplygin's method. Suppose
that Jyy (= 8 2 f/ 8y2) has constant sign is a domain bounded by the curves
y = X(t) and y = x(t) and the lines t = to, and t = to + a, i.e. that sections of
the surface z = J(x,y) cut by planes t = constant are either always convex or
always concave, (fig.I).
z
z
fig 1
Let us consider any section of this surface cut by t = constant (fig 2) with x,
y, X on this section. Construct the chord AB and the tangent AT, if Jyy > 0,
or BT if Jyy < o. In this way we get two surfaces z = F(t, y) and z = r/J(t, y)
with F(t,y) ~ J(t,y) ~ r/J(t,y). We have that F and r/J are linear with respect to
y so the differential equations
z z
B
r Y 0_ Y
x Y X x y X
fig 2
are linear. Let X I and Xl be the solutions of these equations which satisfy the
same initial conditions as above. We can now show that these functions form a
pair of frame curves. Indeed, along the curve y = X(t) we have
dX
-dt -J(t,
X)->0
324 CHAPTER XI
and also f(t,X) = F(t,X). So we have dd~ -F(t,X) ~ 0 and X ~ Xl, and from
the inequality f(t,y)::; F(t,y) we have Xl ~ y. There is an analogous prooffor
x.
Now we formulate Caplygin's Method analytically. Suppose that fyy > O.
Then y' = f(t, y), x' = f(t, x) - aCt) (a(t) ~ 0). Let z = y - x. Using the Taylor
formula for the difference f(t,y) - f(t,x) we get
If we also want to improve the upper estimate we consider the equations X' =
f(t,X) + OCt) (O(t) ~ 0) and y' = f(t, y). Set u = X - y and write u' =
f(t,X) - f(t,y) + OCt). Introduce the notation
and
THEOREM 2. Let functions J(t,y), F(t,y), y(t) and X(t) be defined as in The-
orem 1. H we set
h(t) = X'(t) - J (t,X(t)) ,
then the function
Xl(t) = X(t) - ! t
e-L(t-s)h(s)ds,
to
where L is the Lipschitz constant for the function J(t, y), is an upper fraIlle curve
on the interval [to, to + a] and on that interval we have the inequalities
y(t) ::; Xl(t) ::; X(t).
PROOF: The inequality X1(t) ::; X(t) is obvious. On the other hand we have
!
t
then
Xl(t) = x(t) - ! t
e-L(t-s)h1(s)ds
to
is also a lower fraIlle curve and we have x(t) ::; Xl(t) ::; y(t).
In this case we can continue the process, so we have the following
THEOREM 3. Let J(t, y), F(t, y), y(t) and X(t) be defined as in Theorem 1. Set
ho(t) = X'(t) - J (t, X(t))
and define the sequence
!
t
(6.1) v"(t) - PI(t)V'(t) - Po(t)v(t) - get) < 0, v(O) = uo, v'(O) = u~,
then
(6.4)
u"(t) - V"(t) + er(t)(u'(t) - v'(t)) - (er(t) + PI(t)) (u'(t) - v'(t))
- Po (t)(u(t) - vet)) > 0,
t
Multiplying the inequality (6.4) by exp Jer(s)ds and using (6.5) we get
i
o
(7.1)
This is the Riccati equation associated with the Lagrange adjoint of the operator
The maximal interval over which a solution of (7.1) is continuous is the interval
of disconjugacy of the equation L+u = 0, i.e. the largest interval over which
L+u = 0 has a solution with just one zero. If u is a solution of L+u = 0, then
17 = U' /u satisfies (7.1). The operator L is given explicitly by
EXAMPLE: Let the functions Pl, and Po be constants. Then a solution of (7.1)
is given by
(7.2) J 17 2
dl7
+ Pll7 - Po
= t + C.
(7.3)
are equal so that (7.3) is (17 - a)2 = 0, and an integral of (7.2) is given be
l7(t) = a + (t + C)-l. In this case we can choose C so that 17 is defined
on [0,00), so T is arbitrarily large.
INEQUALITIES OF CAPLYGIN TYPE 329
2. The roots are real and distinct, say a and b. Then an integral of (7.1) is
given by
aCt) = aC - cexp(b - a)t
C - exp(b - a)t
and again we may choose C so that T is arbitrarily large.
3. The roots are complex, say a ± bi. Now the solution of (7.1) is aCt) =
a + b tan( bt + c) and the maximum interval is 7r lb. The method of Riccati
equations gives only T 2: 7r I b.
then also
(8.1') vet) < u(t) on (0, Td (T1:S; T)
where u( t) is a solution of the problem
n
(8.2) u(n)(t) - LPn_k(t)U(n-k)(t) - get) = 0,
k=l
V"(t) - 4> (t, vet), v'(t)) < 0, v(O) = Uo, v'(O) = u~.
330 CHAPTER XI
9. The method of associated equations give only sufficient conditions for the
corresponding inequalities of functions to follow from the differential inequali-
ties. Here we shall give necessary and sufficient conditions for linear differential
inequalities of the nth order.
Consider the linear differential equation
J
t
Yet) = k(s,a)</J(s)ds.
to
J
t
W = K(s,a)</J(s)ds,
to
INEQUALITIES OF CAPLYGIN TYPE 331
Yl (0:) Yn(O:)
y~ (0:) y~( 0:)
(n-Z) ( )
Yl 0: Yn(n-2)( 0: )
Yl(t) Yn(t)
(9.2) K(t, a) =
Yl (a) Yn( 0:)
y~(a) y~( 0:)
(n-2)( ) Yn(n-z\ 0: )
Yl a
(n-l) (a )
Yl Yn(n-l) (0: )
REMARK 4: In the case of a linear differential inequality with constant coeffi-
cients, if the roots of the characteristic equation are all real, then we have an
infinite bound of application since K(t,o:) 2: 0 for t 2: 0: (see Bertolino [11, pp.
251-252]). He gave a very simple proof in the cases when the roots are either all
equal or all distinct (ibid pp. 229-232) that the Cauchy function given by (9.2)
is positive.
Similarly for an equation of the second order Y" + py' + qy = 0 with constant
coefficients p, q, whose roots of the characteristic equation are a±bi, we have that
the Cauchy function is (1 I b) exp( a( t - 0:)) sin b( t - a) and the bound of application
is T* = 7r lb. This is the same result we obtained by using the Riccati equation.
The case of complex roots of the characteristic equation for linear differential
equations of the third and fourth order with constant coefficients was considered
by Z. Radisin (Mat.Vesnik 10 (25)(1973),75-81). For this see also Bertolino [11,
pp. 233-234].
332 CHAPTER XI
YI Yrn
y~ Y:"
(~-I) (rn-I)
YI Yrn
Then (see P6lya [23], Mammana [24], or Rabczuk [17, pp. 101])
implies that yet) > 0 on [to, tl} provided there exists a linear independent system
of solutions Yi(t) of L[y} = 0 such that
On the other hand, the inequality L[yJ > 0 is a generalization of the inequality
y" ~ O. This latter inequality is the differential definition of convex functions,
see for example Mitrinovic [AI, pp 17-18}. The inequality L[y} ~ 0 has a close
relation with convex functions with respect to ECT- systems of functions, see
Karlin and Studden [25} or Pecaric [26}. Note also that the inequality (9.5) is a
necessary and sufficient condition that the system offunctions YI (t), ... , Yn(t) is an
ECT-system(Extended Complete Tchebycheff System) of functions (see Karlin
and Studden [25, pp. 375-466, especially pp. 376 and 379-381).
10. In this Section we shall first give two theorems of B. N. Babkin [27]'[28}.
Both this theorem and another result on differential inequalities of higher order
are given in Mamedov, Asirov, and Atdaev [12}.
THEOREM 1. Let wet) be an arbitrary function defined on [0, T} which satisfies
the conditions
n-2
L[v] - get) > «) L (Mk + IMk I) (v - w)(k) on [0, T],
k=O
where Mk = O<t<T
max Pk( t), and a function u is a solution ofthe equation L[u] = g( t)
with initial co~Jitions (10.1), then
then
where u is a solution of
T T
(10.6) - j(w')2dt - j 4> ... w 2 = O.
o 0
334 CHAPTER XI
From this equation we infer that w = 0, since <Pu 2: O. For the existence of a
solution of the boundary value (1004)-(10.2) we refer to Babkin [27]. Now to
prove the inequality (10.3) we let z = u - v and we get
(10.7) z"-<Pu(t,u+Oz)=o(t)
where oCt) = vlIet) - <p(t,v(t»::; o.
As above we multiply by z and integrate to arrive at
T T T
11. The following result given in Grace and Lalli [29] is a generalization of a
result of Kiguradze [30]:
INEQUALITIES OF CAPLYGIN TYPE 335
where the ai, i = 1,2, ... , n - 1, are positive continuous functions defined on
00
THEOREM 2. Let cfJ(t,u) satisfy the hypotheses of Theorem 1 and suppose the
continuously differential function v satisfies
Then
where en < 0 and limen = o. On the basis of Theorem 1, we have vn(t) < u(t).
By taking limits we get that vn(t) -+ v(t) so that v satisfies (12.5).
REMARK 1: The previous two theorems are given in Gel'fand [37] (see also
Mamedov, Asirov, and Atdaev [12, pp. 58-61].
In the same book [pp. 61-63] the following two theorems are also given.
THEOREM 3. Let cfJ(t, u) satisfy the conditions of Theorem 1 and let v be a
continuously differentiable function that satisfies
then
vet) ~ u(t) on [0, T]
where u is the maximal solution of (12.3) defined on [0, Tj.
The previous two theorems are not direct generalizations of the related results
for functions of one variable since here we do not assume the monotonicity of 4>
with respect to u. Moreover the conditions on the function 4> can be relaxed.
13. Let mine u, v) and max( u, v) be defined as the coordinate wise minimun and
maximum respectively, and let [u, v] denote the interval [UI, VI] X .•. X [Urn, vm].
We say that the function satisfies the Kamke-Wazewski condition (K-W) in u if
4>i is nondecreasing in each Uj for each j #- i. The following Theorem was proved
in Perov [38] (see also Wazewski [39], [40] or Mamedov, Asirov and Atdaev [12,
pp. 63-64]).
then
Suppose the contrary. Then for some i and some r we have Wi( r) < Vie r).
Let ro be the smallest value of s (> 0) for which Wi(t) < vet) for s ~ t ~ r.
Then vi(ro) = wi(ro). On [ro,r] we have 'lj;i(t,W(t)) = 4>i(t,max(w(t),v(t))) 2:
338 CHAPTER XI
s
cPi (t,v(t» since Vj(t) max (Vj(t),Wj(t)) for j 1= i and viet) = max (Vi(t),W(t»
by the K-W condition. From this condition we derive the inequality
J J
T T
J
T
and hence Vie r) S Wier) contrary to our assumption. We thus have proved (13.3).
But now W is also a solution of (12.3) so the Theorem follows.
REMARK 1: If the function cP( t, u) satisfies the K-W condition in u then the
solutions u of (12.3) lie between the minimal solution y and the maximal solution
ti, so that we may replace the inequality (13.2) by
14. The K- W condition for the vector function cP( t, u) in u is essential for the
validity of Theorem 4 of 12. We shall give an example. Consider the problem
u~ = UI - U2 + 2t + 1, UI (0) = 0,
u~ = -UI + U2 + 2t - 1, U2(0) = 1.
This problem does not satisfy the K-W condition in u. The solution of the
problem is UI = t 2, and U2 = 1 + t 2. For a function to compare this with we take
VI = t/5, and V2 = 1 + 3t. Then we have that for 0 < t < 5/8
V~ - VI + V2 - 2t - 1 = (1 + 4t)/5 > 0,
v~ + VI - V2 - 2t + 1 = 3 - 24t/5 > o.
But VI - Ul = t(1/5 - t) < 0 for t > 1/5, i.e. the differential inequality is not
satisfied.
THEOREM 1. Let </>(t, u) satisfy the hypotheses of Theorem 1 of 13. If the con-
tinuous function v satisfies
min (D-v(t), D+v(t)) ~ </>(t, v(t)), v(O) < uo, on (0, Tj or
min (D_v(t), D+v(t)) ~ </>(t, v(t)), v(O) < uo, on (0, TJ,
then
v(t) ~ u(t) on [O,Tj,
where u is the maximal solution of the problem (12.3) on (0, Tj.
i
'¢( t, s, u) is decreaBing in u. Let v( t) be a continuous function on [0, T) sum that
(16.1) v'(t) < (»~ {t, vet), >I' It, " v(,)J dS} on (0, tJ,
i
If u( t) is any solution of the problem
(16.2) u'(t) ~ ~ { t, u( t), >I' It, x, u(,)J ds } , u(O) ~ u" on [0, T],
then
(16.3) v(t) < (»u(t) on (O,Tj.
PROOF: Since v(O) < Uo or v'(O) < </>(0, uo, 0) = u'(O) we have the conclusion
(16.3) on some interval containing zero. If (16.3) is not true then there is a first
point t* where u(t*) = v(t*). Then we have
Therefore to the left of t* we have vet) > u(t) which is a contradiction. Thus
(16.3) holds on (0, T].
THEOREM 2. Let the functions 1/>( t, u, v) and .,p( t, s, u) satisfy the conditions of
Theorem 1 and vet) be a continuous function that satisfies the integro-differential
inequality
(16.4) v'(t) ::; Co,). { t, vet)) .p It, s, v(s)] dS} on (0,1'], v(O)::; (~)u.
Then
(16.5) vet) ::::; (~)u(t) on[O, T],
where u(t) is the maximal (minimal) solution of (16.2) on [0, T].
PROOF: For any fixed n let un(t) be a solution of the problem
(16.7)
(16.8)
vet) < un(t) on (0, h].
The last inequality is by the previous Theorem. Also by the previous Theorem we
have Un+I (t) < u n(t). It follows that lim u n(t) = u( t) exists for [0, h]. By writing
the defining equation for Un(t) in integral form we get the limiting function u(t)
to satisfy (16.2). In fact it is the maximal solution and by (16.8) we get (16.5)
on [0, h]. If [0, T I ] is the maximal interval on which (16.5) holds and TI < T, we
have V(TI) = U(TI) and
Tl t }
v'(t)<1/> { t,v(t),J .,p[t,s,u(s)]ds+ J .,p[t,s,v(s)]ds
o Tl
INEQUALITIES OF CAPLYGIN TYPE 341
on [T!, T]. We may repeat the above argument to get (16.5) on an interval
[TI' T! + h] contradicting the maximality of T. Thus (16.5) is proved for [0, T].
v'(t) ,; 1 {t, vet)}>/> It, x, v(,)[ <Is} on (0, T[, viol < no,
then v(t) ::; u(t) on [0, TJ, where u(t) is the maximal solution of
v'(t) < (> )1 { t, v( t).j>/> It, ", v(,») d'} on (0, TJ,
i
solution of
.'(t) ,;; • { t,.( t), >P It, s, .(s)) ds } on (0,1'], .(0),;; ....
Then vet) ::::; u(t) on [0, T] where u(t) is the maximal solution of (18.1) defined
on [O,T].
REMARK 1: Generalizations of the previous results are also given in the same
book.
19. Not all integro-differential inequalities are included in (16.1) or (16.4). For
example, P. G. Pachpatte [41] proved the following result:
THEOREM 1. Let x(t), x'(t), aCt) and bet) be nonnegative continuous functions
defined on I = [0,(0) such that
J
t
i
Then on I we have
(19.2) x'(t) ,;; a(t) +b(I) 0( s )[A(s) + B( s)] exp (j c(r )[I>(r) - I] dr) ds
where
J
t
i
./
t
PROOF: Let met) = J c(s)(x(s) + x'es)) ds, m(O) = 0, so that (19.1) becomes
o
(19.3) x'(t) ::::; aCt) + b(t)m(t).
INEQUALITIES OF CAPLYGIN TYPE 343
J J
t t
t
Introducing the function v( t) = m( t) + J b( s )m( s )ds this inequality may be
o
rewritten
By observing that m'(t) = v'(t) - b(t)m(t) we combine the latter inequality with
(19.6) to arrive at
J
t
r
whereG(1') = J ds/(1'os + W(s)), l' ~ 1'0> 0, and II ischosensothatG(x(O) +a)-+
ro
t
J b(1' )d1' is in the domain of G- l for t E II'
o
There are many other results for integro-differential inequalities of the first
order as well as some for higher order. For example, Pachpatte [42] considered
the following inequality in connection with the previous results,
J
t
J
t
Then on I
INEQUALITIES OF CAPLYGIN TYPE 345
J
8
REFERENCES
J+
t
J
t
then
Moreover, the essential idea for such inequalities (together with their appli-
cations to differential equations) dates as do related differential inequalities to
1885. For example, Peano [16] proved a special case of (1.1) for a = O. In 1930
E. Kamke [17] proved and made use of an inequality which is essentially that
of Gronwall. Similarly, Reid [18] before Bellman's result, (see also [5]) proved a
result which is essentially the same as that of Bellman except that the hypothesis
on k was relaxed to be Lebesgue integrable.
J
t
Define the function R(t) by the right hand side of this inequality. Then R(t)
satisfies the differential inequality
3. Inequalities of this type, which give explicit bounds for functions which satisfy
a Volterra integral type inequality, have numerous applications to boundedness,
uniqueness, and stability theorems in differential equations and to Volterra type
integral equations. Some of these applications appear in Hille [6], Reid [5],
Filatov and Sarova [19], Martynjuk and Gutovski [20], and Demidovic [21].
Both of the Theorems cited above have hypotheses which are not necessary.
In this section we give various generalizations of Gronwall's inequality involving
an unknown function of a single variable. Later we will consider other general-
izations.
A. Filatov [22] proved the following linear generalization of Gronwall's inequal-
ity.
THEOREM 1. Let x(t) be a continuous nonnegative function such that
J
t
J
t
Ix(t)1 ~ Ix(to)1 exp (-a(t - to» + b( a - a)-l (1- exp (-(a - a)(t - to))).
In the book [24] R. Bellman cites the following result (see also Filatov and
Sarova [14, pp. 10-11]]:
J
t
x(t) ~ /t b(s)ds +
o
sup la(t)lexp
O<t<h
- -
(/t0
a1 (S)ds) .
4. A more general result was given in Willett [26] and Harlanlov [27]. Here we
shall give an extended version due to Beesack [28, pp. 3--4].
THEOREM 1. Let x and k be continuous and a and b Riemann integrable func-
tions on J = [a, Pl with b and k nonnegative on J.
(i) If
t
(4.1) x(t) ~ aCt) + bet) / k(s)x(s)ds, t E J,
01
i
then
(4.1)' x(t) " a(t) +b(t) a(. )k(s) exp (i b(r )k(r )dr) d., t E J.
PROOF: This proof is typical of those for inequalities of the Gronwall type, we
set
J
t
and
U'(t) = k(t)x(t).
Since b ~ 0 substitution of (4.1)" into (4.1) gives (4.1)'. The equality conditions
are obvious and the proof of (ii) is similar or can be done by the change of
variables t - -t.
REMARK 1: Observe that the stated hypotheses assure the existence of all the
integrals that appear. In case the integrals are interpreted to be Lebesgue inte-
grals (Willett [26]) the hypotheses can be relaxed to: x, a, b, k are measurable
with kx, ka, kb being integrable. The equality and inequality conditions then are
to be interpreted as a.e. as is usual. In this sense the stated conditions for equal-
ity are necessary and sufficient. Similar remarks apply to the other inequalities
considered in this Chapter.
REMARK 2: B. Pachpatte [29] proved an analogous result on R+ and (-00,0].
REMARK 3: Willett's paper [26] also contains a linear generalization in which
n
E
v
THEOREM 1. H
J J
t n ~
for t E [a, b], where a = to < ... < tn = b, Ci are constants and the functions
appearing are all real, continuous and nonnegative, and if
then
where
and
J
t
THEOREM 2. Let x(t) be real, continuous, and nonnegative on [c, d] such that
J
t
°
o
i
where aCt) ~ 0, bet) ~ 0, k(t,s) ~ and are continuous functions for c ~ s ~ t ~
d. Then
where A(t) = sup a(s), B(t) = sup b(s), K(t,s) = sup k(u,s).
C~8~t C~8~t 8~u~t
J
t
7. S. Chu and F. Metcalf [33] proved the following linear generalization of Gron-
wall's inequality:
THEOREM 1. Let x and a be real continuous functions on J = [a,,8] and let k
be a continuous nonnegative function on T : a ~ s ~ t ~ ,8. If
J
t
(7.1) x(t) ~ aCt) + k(t,s)x(s)ds, t E J,
a
then
J
t
(7.1)' x(t) ~ aCt) + R(t,s)a(s)ds, t E J,
a
= E K;(t,s),
00
where R(t,s) with (t,s) E T, is the resolvent kernel of k(t,s) and
;=1
K;(t,s) are iterated kernels of k(t,s).
REMARK 1: P. Beesack [34] extended this result for the case when x,a E L2(J)
and k E L2(T), and he noted that the result remains valid if ~ is substituted for
~ in both (7.1) and (7.1)'.
360 CHAPTER XII
n
REMARK 2: If we put k(t,s) = b(t)k(s) and k(t,s) = E bi(t)ki(S) we get the
i=l
results of D. Willett [26].
8. G. Jones [35] extended Willett's result in the case of Riemann-Stieltjes inte-
grals. This result is related to discrete results so it will be given in Chapter XIV.
For some analogous results see Wright, Klasi, and Kennebeck [36], Schmaedeke
and Sell [37], Herod [38], and B. Helton [39],[40].
f
t
(9.1) u(t) ~ c+ (a(s)u(s) + b(s)uat(s»ds, c ~ 0, a ~ 0,
to
°
where a(t) and b(t) are continuous nonnegative functions for t
For ~ a < 1 we have
~ to.
(9.1')
j
1
PROOF: For 0: = 1 we get the usual linear inequality so that (9.1') is valid.
Assume now that 0 < 0: < 1. Denote by v a solution of the integral equation
J
t
J
t
(i ~".(')ds
then
THEOREM 1. If
J J
t h
Cl ~ 0, c2 ~ 0, c3 > 0, and the functions u(t) and ,pet) are continuous and
nonnegative on [0, h], then for 0< 0: < 1 we have
('i-- i
1
t
If 0: > 1 and C2 (0: - 1) f ,p( s )ds < c~ -0:, there exists an interval [0,8] C [0, h]
i
o
where
1
J
t
then
REMARK 1: The inequality (11.1) was considered by P. Maroni [46]' but without
the assumption of the monotonicity of the ratio a/b. He obtained two estimates,
one for n = 2 and another for n ~ 3. Both are more complicated than (ILl'). For
n = 2 and alb nondecreasing, Starchurska's result can be better than Maroni's
on longer intervals.
THEOREM 1. Let u(t) and k(t) be positive continuous functions on [c, dj and let
a and b be nonnegative constants. Further let g( z) be a positive non decreasing
function for z ~ o. If
J
t
then
A
where G(..\) = J ds/g(s) (~ > 0, ..\ > 0) and d l is defined such that G(a) +
~
t
b J k(s)ds belongs to the domain ofG- l for t E [c,d l ].
c
J
t
If
(13.2)
where
J
6
ds
(13.3) G( c5) = g( s ) , c: > 0, c5 > 0,
and (13.1') holds for all values of t for which the function
J[a(s)~(s) +
t
J
t
14. K. Ahmedov, A. Jakubov and A. Veisov [48] proved the following theorem:
INEQUALITIES OF GRONWALL TYPE 365
to
where
1) f(t) is continuous, nonnegative, and nonincreasing;
2) ljJ(t) is differentiable, ljJ'(t) ~ 0, ljJ(t) ~ t, ljJ(to) = to;
3) g( u) is positive and non decreasing on R; and
4) k( t, s) is nonnegative and continuous on [to, T] x [to, T] with ~~ (t, s) non-
negative and continuous.
Then for G defined by (13.3) we have
where
J
</>(t)
ak
F(t) = k (t, ljJ(t» ljJ'(t) + at (t, s)ds.
to
J
x
where G is defined by (13.3) and we assume that the term in the { } is in the
domain of G-l.
x
PROOF: We define R(y) = J a(r)g (u(r»dr so that the inequality becomes
y
J
x
But Gis nondecreasing so G (u(y)) ~ G (u(x) + R(y)). Combining the last two
inequalities and rearranging we arrive at
J
x
16. The following result is proved in Ahmedov, Jakubov, and Veisov (48].
THEOREM 1. Suppose that
where u(t), J(t), and ,8i(t) are positive and continuous on [to, 00), ai(t) > 0 while
aHt) ~ 0; 9 is a non decreasing function that satisfies g(z) ~ z for z > o. Then
u(t) ~ J(t) -1 + G- 1 [
G(f) + log gn
ai(t) -In g
n
ai(t O)
+ /. t, Oi(S)!ii(S)ds 1
INEQUALITIES OF GRONWALL TYPE 367
4>l(t) 4>2(t)
Ft(z) j C
G(z) = F2(z) = G(zo) + F2{S) ds,
%0
J
n t/>i(t)
J
t
i
we also have on [a, P)
t
PROOF: Let yet) = J L(s,x(s))ds. Then y'(t) = L(t,x(t)) and yea) = O. Using
a
the integral inequality that x satisfies and then condition (17.1) we have
y'(t) :::; L (t, A(t) + B(t)y(t)) :::; L (t, A(t)) + M (t, A(t)) B(t)y(t)
which gives the above estimate.
Some related results are given in Pecaric and Dragomir [51].
18. In the previous results we have a nonlinearity in the unknown function only
under the integral sign. In the next few results we allow the nonlinearity to
appear everywhere.
THEOREM 1. Let the positive functions u( t), a( t), and b( t) be bounded on [c, d);
k( t, s) be a bounded nonnegative function for c :::; s :::; t :::; d; u( t) is a measurable
function and k(., t) is a measurable function. Suppose that f ( u) is strictly in-
creasing and g(u) is non decreasing. If A(t) = sup a(s), B(t) = sup b(s), and
c$s$t c$s$t
K(t,s) = sup k(a,s), then from
s$O'$t
follows
u
where G(u) = Jdw/g (I-l(w)) (e > 0, u > 0) and
e
This theorem was proved in Butler and Rogers [52] and the following one in
Gyori [12]:
370 CHAPTER XII
THEOREM 2. Suppose
J
t
where the functions I(t), aCt), .B(t), and g(u) satisfy the conditions of Theorem
1 in 13., and the function F(u) is monotone decreasing and positive for u > o.
Then on [to, d']
t
where G(z) = J dsjg [F-l(s)] , z > c ~ 0 and d' is defined such that the function
e
c5(t) defined in Theorem 1 of 13. belongs to the domain of definition of the
function F- l 0 G- l .
19. The next result allows the integral to appear in the nonlinearity. It is due
to Willett and Wong [43].
THEOREM 1. Let the functions x, a, b, and k be continuous and nonnegative on
J = [a,.B], 1 ~ p < 00, and
t ) IIp
x(t) ~ aCt) + bet) ( [k(S)XP(S)dS , t E J.
Then
(j k(s)e(s)ap(s)dS) IIp
x(t) ~ aCt) + bet) a 11' t E J,
1 - [1 - e(t)] P
20. Generalizations of this result were given in Beesack [28, pp. 20-30]. Here
we shall give some results obtained in Filatov and Sarova [19, pp. 34-37] and
from Deo and Murdeshwar [54].
INEQUALITIES OF GRONWALL TYPE 371
THEOREM 1. Suppose
1) u(t), J(t) and F(t,s) are positive continuous functions on R, and s:S t;
2) aF(t,s)JOt is nonnegative and continuous;
3)
4) °
g( u) is positive, continuous, additive, and non decreasing on (0,00);
h(z) > and is non decreasing and continuous on (0,00).
(i
If
l
t
aF
<p(t) = F(t,t) + 7it(t,s)ds,
o
and
PROOF: Using the additivity of the function 9 and that nondecreasing nature of
F(t,s) in t, we have
u(t) - J(t) :S h(v(t)) ,
where
1 1
t T
[1 1
and since u(T) - J(T)
(i
IT
21. Further generalizations of this result are given in Deo and Dhongade [55],
[56] and Beesack [28, pp. 65-86]. Bessack has also given corrections of some
results from Deo and Dhongade [55],[56]. Here we give only one result of P.R.
Beesack (this result for f(x) = x, h(u) = u, and aCt) = a becomes Theorem 2 in
Deo and Dhongade).
THEOREM 1. Let x, a, k, and k1 be nonnegative continuous functions on J =
[a, (3), and let aCt) be non decreasing on J. Let g and h be continuous nondecreas-
ing functions on [0,00) such that g is positive, subadditive and submultiplicative
on [0,00), and h( u) > 0 for u > O. Suppose f is a continuous strictly increasing
function on [0,00) with feu) 2: u for u 2: 0 and f(O) = O. If
then
where
Yo
dy/ j-l(y), y> 0, (Yo > 0)
and
J
u
with
If a(t) = a then we may omit the requirement that 9 be subadditive and then
[or a ~ t ~ /32 we have
where Ga(u)
o
22. In previous sections we have given explicit estimates for Wlknown functions
which satisfy integral inequalities. Several of these results may be given in terms
of solutions of some differential or integral equation and are similar to results for
differential inequalities exposed in Chapter XI. The first one we give is due to B.
N. Babkin [57].
THEOREM 1. Let ¢( t, u) be continuous and nondecreasing in u on [0, T] X ( -6,6)
with 6 ~ 00. I[v(t) is continuous and satisfies
J
T
v(t) ~ Uo + ¢(t,v(s))ds,
o
INEQUALITIES OF GRONWALL TYPE 375
23. The following two theorems are generalizations of the preceding result.
THEOREM 1. Let ¢(t, s, u) be continuous and non decreasing in u for ~ t, s ~ T
and lui ~ 6. Let uo(t) be a continuous function on [0, Tj. If vet) is a continuous
°
function that satisfies the integral inequality (on [0, T])
J
t
J
t
J
to
+J
to
J
t
J
t
J
t
J J
t T
J J
t T
v(t) ~ uo(t) + (,61 (t,s,v(s»ds + (,62 (t,s,v(s»ds,
o 0
then
v(t) ::; u(t) on [0, T)"
where
J J
t T
REMARK 1: The special cases when (,61 == and (,62 == in the previous theorem
gives integral inequalities of Fredholm and Volterra respectively.
REMARK 2: The previous theorem is given in Mamedov, Asirov and Atdaev [58).
This book also contains the following result of Ja. D. Mamedov [59).
THEOREM 2. Let the function (,6(t, s, u) be continuous in t (in [0,00») for almost
all s E [0,00) and u with lu I < 00. Suppose that for fixed t and every continuous
function u(s) on [0,00) the function (,6(t,s,u(s» is mesurable in s on [0,00).
Further let (,6 be non decreasing in u and Uo (t) be a continuous function on [0, 00 )
J
00
then
J
00
REMARK 3: Mamedov [59] (see also Mamedov, Asirov, and Atdaev [58]) also
considered (24.1) with "~" instead of " < ", as well as the inequality
J J
t 00
if Wet, oS,.) and I are monotonic in the same (opposite) sense, where ,81 > a is
chosen so that the maximal (minimal) solution exists for the indicated interval.
Then if W(., s, u) and H( t, .) are monotonic in the same sense,
J J
t t
(25.6) v'(t, T):::; (~) W (T, t, I-I [aCt) + H (t, vet, t»)]) ,
if the functions W (t, s, .) and I are monotonic is the same (opposite) sense. On
the other hand by (25.4) we have
(25.4') H (t, vet, t»:::; (~) H (t, vet, T», a:::; t :::; T,
if (a) W(t,s,.) and H(t,.) are monotonic in the same «b) opposite) sense. Thus
I-I [aCt) + H (t, vet, t»):::; (~) I-I [aCt) + H (t, vet, T»] ,
on a:::; t:::; T if (a'): I is increasing and (a) or I is decreasing and (b) «b') I is
increasing and (b) or I is decreasing and (a». This in turn implies that
if (a"): Wet, s,.) is increasing and (a') or Wet, s,.) is decreasing and (b') ( (b ll ):
Wet, s,.) is increasing and (b' ) or Wet, s,.) is decreasing and (a'». Combining
this with (25.6) we see that if Wet, s,.) and H(t,.) are monotonic in the same
sense then
(25.8) v'(t, T) ~ (?) W (T, t, 1- 1 [aCt) + H (t, vet, T»)) , a ~ t ~ T < /3,
provided that (A): Wet, s,.) and I are monotonic in the same sense «B): Wet, s,.)
and I are montonic in the opposite sense).
As in the analysis of (25.4) and (25.4') we have on [a, /31]
if (A'): H(t,.) is increasing and (A) or H(t,.) is decreasing and (B), «B'): H(t,.)
is increasing and (B) or H(t,.) is decreasing and (A». Now if (A"): I is increasing
and (A') or I is decreasing and (B') «B"): I is increasing and (B') or I is
decreasing and (A'» then
on [a, .81]. The conclusion (25.2') now follows in cases (i) or (ii) from (25.5) and
(25.1).
In the same way one can prove:
INEQUALITIES OF GRONWALL TYPE 381
and that W(., s, u) and H(t,.) are monotonic in the opposite sense. Let ret) =
ret, t, a) where ret, T, a) is the maximal (minimal) solution of problem (25.3) and
suppose that W(t,s,.) and I are monotonic in the opposite (same) sense. Then
I
()(~s~t s~u~t
then
27. P. R. Beesack [2, pp. 56-65] showed that a sequence of well-known results
can be simply obtained by using the previous results. Here we shall give one
example. We consider the following inequality of Gollwitzer [53]:
J
u
(27.2')
then
where
J
u
i U
tinuous functions defined on R+ such that for t E R+,
x(t) ,; a(t) + b(t) (i c(')x(, )d, + c(, )b(') d( u)x( U)dU) d') .
i
Then on the same interval we have
29. The following three theorems from Bykov and Salpagarov [61) are given in
the book Filatov and Sarova [19).
THEOREM 1. Let u( t), v( t), h( t, 1'), and H( t, 1', x) be nonnegative functions for
i [v(. i
t ~ l' ~ X ~a and Cl, cz, and C3 be nonnegative constants not all zero. If
JJJ
t r s
+ C3 H(s, 1',x)u(x)dsd1'ds,
a a a
384 CHAPTER XII
then for t ~ a
e, {c, i["(8) + i e) i!
(29.1')
PROOF: Let the right hand side of (29.1) be denoted by bet). Then b(s) :::; bet)
for s :::; t since all the terms are nonnegative. We have
b'(t) _ ( )u(t)
bet) - C2 V t bet) + C2
1 t
h(t, r)u(r) d
bet) r
a
11
t r
H(t,r,x)u(x)d d
+C3 bet) X r
a a
! !!
t t r
i i
Integration from a to t yields
!!!
t 8 r
+ C3 H(s, r, x)dxdrds.
a a a
Writing this in terms of bet) and using u(t) $ bet) completes the proof.
THEOREM 2. Let the nonnegative function u(t) defined on [to, 00) satisfy the
inequality
! !!
t t 8
u(t) $ cexp {j to
[k(S'S)+ j (ak~;O')
to
+G(S,s,O'») dO'
+ j j aG(~sO"
to to
r) drdO'] dS} .
INEQUALITIES OF GRONWALL TYPE 385
THEOREM 3. Let the functions u(t), a(t), v(t), and w(t,r) be nonnegative and
continuous for a ~ r ~ t, and let Cl, C2, and C3 be nonnegative. H for tEl =
i i
~,oo) .
30. Other related results exist. Here we shall mention one which appears in E.
H. Yang [62].
THEOREM 1. Let x(t) be continuous and nonnegative on I = [0, h) and let p(t) be
continuous, positive, and non decreasing on I. Suppose that J;( t, s), i = 1, ... , n,
are continuous nonnegative functions on I X I, and non decreasing in t. If for
tEl
J J
t tl t n -l
then
x(t) ~p(t)U(t), tEl,
where U(t) = Vn(t, t) and Vn(T, t) is defined successively by
REMARK 1: In the special case of this theorem for n = 2, one can get the
i
conclusion for tEl
31. In the previous theorems we have had linear inequalities. We now turn to
some nonlinear inequalities. B. Pachpatte in [63] and [29] proved the following
two theorems.
THEOREM 1. Let x(t), aCt), and bet) be real nonnegative and continuous func-
tions on I = [0,00) such that
x(t):::; Xo + j a(s)x(s)ds +
0 0 0
j (j
a(s) b(r)XP(r)dr) ds, tEl,
i
where Xo is a nonnegative constant and 0 :::; p :::; 1. Then for tEl
I
1
THEOREM 2. Let x(t), aCt), bet), and c(t) be nonnegative and continuous on R
i
such that for tEl
THEOREM 1. Let u(t), aCt), J(t,s), gi(t,S), and hi(t,S), i = 1, ... ,n, be non-
negative continuous functions defined on I = [0, h) and I x I. Let aCt) be
nondecreasing and J(t, s), 9i(t, s) and hi(t, s) be nondecreasing in t. liO < p ::; 1
and
t t
wbere F( t) = exp JJ( t, s )ds, and Gi(t) = Jgi( t, s )ds, i = 1, ... , n.
o 0
THEOREM 1. Let x(t), aCt), and bet) be nonnegative continuous functions defined
on I = [a, bj, and let g( u) be a positive continuous strictly increasing subadditive
function for u > 0 with g(O) = o. If for t E I
i
then for t E Io we have
where
J
u
and
THEOREM 2. Let x(t), aCt), bet), e(t), and k(t) be continuous on I = [a, bj and
f( u) be positive, continuous, strictly increasing, submultiplictive, and subaddi-
tive function for u > 0, with f(O) = o. If for t E I we have
J
t
J
8
i i
where
f f~:),
U
i
and
!
previous theorem. IT for tEl we have
x(t) " a(t)+b(tlr' (j c(s)f (x(s)) ds + c(,)f (b(s)) (i k(r)f (x(r)) dr) dS) ,
!
t S b we have
x(t) '" x(,) - b(tlr' (j c(r)f (x(r)) dr + c(r) (i k(u)/ (x(u)) dU) dr) ,
!
then for the same range of values we also have
x(t) ;, xes) (r' (1 + f (b(t)) c(r)exp (i (c(u)/ (b(t)) + k(u)) dU) dr) ) -,
390 CHAPTER XII
34. In Bykov and Salpagarov [61) the following theorem was proved:
THEOREM 1. Suppose that the functions u(t), aCt), and f3(t,s) are nonnegative
for 0 < s < t < b and cjJ( s) is positive, non decreasing and continuous for s > o.
H
J
t
== C + LcjJ(u)dr = bet),
a
! cjJ~:)::;!
u( t) t [
a(r) + !
r 1
f3(r,s)ds dr = pet).
b'(t) cjJ(u(t))
cjJ (b( t)) = a( t) cjJ (b( t)) +
J t
cjJ(u(s))
f3( t, s) cjJ (b( t)) ds
a
J
t
J
t
J
t
36. Further bibliography for this Chapter can be found as items [68]-[151].
392 CHAPTER XII
REFERENCES
129. PERESTYUK, N.A., S.G. HRISTOVA and D.D. BAINOV, A linear in-
tegral inequality for piecewi.'le continuoU8 function.'l (Bulgarian), Godisnik
Viss. Ucebn Zaved. Prilozna Mat. 16 (1980), 221-226.
130. RAGHAVENDRA, A remark on an integral inequality, Bull. Austral.
Math. Soc.23 (1981), 195-197.
131. ROGERS, T.D., An integral inequality, J. Math. Anal. Appl. 82 (1981),
470-472.
132. SATO, T., Sur l'equation inUgrale non lineaire de Volterra. Compositio
Math. 11 (1953), 271-289.
133. SOBEIH, M. and G. HAMAD, On a generalization to Gronwall'8 integral
inequality. Afrika Mat. 6 (1984), 5-12.
134. STORCHI, E., Su una generalizzazione dela lemma di Peano-Gronwall.
Matematiche 16 (1961), 8-26.
135. THANDAPANI, E., A note on Bellman~Bihari type integral inequalities.
J. Math. Phys. Sci. 21 (1987), 1-3.
136. , On .'lome new integrodifferential inequalitie.'l theory and ap-
plications. Tamkang J. Math. 11 (1981), 169-184.
137. VERLAN', A.F. and V.S. GODLEVSKll, Some integrale.'ltimates of the
type of Gronwall-Bellman and Bihari inequalitie.'l (Russian), Tocn. i Nadezn.
Kibernnet. Sistem. No 2 (1974), 3-8, 140.
138. VOLTERRA, V., "Theory of Functionals and of Integral and Integrodif-
ferential Equations," New York, 1959.
139. WANG, C.L., A short proof of a Greene theorem, Proc. Amer. Math.
Soc. 69 (1978), 357-358.
140. WANG, C.L. and L.C. ZHONG, Some remark.'l on integral inequalities of
Bellman-Gronwall type (Chinese), Sichuan. Daxue Xuebao 1984, No 1,
18-27.
141. WATTAMWAR, M.J., On some integral inequalities of Gronwall-Bellman
type, J. Maulaua Azad. College. Tech. 15 (1982), 13-20.
142. YANG, C.C., On some nonlinear generalizations of Gronwall-Bellman's
inequality, Bull. Inst. Math. Acad. Sinica 7 (1979), 15-19.
143. YANG, E.H., A generalization of Bihari'.'l inequality and its applications
to nonlinear Volterra integral equations (Chinese), Chinese Ann. Math. 3
(1982), 209-216.
144. YANG, G.S., A note on an integro-differential inequality, Tamkang J.
Math. 12 (1981), 257-264.
145. , A note on .'lome integro-differential inequalities. Soochow
J. Math. 9 (1983), 231-236.
400 CHAPTER XII
JJv(r,s)u(r,s)drds
x
where a(x), bey) > 0, a'ex), b'(y) 2:: 0, u(x, y), vex, y) 2:: 0, then
2) If
J J
x Y
u(x,y):::; c+a u(s,y)ds+b u(x,s)ds
o 0
J J
x Y
u(x,y):::; a(x) + bey) + a u(s,y)ds + b u(x,s)ds,
o 0
PROOF:: We shall only prove the first conclusion. Denote the expression on the
right hand side of (1.1) by b( x, y). Then we have
(1.2)
bx(x,y) a'(x)
b(x,y) ~ b(x,y) +
J x
v(x,s)ds
o
+J
Y
a'(x)
~ b(x,O) v(x,s)ds.
o
lnb(x,y)-lnb(O,y)~
a'(x)
J (a(x)+a(O))dx+
x JX JY v(r,s)drds.
o 0 0
JJ
x Y
(2.1 ) u(x, y) ~ a(x, y) + b(x, y) c(z, t)u(z, t)dtdz,
o 0
JJ
x Y
a(x,y) -t a(x,y) + b(x,y) d(x,y,z,t)dtdz.
o 0
GRONWALL INEQUALITIES IN HIGHER DIMENSIONS 403
REMARK 2: Note that in the case when a(x,y) == C and b(x,y) 1, (2.1')
becomes u(x, y) :::; Cw(x, y) where
w(x,y) = 1+ JJ JJ
o
x
0
y
exp
(
z
x
t
y
c(r,s)drds
)
c(z,t)dtdz.
This inequality is better than Wendroff's obtained from (1.1') by taking a(x, y) ==
C and b( x, y) == 0, since
JJ
00 00
Then if c > 0, and a(x, y) is nonnegative, continuous and J J a(z, t)dz dt < (Xl,
x y
we have
u(x,y):::; v(x,y)
where
REMARK 3: Note that this is better than the corresponding Wendroff inequality
since
and
zt Oy Ot
JJ
x y
J J
x y
JJ ff
x y x y
o 0 0 0
J J
x y
o 0
then
s f(x, y) + L
00
where
+~
n-
I
a(x)b(y)
cn,1I (n _ v + l)!(v _ I)!
JJ x y
fez, t)r/J(z)1jJ(t)
[! [!
11-1 0 0
and
REMARK 5: Previously in [2] Nurimov had proved the special case: J(x,y) =
cxayP, a(x) = ax"f, bey) = by6, ¢J(z) = z-r, '¢(t) = r s , with c,a,b ~ 0, a,{3 > 0,
"'1, 6 ~ 0, r - 1 < a, s - 1 < (3, r - 1 < "'1, s - 1 < 6. In this case one obtains the
result with
Gn(x,y)
v=O
X {[a - (r - 1)] ... [a + (n - v + Ih - (n - v)(r - 1)]
X [{3 - (s - 1)] ... [{3 + (v - 1)6 - v(s - I)]} -1.
u(x,y)::; a + b JJ
x
o 0
Y [
¢J(z,t)u(z,t) + JJ
0
z
0
t ]
J(z,t,s,r)u(s,r)dsdr dzdt,
4. The final result that we give for functions of two variables is from Bykov and
Salpagarov [5]:
THEOREM 1. Let the functions u(t, x), it (CT, 6), h( CT, 6, s), h( CT, 6, r), and h( CT, 8, s, r)
406 CHAPTER XIII
ff
x t
f
6
+ f2(O',8,s)</l[u(O',s))ds
a
f
u
+ h(O',8,r)</l[u(r,8))dr
b
ff
6 u
u(t,x) x t { 6
u {j
f f f
U }
Chapter XII (see for example Mamedov, Asirov and Atdaev [11, pp. 129):
GRONWALL INEQUALITIES IN HIGHER DIMENSIONS 407
on ([0, TJ) where uo(t), ai(t), i = 1,2 are nonnegative and continuous and p > D.
If p > 1 then
(i )d')
1r-I
v(t) '" u,(t) - "'+ exp a, (,
~ [W1(U)] _ ~ (C i- 0 constant).
du W2(U) - W2(U)'
If v( t) is a continuous nonnegative function such that
2 t
then
Uo g(t)
v(t) ::; uo(t) - g(t) go + go G- 1 { exp ( cgo!t al(s)ds ) [G(UO)
+ Cgo j
o
a2(s)exp (-cgo J] . . ]
0 0 0
a1(i)di) dS] }
J
t
(5.1')
u
where G(u) = J du/w(u) (0 < a::; u).
(r
PROOF: Denote the right hand side of (5.1) by z(t) - a for a a sufficiently small
positive number. Assume that Uo is differentiable in t 1 • Then
J... J
t2 tn
J... J
t2 In
and by integration
J
t
J
s
where G is defined as in the previous Tbeorem and tbe term in tbe [ ) is in tbe
domain of G for 0 :s; x ::; 8 ::; so.
where ai(xi) >0 and aHx) ~ 0, i = 1, ... , n. Then for Xo ::; x ::; x we have
)
Xo
+ J
XOl
tl
J
t
+ p(s)ds }dt
Xo
r
where G(r) = J dsJ (s + H(s)), 0 < ro ::; r, and x is taken such that the expres-
ro
sion in { } is in the domain of G- I .
J
t
vet) ::; uo(t) + get) a(s)v(s)ds.
o
Then
Onv(s' x)
(8.1) (_I)n , - b(s)v(s;x) =0 on Q,
OSI ••• oSn
v(s; s) = 1,
GRONWALL INEQUALITIES IN HIGHER DIMENSIONS 411
and let D+ be the corresponding subdomain of Q which contains all x such that
v 2: 0 for all s E D+. If D C D+ is a parallelepiped defined by Xo < t < x and
J
x
then
J
t
J
x
J JJ... J II
x x x x k
is given by Beesack [17]. This inequality is also established in Fink [19] without
the continuity hypotheses. He merely requires that u be bounded and measurable
on bounded sets and that b is integrable. We give the proof.
412 CHAPTER XIII
v(t;x) = 1 + J J J
C(t)
b(s)ds +
C(t)
b(s)
C(s)
b(u)v(u;x)duds
= 1 +J +J J b( s )ds b( s) b( u )du ds
C(t) C(t) C(s)
= 1+ J +J J
C( t)
+ ...
b(tl)dtl
C( t)
t(tt}
C(td
b(t2)dt2 dt l
Rn = J J J
C(t)
b(tt)
C(td
b(t2)
C(t2)
b(t3)'"
C(t n)
J b(u)v(u;x)dudtn ... dit.
Now
h = J J C( t)
b(tt)
C( ttl
b(t2)'" J
C( tk)
b(tkH)dtk+l'" dt l
Rn :::; (n~l)!
(
J b(s)ds
)
and so Rn -+ 0 as n -+ 00 and
G(t)
J b(s)ds
=&(t)
9. Similar results for various other integral inequalities are given in Yeh [20].
Here we shall give the following one:
THEOREM 1. Suppose tbat u(x), a(x), b(x), and c(x) are nonnegative continuous
functions defined on Q witb D and D+ defined as in Theorem 2 of 8. Let v( S; x)
be tbe solution of tbe characteristic initial value problem
anv(s· x)
(-lt aS1 ... 'as n -[b(s)+c(s)]v(s;x) =0 onQ, v(s;x)=lfors=x.
j [U(') + j «i)u(i)di1d"
H
tben
u(x) :;; a(x) + j b(,) [a(,) + 1 a(i) [b(i) + e(t)[ v(i; ,)dil d,.
J
t
then
J
t
PROOF: The parallelepiped ([0, T]) is a convex set and (10.1) and (10.2) imply
(10.1') on any face of dimension r ~ n - 1 that has one coordinate zero. Denote
by G the set of points in in ([0, T]) for which vet) f:.. u(t). This set is closed and
by the above remark does not contain any points of the faces mentioned above.
Let t denote a point which is the upper limit of values a such that the interval
o ~ t ~ (a, a 2 , ••• , an) does not intersect G. There exists a point r such that
vCr) f:.. u(r) and r ~ a. On the other hand, for 0 ~ t ~ (rl,a2, ... ,an) we have
the vet) < u(t) so as a consequence of (10.1) and (10.2) we have
J J
T T
J
t
J
t
then vet) ~ u(t) where u(t) is the maximal solution of (10.2) defined on ([0, T]).
The proof is similar to that of Theorem 2 from 23. of Chapter XII.
GRONWALL INEQUALITIES IN HIGHER DIMENSIONS 415
REMARK 1: With respect to condition (/-L) note that the following result holds
(see Beesack [21, pp. 88-89]):
Let G be an open set in Rn, and for x, y E G let G( x, y) denote the par-
allelepiped with diagonal joining x to y. Let the points Xo, Y E G such that
Go = G(XO,y) c G and suppose the functions a(x) and k(x,t,z) be continuous
on Go and GT X R respectively where GT = {(x,t): x EGo, t E G(xo,x)}.
Suppose further that k is nondecreasing in z for (x, t) E G T . If
(10.3) Ik(x, t, z)1 < h(t)g (Izl) for (x, t, z) E GT X R,
00
J
x
J
x
J
x
11. K. M. Das [23] in 1979 gave a short and elegant proof of an inequality of
Gronwall type for systems of two integral inequalities
J J
t t
+J J
t t
get) ~ K2 e-Il- S
h3 (s)J(s)ds + h4 (s)g(s)ds
o 0
416 CHAPTER XIII
where K 1 , K 2 , and J-l are nonnegative constants and I, g, and hi are nonnegative
continuous functions on [0,00).
T)lls result is due to D. E. Greene [24] and says that there are constants Cj
and Mi such that for t ~·O
(11.2)
J
t
x(t):::; c+ I(s,x(s))ds,
a
for some c in Rn and if r is the maximal solution on J 1 = [a, /31) with /31 < /3) of
J
t
r(t)=c+ I(s,r(s))ds,
a
12. Mamedov, Asirov and Atdaev [11, pp. 101-102] proved the same result by
replacing I by a more general function:
THEOREM 1. Let cf>(t, s, u) be continuous and nondecreasing in u on 0 S; s, t S; T,
IUil < Oi, and let w(t) be a continuous vector function on [0, T]. If v(t) is a
continuous vector function such that
J
t
then v(t) < u(t) on (0, T] where u(t) is the minimal solution of
J
t
for fixed t or t I .
Denote by UTe t) the solution of the system (12.2) defined on [0, r) C [0, T). If
r < ~ and ue(t) = uT(t) for t E [0, r) then we say that ue(t) is an extension of
uT(t) on [o,~), and uT(t) is part of ue(t). Finally, if uT(t) is not part of another
solution, we say that it cannot be extended.
The following theorem is also given in Mamedov, Asirov, and Atdaev [11, pp.
103]:
THEOREM 2. Let the vector function cfJ(t, s, u) satisfy the condition (K) fort, s E
[0, T] (T ~ 00) and lui ~ 0 ~ 00. Moreover assume that it is nondecreasing in u
and that wet) is continuous and Ilw(O)1I < o. Let u,,(t) be the minimal (maximal)
solution of the system (12.2) which cannot be extended. If vet) is a continuous
vector function on [0, d) C [0, T) with IIv(t)11 < 0, and satisfying
t
13. Let us return to inequality (11.1). B.B. Shinde and B.G. Pachpatte [27,
Th.1] dealt with a generalization of (11.1) where the independent variable is n-
dimensional. Their result is in a certain sense included in a general theorem
418 CHAPTER XIII
proved in 1976 by J. Chandra and P. W. Davis [28], which we give in our next
theorem. Moreover, this latter result is turn included in an abstract version of a
general linear Gronwall inequality formulated at least as early as 1969 by E. Hille
[29, pp. 18-20],and [30, pp. 364-376]. Of course, comparison theorems give the
best possible results for the considered inequalities. From the practical point of
view the problem of solving a system of inequalities has only been replace by
the possibly more difficult problem of solving a system of equations. This is not
of much use to someone who is seeking explicit upper bounds for the functions
satisfying the inequalities.
THEOREM 1. Let G(x) and H(x) be N X N matrices and a(x) and u(x) be N-
vectors that are continuous functions for Xo ~ x. Let all the components of H
and G be nonnegative. H
J
x
then
J
x
where
J
x
14. Using this best possible estimate for u, we can obtain various explicit exti-
mates (see Chandra and Davis [28], and especially Beesack [31, pp. 11-17]). Of
course, some explicit estimates can be obtained directly. Here we shall give some
results of P. R. Beesack [31, pp. 4-11]:
THEOREM 1. Suppose Ui, ii, gij, and h ij are all continuous functions with Ui,
gij, and hij all nonnegative for x ~ O. H G [gij] and H = =
[hij] and u satisfy
the inequality (13.1) for x ~ 0 then for such x and i = 1, ... , N we have
(14.1)
GRONWALL INEQUALITIES IN HIGHER DIMENSIONS 419
where
(14.2)
.2: Jhkj(Xl,t)h(Xl,l)dt,
N x
(14.4) A(Xl'X) =
J,k=lo
and
(14.5)
n.(x) ~ exp
N
(J P(8)d') I A(t)exp (- ! N
P(')d') di,
REMARK 2: Using the above notation, the system (11.1) can be rewritten in the
form
J J
X X
+J +J
X X
U ~ f{2 e-l' t h3 dt h4 U 2 dt ,
o 0
420 CHAPTER XIII
so that G(x) = I, hll = hI, h12 = el'thz, h21 = e-I' t ha, hzz = h4' so by Remark
1 we have
1 (!
pes) = max{h l + e-I' Bh3, el'Bh z + h4}, and
For a crude estimate note that A(s) ~ (Kl + Kz)P(s), so (14.6) becomes
(I
(14.7)
(J
(14.8)
where h = max(hl + h3' h z + h4). The crude bound (14.7) is thus better than
(14.8) for small x. However in case hz + h4 ~ hI + h3 then h ~ P and Das's
bounds are better for large x.
THEOREM 2. H Ui, fi, gij, and hij are defined as in Theorem 1. and U satisfies
the inequality (13.1) for x ~ 0 then
N N
(14.10) G(x) =L max gik(X), H(x) = max L hkj(x).
k=l 19~N 19~N ;=1
GRONWALL INEQUALITIES IN HIGHER DIMENSIONS 421
N
Ui(X) ::; Ji(X) + 'L gik(X)Rk(X),
k=l
where
'L Jhkj(t)Uj(t)dt,
N x
as well as
L Jhkj(t)u(t)dt = JHk(t)U(t)dt,
N x x
Rk(X) ::;
)=1 0 0
N
where Hk = L: hkj; that is we get
j=l
J
x
15. Some generalizations of Greene's Theorem are also given in G. S. Yang [32]
and Pachpatte [33].
16. The method used in either Theorems of 14. can also be applied to nonlinear
systems of inequalities. Beesack illustrated this by using the method of Theorem
2 of 14. for the system
J
x
?= Jkij(X, t)uj{t)dt,
N x
or in matrix form
J
x
has a unique solution u E L2( J). This solution is given be the Neumann series
=L
00
where
J
x
J
x
00
J
x
L K(r)(x, t).
00
(17.7) M(x, t) =
r=l
THEOREM 2. Let J = [a, b] be compact in Rn, and suppose that f and K are
bounded and measurable (or continuous) on J and T respectively (as in Theorem
1). Then the system (17.1) or (17.2) has a unique solution which is bounded and
measurable (continuous) on J. The solution is given by the Neumann series
(17.3) which is uniformly absolutely convergent on J, and is also given be the
representation (17.6) where is bounded and measurable (continuous) on T. The
Neumann series (17.7) for is uniformly convergent on T.
Note also that
JJ... J
X tl t r -2
JJ... J
x x x
and
z
K(r)(x,t) =/ K(i)(x,s)KU)(s,t)ds, r ~ 2, i + j = r,
t
and 1::; j ::; r - 1.
z z
then
z
Since j - f is a function of the same class as f and u, the basic Theorems 1 and
2 yield
J
x
REFERENCES
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tions of several variables (Russian), Bull. Math. Soc. Sci. Math. R.
S. Roumaine (N.S.) 2674 (1982), 15-20.
47. BEESACK, P.R., On some Gronwall-type integral inequalities in n inde-
pendent variables, J. Math. Anal. Appl. 100 (1984), 393-408.
48. BOBROWSKI, D., J. POPENDA and J. WERBOWSKI, On the systems
integral inequalities with delay of Gronwall-Bellman type, Fasc. Math. 10
(1978), 97-104.
49. BONDGE, B.K. and B.G. PACHPATTE, On nonlinear integral inequali-
ties of the Wendroff type, J. Math. Anal. Appl. 70 (1979), 161-169.
50. , On some fundamental integral inequalities in two indepen-
dent variables, J. Math. Anal. Appl. 72 (1979), 533-544.
51. , On Wendroff type integral inequalities in n independent
variables, Chinese J. Math. 7 (1979), 37-46.
52. , On generalized Wendroff-type inequalities and their appli-
cations, Nonlinear Anal. 4 (1980), 491-495.
53. , On some fundamental integral inequalities in n independent
variables, Bull. Inst. Math. Acad. Sinica 8 (1980), 553-560.
54. , On some partial integral inequalities in two independent
variables, F\mkcion. Ekvac. 23 (1980), 327-334.
55. BURLACENKO, V.P. and N.J. SIDENKO, A generalization of the Wen-
droff inequality (Russian), Question of the theory of the approximation of
functions (Russian), pp. 14-21, 193. Akad. Nauk USSR, Inst. Mat. Kiev,
1980.
56. CHEN, L.S. and C.C. YEH, Some integro-differential inequalities in n
independent variables, Proc. Roy. Soc. Edinburg. Sect. A. 89 (1981),
347-353.
57. CHOSHAL, S.K. and M.A. MASOOD, Generalized Gronwall's inequal-
ity and its applications to a class of non-selfadjoint linear and non-linear
hyperbolic partial differential equations, to establish their uniqueness con-
tinuous dependence and comparison theorems, Indian J. Pure Appl. Math.
5, (1974), No 3.
58. CORLAN, J. and C.L. WANG, Gronwall-Bellman type inequalities for
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59. , Higher dimensional Gronwall-Bellman type inequalities, Ap-
plicable Analysis 18 (1984), 1-12.
60. CONSTANTIN, A. A Gronwall-like inequality and its applications, Rend.
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61. CORDUNEANU, A., A note on the Gronwall inequality in two indepen-
dent variables, J. Integral. Equations 4 (1982), 271-276.
GRONWALL INEQUALITIES IN HIGHER DIMENSIONS 429
113. SNOW, D., A two independent variable Gronwall-type inequality. In: "In-
equalities," Vol. III, 333-340, New York 1972.
114. THANDAPANI, E. and R.P. AGARWAL, On some new inequalities in n
independent variables, J. Math. Anal. Appl. 86 (1982), 542-561.
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Polon. Math. 3 (1957), 210-212.
116. YANG, E.H., On some new integral inequalities in n independent vari-
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117. , On some new n-independent-variable discrete inequalities
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118. YANG, Y.S., On Bellman-Gronwall's inequality, J. Math. (Wuhan) 2
(1982), 269-273.
119. YEH, C.C., Bellman-Bihari integral inequalities in several independent
variables, J. Math. Anal. Appl. 87 (1982), 311-321; Errata. 90 (1982),
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120. YEH, C.C. and M.N. SHIH, The Gronwall-Bellman inequality in several
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121. YOUNG, E.C., Functional integral inequalities in n variables, Chinese J.
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122. ZAHARIEV, A.1. and D.D. BAINOV, A generalization of the Bellman-
Gronwall inequality (Bulgarian), Godisnik Viss. Ucebn. Zaved. Prilozna
Mat. 12 (1976), 207-210.
123. , A note on Bellman-Gronwall's inequality, J. Math. Anal.
Appl. 80 (1981), 147-149.
124. PECH, Pavel, Inequality between sides and diagonals of a space n-gon and
its integral analog, Casopis Pest. Mat. 115 (1990), 343-350.
CHAPTER XIV
1. There are Gronwall type inequalities in which the unknown function is not a
function on R n , rather in some other space. This Chapter is devoted to these
kinds; on discrete, functional and then abstract spaces.
with a(x) ::; rj>(x), 'IjJ(x) ::; b(x) for x = Xo + jh, j ::; N (N is a natural number).
The following Theorem is valid, see Rabczuk [1, pp. 200-202]:
THEOREM 1. !ffor x = Xo + jh, j ::; N -1,
433
434 CHAPTER XIV
so on the basis of (2.1) we have I':l.¢(xo) < I':l.tP(xo) and thus ¢(xo +h) < tP(xo + h)
and the assertion is true for n = O. Suppose by way of proof that the result is valid
for an index k, i.e. that ¢(Xk) < tP(Xk) for k < N. Combining the inequalities
f (Xk' ¢(Xk» < 9 (Xk,tP(Xk» and 1':l.¢(Xk) < I':l.tP(Xk) we get ¢(Xk+h) < tP(Xk+h)
which is the assertion for k + 1. The Theorem 1 is proven.
Further let the functions u( x) and v( x) be solutions of the difference equations
with u(xo) = Yo = v(xo) and a(x) 5, u(x), v(x) 5, b(x) for the same x. (¢(x) will
be defined as above.)
COROLLARY 1. Let the functions fI(x, y) and h(x, y) be defined for x = xo+jh,
j 5, N - 1, and a(x) 5, y 5, b(x) and suppose they are increasing in y. If
where y = y(x) is the solution of the difference equation I':l.y = hf(x, y), y(xo) =
Yo. The assertion is also valid if both (2.6) and (2.7) reverse the inequalities.
or
(B) f(t,u) is decreasing in u and
~v(t) - f (t, v(t + 1)) ~ ~w(t) - f (t, w(t + 1)) for t E R+,
then from v(t) ~ w(t) on [0,1) we conclude the same inequality on R+.
t-l
(4.1) x(t) = g(t) + ,Lf(t,s,x(s)), t=I,2, ... ,
8=0
with the initial condition x(o) = g(O), where x, g, and f can be scalar or vector
and f(t, s, x) is defined for t, s = 0,1,2, ... and for all x.
The following theorem was proved in Pachpatte and Singare [3]:
THEOREM 1. Let f(t,s,x) be defined as above with f non decreasing in x. If
u( t) satisfies the inequality
t-l
(4.2) u(t)~g(t)+ ,Lf(t,s,u(s)), u(O)~g(O),
8=0
then u(t) ~ x(t), t = 0,1, ... , where x(t) is the solution of the difference equation
(4.1).
PROOF: First u(O) ~ g(O) = x(O). Suppose that the inequality u(t) ~ x(t) is
not valid for for some to, and suppose it is the first one for which the inequality
fails. Then
t-l
x(to) -u(t o) ~,L [f(to,s,x(s)) - f(to,x,u(s))] ~ 0,
8=0
i-I
(5.1) Xi S ai + bi L kjXj, i = 0,1, ... , N,
j=O
jEep
Set
X(t) = { Xi, i St < i + 1, 0 SiS N - 1,
XN, t=N,
and analogously for a, b, and k, we get (5.1) from
J
t
if t = i. Similarly from
x(t) <; a(t) + b(t) ia(,)k(,) exp (i b(r)k(r)dr) <is, 0 <; t <; N,
6. None of the above results is best possible. The best possible results for the
inequality (5.1) was given by G. S. Jones [6]:
THEOREM 1. Let X, a, b, and k be real-valued functions on J = [a,,8], with b 2:: 0
and k 2:: 0, and let {to, t l , ... , tN} be a strictly increasing sequence in (a, ,8). If
PROOF: Denote the right hand side of (6.1') by yet). Then the function y is a
solution of the equation
This is clearly the case for a :s; t :s; to, and for all t such that b( t) O. If
to :s; t :s; t 1 , then
since y(to) = aCto). One can prove that if yet) satifies (6.2) for a :s; t :s; tj, then
yet) also satisfies (6.2) for ti :s; t:s; ti+l if i < N, or for ti < t :s; (3 if i = N. The
desired conclusion now follows by mathematical induction.
A similar induction shows that if w = x - y, then wet) :s; 0 for a :s; t :s; ti,
o :s; i :s; N + 1 (tN+l = (3). This proves the Theorem 1 and shows that (6.1') is
best possible.
REMARK 1: To reduce (6.1) to (5.1) it suffices to set
with a corresponding definition of a, b, and k. Then (5.1) implies (6.1) and hence
also (6.1'). Setting i = ti we obtain the best possible result
i-I
(6.3) Xi :s; ai + bi L ajk II j [1 + bmkm) , O:S; i :s; N.
j=o j<m<i
7. T. E. Hull and W. A. J. Luxemburg [7) (see also Filatov and Sarova [8, pp.
42-45)) obtained the following discrete version of Bihari's inequality but with all
k j = k > O.
438 CHAPTER XIV
i-1
(7.1) Xi':::; a + 2:= kjg(Xj), i 2:: 0,
j=O
then
u i-1
where G(u) = J dyjg(y), u 2:: Uo, and N = sup{i: G(a) + 2: k j E G(I)}.
Uo j=O
8. The following two theorems give discrete analogues of Theorems 1 of 19. and
Theorem 1 of 9. in Chapter XII (see Willett and Wong [9]):
THEOREM 1. Let {x n }, {an}, {b n }, and {k n } be nonnegative sequences and
suppose that 1 .:::; p < 00. If
then
where
i-1
(8.2) ej = II (1 + kjlJj) -1 , i = 0,1, ....
j=O
(8.3)
GRONWALL INEQUALITIES ON OTHER SPACES 439
i-1
(8.4) Ci = ei L kjx~, i ~ 0,
j=O
(8.5)
n-l
where A =E kibf cd (1 + kill;).
i=O
Since the function f(x) = (c+x)llp- x l /p is decreasing for x> 0, (8.5) remains
valid when A is increased. By (8.4)
by (8.2)
by (8.3),
so
n-l
then
(8.6') n '? 0,
where
;-1
(8.7) E; = II (1 + k )-1,
j i '? 0.
j=O
(8.8)
Denote the expression on the right hand side of (8.6) by Cn and consider the
sequence {Cn}. Note that Co = a and by (8.6) Xi :S C j for i '? 0. Since
C i +1 - C j = kjxj + Kjxf and p '? 0, it follows that
(8.10)
(8.11)
where X lies between EiCi and Ei+1Ci+1. Using the fact that {Ci} is nonde-
creasing while {Ei} is nonincreasing, it follows that
(8.12)
where ~ or 2:: holds according to whether q > 0 or < o. Now sum (8.12) for
i = 0,1, ... ,n - 1 using Co = a to obtain
n-l
9. The following two theorems were proved in Singare and Pachpatte [11]:
THEOREM 1. Let x(n), ken), pen), fen), g(n), and hen) be non-negative
[unctions on N = {1, 2, ... } [or which
+ s~o g(s) (~o k(t) (f(t) + h(t)) m[1 (1 + p(m) (f(m) + h(m)) + g(m)))
for n E N.
THEOREM 2. Let x(n), f(n), g(n), and h(n) be nonnegative functions defined
on N such tbat
n-l
x(n)::; Xo II (1 + f(s) - g(s))
s=no
n-l n-l n-l
+ L g(s) L (1 + f(m) - g(m)) x II (1 + f(m) - g(m))
s=no m=s+1 m=no
X
[
8-1
(1 + f(t) - g(t))P-l
]1/(l-P)
10. Of course there exist discrete analogues for other known inequalities. So
in Singare and Pachpatte [12] the following discrete version of a Wendroff type
inequality is proved:
THEOREM 1. Let u(x, y) and c(x, y) be nonnegative functions defined for x, y ~ 0
and satisfy
x-ly-l
u(x, y) ::; a(x) + b(y) + L L c(s, t)u(s, t), x, Y ~ 0,
s=o t=o
GRONWALL INEQUALITIES ON OTHER SPACES 443
where a(x) and bey) are positive functions defined for x, y > 0 with 6.a(x), 6.b(y) 2::
O. Then for x, y 2:: 0
11. The previous theorem gives discrete inequalites for two variables. There are
analogous results for more than two variables. Here we shall give a result from
Yeh [13, II].
Let N be a set of nonnegative integers, and n a positive integer. If 1 =
(1,1, ... , 1) and u is a function defined on N to R+ define: x = (Xl, x) where x =
x-l xl-1 x" -1 x-l
(X2, ... ,x n); and E u(y) = E··· E U(Y1,···,Yn); E u(y) = 0 whenever
y=O Yl=O y,,=O y=o
any Xi = O.
THEOREM 1. Let the functions u(x) and k(x) be defined on Nn to R+ and
f(x; s) be defined on N2n to R+ for s ~ x. H
x-I
u(x) ~ k(x) + L f(x, s)u(s) for X E N n,
8=0
then
where
then
or
x-I
(ILl") u(x) :S k(x) +L f(x; s)K(s)
E
8=0
12. There are also results which are discrete versions of integra-differential in-
equalities. For example, Pachpatte [14] has the following theorem:
THEOREM 1. Let x(n), ~x(n), f(n) and g(n) be nonnegative functions on N,
for which
fo
Then
14. Finally, note that there are results which incorporate both discrete and
continuous results. For example, such theorems are given in the following result
of G. S. Jones [6]:
THEOREM 1. Let the functions x, a, b, and k on J to R be either, continuous
or of bounded variation and a, b, and k be nonnegative, (J = [a,p]). Let f-t be
non decreasing and left continuous on J 'and suppose that x, a, b, and k are all
continuous from the right at all points of discontinuity of f-t. If
r
x(t) ::; aCt) + bet) J k(s)x(s)df-t(s), t E J,
0<
1 (1
then for t E J
Moreover, if (To =a and (T1, (T2, ... are the points of discontinuity of f-t in (a,p]
and
16. Similar results for Stieltjes integrals are given in Das and Sharma [27], and
Mingarelli [28].
for every fixed s E [0, T], then lim ¢(t, u~n+l») = ¢(t, Ut) for each t E [0, T];
n
b) if u(s) E Cr, then ¢(t,ut) = u(t) E Cr.
We say that the Volterra operator ¢(t, Ut) is nondecreasing in the second ar-
gument if from u(s) ::; v(s), 0::; s ::; t ::; T, it follows that ¢(t,Ut) ::; ¢(t,Vt) on
[O,T].
THEOREM 1. Let the Volterra operator ¢(t,ut) be non decreasing in the second
argument and suppose that u(t) is the lower solution of the equation
(17.2) vet) < ¢(t, Vt) on (0, T], v(O) < u(O),
then
PROOF: The inequality (17.2') is true for t = 0, and hence on some nontrivial
interval (0, t*). If we select t* to be the largest number for which (17.2') holds
GRONWALL INEQUALITIES ON OTHER SPACES 447
and t* < T then we must have vet) < u(t) on [0, t*) and v(t*) = u(t*). But we
then have
v(t*) < cjJ(t*,Vt.):-:; cjJ(t*,Ut.) = u(t*),
providing a contradiction, and proving the Theorem.
REMARK 1: Theorem 1 is also valid if in (17.2) and (17.2') all inequalities are
reversed and u is the upper solution of (17.1).
We say that the Volterra operator cjJ( t, Ut) satisfies the condition (p) if the
equation
Wet) = cjJ(t, W,) + 8, W(O) = u(O) + 8,
has a solution Wet) E CT (or ST) for every fixed 8 in [O,p].
THEOREM 2. Let the Volterra operator cjJ(t, Ut) be nondecreasing in the second
argument and satisfy the condition (p) for sufficiently small p. If the function
v( t) E CT (or ST) satisfies
(17.3) v(t):-:; cjJ(t,Vt) on (O,T], v(O):-:; u(O),
and ifu(t) is the upper solution of(17.1), then
(17.3') vet) :-:; u(t) on [0, T].
PROOF: For every fixed n = 1,2, ... , let u(n)(t) denote the solution of the equa-
tion
(17.4)
defined on [0, T], where c: >
obtain
° is a sufficiently small number. From (17.4) we
(17.5) u(n)(t) > cjJ(t, u~n») on (0, T], u(n)(O) = u(O) + c:/n,
and
u(n+l)(t) < cjJ (t, u~n+l») + c:/n on (0, T],
(17.6)
u(n+l)(o) = u(O) + c:/(n + 1).
From (17.1), (17.5) and (17.6) and Theorem 1 we get
u(t) < u(n+l)(t) < u(n)(t) :-:; u(1)(t) on (0, T].
From inequality (17.3) we find that
(17.7) vet) < u(n)(t) on [O,T].
Now lim u(n)(t) = fi(t) exists on [0, T] and u(t) :-:; fi(t) and by taking limits in
n
(17.7) we get the result of the Theorem.
448 CHAPTER XIV
then
18. We need the following notations to discuss the generalizations of the previous
results:
1) 4>(t,u,Vt) is a nonlinear Volterra operator for fixed u in [-r,rj, and for
fixed t E [0, Tj and Vt EST, 4>(t, u, Vt) is a continuous function on [-r, rj
and satisfies the conditions:
GRONWALL INEQUALITIES ON OTHER SPACES 449
then
THEOREM 2. Let the operator ¢>(t, Ut, vr) be non decreasing in the second and
third arguments. Ifv(t) E Sr and
then
(18.2)
450 CHAPTER XIV
19. The above results in 17. and 18. are obtained in Mamedov and Asirov [29]
as well as in Mamedov, Asirov, and Atdaev [30]. Similar results are obtained in
the case when [0, T] is replaced by [0,00).
As examples of the previous theorems we might take
~(t, u" UT) = ~ (t, i f (t, " u(,» d" I 9(t, " *» d8) .
20. Mamedov and Novruzov [31] (see also Mamedov, Asirov, and Atdaev [30])
considered differential inequalities with Volterra operators <p(t, u, Vt).
THEOREM 1. Let the operator <p(t, u, Vt) be continuous in all variables and non-
decreasing in the third argument. Let v( t) E CT (or v( t) E ST) and satisfy
Then
If (20.1') does not hold on (0, Tj, let t* be the first t for which u(t) = vet). Then
using (15) and (18) we get
But then vet) > u(t) to the left of t*. This contradiction proves that (20.1') holds
on (0, Tj.
REMARK 1: The theorem is valid if all the inequalities are reversed and u(t) is
the upper solution of (20.3).
21. We say that the operator 4>( t, u, Vt) satisfies the conditon (J.L) if the problem
then
Again using the comparison theorem we get u(n+l)(t) ~ u(n)(t) on [0, T] and the
sequence in non decreasing.
Next we prove that
v'(t) ::; r/> (t, vet), Vt) ::; r/> (t, vet), b h)' v(O)::; Uo·
From the comparison theorem we get vet) ::; u(l)(t) on [0, T]. Again by way
of induction we assume that (21.4) holds for a certain n. Then from v'(t) ~
r/> (t, vet), u~n»), v(O) ::; uo, and (21.3) and the comparison theorem we get vet) ::;
u(n+1)(t) on [0, T], proving (21.4) for all n.
Since the sequence u(n)(t) is monotonic and bounded below by vet), the lim u(n)( t)
n
exists on [0, T], call it u(t). From the integral equation
J (s,u(n>(s),u~n-l»)
t
°
we get
J
t
u(t) = Uo + r/>(s,u(s),us)ds,
°
so that fi(t) is a solution of (20.3). In fact, fi(t) is the upper solution of (20.3) so
that it is u(t). Clearly (21.1') follows from (21.4) and the theorem is proven.
GRONWALL INEQUALITIES ON OTHER SPACES 453
THEOREM 3. Let the operator ¢(t, u, vd be continuous in all variables and non-
decreasing in the third argument. Let the function v( t) E CT (or ST) and satisfy
D_v(t) < ¢(t,v(t),Vt) on (O,T], v(O) < uo
and in the case v(O) = uo, then D+v(O) < ¢(O,uo,{u}o). Then vet) < u(t) on
(0, T] where u(t) is the lower solution of the problem (20.3) defined on [0, T].
THEOREM 4. Let the operator ¢(t, u, vd be continuous in all variables, nonde-
creasing in the third argument, and satisfy the condition (p) for some smallp.
Let the function v( t) E CT (or ST) satisfy the inequality
or
or
Then
vet) ::; u(t) on [0, T],
where u(t) is the upper solution of (20.3) on [0, T].
(22.2)
Now Tnm x :S x by hypotheses (22.1) and limTnmx = Xo. This implies that
n
Tnm x :S xo, and by (22.2) this can be extended to Tn x :S x. Thus x < Tx :S Xo.
When :S is a partial order relation in a linear space X we always will assume
that :S is compatible with the algebraic operations in X. That is, we assume
that
x :S y implies that x + z :S y +z
(22.3)
and ax :S ay if a > O.
Note that this also means that
x ~0, y ~ 0 imply that x + y ~ 0, and
(22.4)
x :S y is equivalent to y - x ~ 0 or is equivalent to - y :S -x.
If X is a partially ordered Banach space with partial order:S, then X+ = {x E
X I x ~ O} is called the positive cone of X. An operator S: X -+ X is called
positive if S( X+) eX. If S is linear it is clear that S is positive if and only if
it is order preserving.
For our next theorem we require the following lemma (see Hille [351,[361) called
Volterra's fixed point theorem:
LEMMA 1. Let S: X -+ X be a bounded linear operator on a Banach space X
such that
00
(22.5)
For each z EX, the operator Tz: X -+ X defined by Tzx = z + Sx has a unique
fixed point x z E X given by
(22.6)
GRONWALL INEQUALITIES ON OTHER SPACES 455
n n
II L skzll ~ Ilzll L IIsk ll,
k=m k=m
and the completeness of X that the series in (22.6) converges. It is easy to see
that the sum defines a fixed point of T. On the other hand, if x E X is a fixed
point of Tz , then
n-l
X = Z + S(z + Sx) = ... = L Sk z + snx.
k=O
X z - Y = lim(Yn - y) E X+.
n
f
t
Hence if M = IIbllllkll(,8 - a), then IIsnxll S IIxllMnln! for all x E G(J) whence
IISnll < M nIn!, n ;::: 1. Thus (22.5) is satisfied so it follows from Theorem 2 that
if x, a E G (J) and
f
t
then x(t) S xa(t), for t E J, where y = Xa is the unique solution of the equation
f
t
t
Setting U( t) = J ky ds this equation is equivalent to the initial value problem
a
and with the partial order relation x ~ Y if and only if Xij(t) ~ Yij(t) for all i,j
and t E J. The positive cone X+ = MnG+(J) consists of all matrices all of
whose entries are nonnegative functions and thus is clearly closed. For k E X+
I I
define S: X ---t X by
so that
and
J
t
THEOREM 1. (Losonczi [39, Th. 1]) Let X be a Banach space with a partial
order :::; such that the positive cone X+ is closed. Let A, B: X --t X be two
operators such that
(i) x, y E X, and x:::; y imply that Ax :::; By; and
(ii) the equations x = g + Ax, y = h + By have unique solutions Xg , Yh for
arbitrary g, hEX, and these solutions can be obtained as the limits of the
sequence of successive approximations beginning with Xl = g and YI = h.
Then
25. More generally, according to a theorem of D. Boyd and J.S.W. Wong [40],
condition (ii) is satisfied (say for A) if there is a continuous function won R such
that w(r) < r for r > 0 and
For other conditions which assure condition (ii) see Geraghty [41] and Kannan
[42].
If A is an order preserving operator and satisfies (25.1) then
(25.2) u :::; q + Au implies that u :::; Xq (x q = q + Ax q ).
26. The condition that w(r) < r for every r > 0 in (25.1) is weakened in Chandra
and Fleishman [37, Th. 3]:
THEOREM 1. Let the partially ordered Banach space X have a closed positive
cone X+, and let the operator A: X --t X be an order preserving compact
(completely continuous) operator which satisfies conditon (25.1) where w is a
nonnegative non decreasing function on R and there is an r* such that
(26.1) w(r) + IIqll + IIA611 < r ifr > r*.
Suppose that
(26.2) u:::; q+Au.
Then the equation v = q + Av has a solution given by v = lim v n , where
n
(26.3) VI = u, Vn+l = q + Av n , n ~ 1.
Moreover, u :::; v. In particular if v = q + Av has a maximal solution l/J, then
u :::; l/J.
27. P.R. Beesack [5, p. 115] noted that the conditions involving w can be
replaced by the following:
There exist numbers r* and R* such that 0 < r* < R* and a nonnegative
nondecreasing function w on [0, R*] such that
(27.1) IIAxl1 :::; w (lIxl!) for IIxll :::; R*. (See (25.1))
(27.2) w(r)+llqll:::;r ifr*:::;r:::;R*. (See (26.1))
Then the conclusion remains valid for u E X with lIuli :::; R*.
REMARK 1: In Theorem 1 of 26. we understand that it was implicitly assumed
that the equation v = q + Av has a maximal solution.
28. Finally we shall give one result from Chandra and Fleishman [37, ThA].
Let X = SnG(J) be the class of all real n x n symmetric matrices x = (xii)
with Xii E G(J), J a compact interval. X is a Banach space with the sup
norm as before and with the positive cone S+ given by the nonnegative semi-
definite symmetric matrices. This cone is closed since the quadratic form is a
continuous function. Let Sn denote the constant matrices in SnG( J) and note
n
that IIxll = Ixl = sup E Ixl if x E Sn.
i i=l
460 CHAPTER XIV
If x E SnC( J) and
J
t
tben x(t) ::; yet), t E J, wbere yet) is tbe unique solution of tbe corresponding
equation on J.
29. The natural conception for integral inequalities of interval functions is theory
of interval analysis, initiated by R. E. Moore in 1979. Formally, we have substi-
tution of sign ::; by ~,i.e. connection between intervals and classical inequalities
are given by:
[a, b) ~ [e, d) if and only if e ::; a ::; b ::; d.
A short exposition of theory of integral inequalities by using interval functions
is given in the book [43) by A.A. Martynyuk, V. Lakshmikanthan and S. Leela.
This is a new theory in mathematical literature, and it started by the papers of
R.E. Moore [44)-[46), O. Caprani, K. Madsen and L.L. Ran [47), and G. Alefeld
and J. Herzberger [48).
REFERENCES
L Ostrowski's inequality. A. Ostrowski [1] has proved the following result (see
also 11., 17., and 52. from Chapter I):
THEOREM 1. Let f be a differentiable function on (a, b) and let, on (a, b),
1J'(x)1 ::; M. Then, for every x E (a, b),
(1.1 ) f(x) - - -
b-a
1 Jb f(x)dx::;
(1- -
4
(x _
(b-a)2
a+b)2) (b -
2 a)M.
a
{(x1, ... ,x m )l ai ::; Xi::; bi (i = 1, ... ,m)}, and let Mi (Mi > 0;
i = 1, ... , m) in D. Furthermore, let function x I---t p( x) be defined integrable
and let p(x) > 0 for every xED. Then for every xED,
m
I p(y)f(y) dy 2: Mi I p(y)IXi - Yildy
< i=l D
(1.2) f(x) - D I p(y)dy I p(y)dy
D D
468
INEQUALITIES INVOLVING FUNCTIONS WITH BOUNDED DERIVATIVES 469
1 (
;;: f(x)
n-l) -
+ t;Fk 1
b-a
Jb f(y)dy
(1.3) a
Fk = Fk(fjnjx,a,b)
(1.4) n - k f(k-l)(a)(x - a)k - f(k-l)(b)(x - b)k
k! b- a
PROOF: Integrating by use of Taylor's formula
(1.5)
J
b
= -\ f(n)(e)(x - yt dy.
n.
a
b b
Putting 1k = -b J f(k)(y)(x - y)k dy (k 2:: 1) and 10 = J f(y)dy, by means
a a
of partial integration on 1k, we have
I.e.
(n - k)(1k - 1k-l) = -(b - a)Fk (1::::; k ::::; n - 1),
where Fk is defined by (1.4).
Since
n-l n-l
I.e.
n-I n-l
J
n-I b
! (f(X) + (x -
2
a)f(a) + (b - X)(b») __1_
b-a b-a
J b
f(x)dy
a
<M(b-a)2(-.!.. (X_~)2)
- 4 12 + (b - a)2
In [5] is also given a generalization in which f(n-l) satisfy the Lipschitz's condi-
tion of order a (0 < a ~ 1). Generalizations for positive linear functionals are
given in [6]. Applications of previous results to quadrature formulas are given in
the same papers.
2. A. M. Fink [7] has generalized the above to prove the following result.
INEQUALITIES INVOLVING FUNCTIONS WITH BOUNDED DERIVATIVES 471
1 ( 1 J6
(2.1) ;; f(x) + n-l
(; Fk(X) ) - b _ a a f(y)dy ~ K(n,p, x)lIJ<n) lip
where
(2.2)
n+ 1 ~ n+ 1 ] 1/ p'
1 [(x - a) PI" + (b - x) PI" I I 1/ p'
K(n,p,x) = n! b_ a B«n -l)p + 1,p + 1)
if 1 < p ~ 00 and
J
6
1 M(b - a)2 1 2
(3.1) f(x)dx - 2(b - a) (f(a) + f(b)) ~ 4 - 4M (f(b) - f(a)) .
a
(3.2) iA(f') _ ~ (f(a) + f(b»i < M(b - a) . (A + q)(l - q2) + 2(A - l)q
,p 2 - 2 2A(1+q)-(A-1)(1+q2)'
where A and q are defined by
6
J p(x)f(x )dx
A(f;p) = _a- 6 - - -
and If(b) - f(a)1
q= M(b-a) .
J p(x)dx
a
The following result from [10] is a special case of a more general result in which
fen-I) satisfy the Lipschitz's condition of order 0:.
THEOREM 3. Let function f: [a, b] --+ R have a continuous derivative of order
n -1 and bounded derivation of the order n, i.e. If(n)(x)l:::; M for x E (a,b). If
f(k)(a) = f(k)(b) = 0 (k = 1, ... ,n -1), then the inequality
(3.3)
1
b-a
J b
1
f(x)dx- 2 (f(a)+f(b»:::;
M(b-a)n{
(n+1)!
q }
(n-2(1+n(2(-1» ,
a
-1b
-a
J b
f(x)dx - ~ (f(a) + f(b» <
2 -
M(b - a)2
24
1 (f(b) - f(a»2
-2' M(b-a)2
a
4. The inequalities (3.1) and (3.3) were considered in [7]. Fink proves the fol-
lowing result.
INEQUALITIES INVOLVING FUNCTIONS WITH BOUNDED DERIVATIVES 473
Rn,p
() = . lI(b - t)n - (a - t)n - q(t)\\p'
(4.2) Illln
qE,.. .. -2 2(b-a)
1
2n!(b - a)"
a
(b - a)1-t 1
(4.3) R(I,p) = 2(1 + p,)1/p" R(I, 1) = "2;
R(2, 00) = 6-;::)2, R(2, 1) = 6~a, R(2,2) = (6;);/2
(4.4) { 2_l.
R(2 ) < (6-a)
, P - 4(2p'+1)1/p';
P
(4.6)
(b- t-.l ( IIp'
2l~P' B (i- 1)p'+2,P'+1
1 )
R(n,p)~ p , n ~ 4, p> 1;
(4.7)
2) -2-,
n-2
(5.1) J x
f(t)dt ~ M(b; a)2 ;
a
(5.2) J x
f(t)dt ~ M(b1~ a)
2
.
a
p(m,k)(X)
n
=
-
p(m,k)(x· a b)
n "
(_l)n-k(n _ m)! k-m (b _ a)m-n+i
= m!(k-m)!(n-k-1)! ~ n-m-i
X (k ~ m) (x _ a)m(x _ bt-m- i
J
z
(5.6)
a
(5.7)
a
PROOF: In the proof we use the following result from [13] which is also a conse-
quence of (5.3):
Let f ( x) be n- times differentiable function such that If( n} ( X ) I ~ M for x E
(a, b), f(;}(a) = 0 (i = 0,1, ... , k - 1) and f(;}(b) = 0 (i = 0,1, ... , n - k - 1).
Then
(5.8)
a
For a proof of Theorem 2 we may assume that f(c) = 0 for some c E (a, b).
Moreover, by symmetry we may assume that a < c ~ (a + b)/2. We may also
assume that c is the largest zero of f on (a,(a - b)/2]. For a ~ x ~ c, (5.8) for
b = c, k = n - 1, implies that
J
a
z
f(t)dt ~
-Jc
a
M(b-a)n+l
If(t)ldt ~ 2n+ln(n + I)! (= T).
INEQUALITIES INVOLVING FUNCTIONS WITH BOUNDED DERIVATIVES 477
:I:
If f(x) i= 0 for C < x < b, then IG(x)1 = II f(t)dtl would be decreasing on [c,b],
a
so that IG(x)1 ::; IG(c)1 ::; T would follow. We may thus assume that f(cd = 0
for some Cl E (c, b), hence for some Cl E [(a + b) /2, b). Now we may assume that
Cl is the least zero of f on this interval, and with no loss of generality suppose
f (x) > 0 for c < x < Cl. Then
J J
b b
(6.1) !:. f( x) -
n
_1_
b- a
J b
f(y)dy ::; M( n, p, x) IIf(n) lip
n!
a
wbere
and
t- a a::; t ::; x ::; b;
(6.3) k(t,x) = {
t - b a::; x <t ::; b.
b
If in addition I f(y)dy = 0, tben
a
(6.4)
478 CHAPTER XV
where
C( n,p,x ) = . lI(x-t)n-1k(t,x)-q(t)lIp'
(6.5) Ill1n b .
qE1rn - a
All of these inequalities are best possible if 1 < p < 00. If p = 00 (6.1) is always
best possible.
J
a
x
f(y)dy 5,
"
Ilf(n)lIpmin{(x - a)n+1/p , (b - xt+ 1/ P }
/ '
(n -I)! [en -l)p' + 1]1 P (n + lip')
.
Explicit estimates for M(n,p,x) (see (6.1» are given in Theorems 2 and 2'.
THEOREM 2. For 1 5, p ~ 00 and f(a) = feb) = 0, we have the best possible
J
inequality
b
1 (b - a)l/ P'
f( x) - b _ a f(y)dy 5, 2(1 + p')l/p' 1If'lIp
a
. (b_a)l/P' l!!±! .
that lS, M(l,p,x) = 2(Hp')1/pl, and M(l, 1,x) = 2". For x 2:: 2 an extremalls
given by (1 < p)
(x- ~/ - (x-t- b;a/, a < t -< x
-
_ b-a.
2 ,
{
p'f(t) = (x - ~)P' - (t - x + b;a/ , x - b-a
2 -
<t <
-
x·,
, ,
(x-~)P _(x_t+b;a)P, x 5, t 5, b.
A similar formula holds when x < ~. Any other extremal is a multiple of this
one.
THEOREM 2'. For n 2:: 4 and f(j)(a) = f(j)(b) = 0, j = 0, ... , n-1
1 1
;;f(x)-b_a J b
f(y)dy
(n _3)n-3
a
2
5, (b - a)3__ -- max{(x - at- 3,(b - xt- 3 }llf(n)lh.
n-2 n-2
INEQUALITIES INVOLVING FUNCTIONS WITH BOUNDED DERIVATIVES 479
Moreover for n =3
J
6
1
2f(x)- b-a
1 J 6
(b - a)
f(y)dy::; -S-max{(x-a)2,(b-x)2}1If"lh.
a
Hn ~ 3, then for p >1
M(n,p,x) ::; 2(b - a)"'" (H}) B (1 + p'(n - 3),1 + p,)l/ P'
l/p'
x [(x-a)P'(n-2)+! + (b-x)P'(n-2+1]
For C(n,p,x) (see (6.4)) the following are given in [7]:
_ { x(.~=,
- - 4 4'
where
(j -
_ rmn
. (x-b-
- a, -b
b- x)
-
-a -a
and
If(x)1 ::; (x - a)(b - x)IIi"1I1.
Two other results are also given that are related to Mahajani type inequalities.
b
THEOREM 4. Suppose ff = 0. Then the best possible inequalities are
a
and
If(x)::;IIf'lltmin {
x-a
b-a'b-a
b-X} .
Moreover, we also have best possible inequalities
Jf( )
x
t
d
t::;
(b-x)(x-a)
(b _ a)l/p
1If'lIp
(1 + p,)l/p" 1 < p ::; 00,
and
J
x
b
THEOREM 5. Let f f = 0, f(a) = f(b) = 0, and p = 1, or 00. Then
a
J
x
.l.. _l(x-~r
16 2 (b-a)2 , ~a + ~b ::; x ::; ~a + ~b,
(b_x)2
-2- la
4
+;!b
4
<
-
x <
-
b
INEQUALITIES INVOLVING FUNCTIONS WITH BOUNDED DERIVATIVES 481
J
x
J x
f(t)dt ~ (b - x);x - a) 1If'1I<Xl
a
(b ~ a! If(x)I' dx )'" " M(bn~ a)" B (rk+ 1, r(n - k) + 1)'" (r > 0).
From Theorem 1 can be obtained the following inequality
J
b n
f(x )dx - "" (2m - k)! (m) (b - a)k (f(k-l)( a) _ (_1)k f(k-l)(b»)
D (2m)! k
(7.3) a k=l
< M(m!)2(b _ a)2m+1
- (2m)!(2m + 1)!
where If(2m)(x)1 ~ M for x E (a, b).
The foilowing generalization of (4.3) is also given in [13):
482 CHAPTER XV
Po J
a
b
f(x)dx -
n
J
I
J J
I I
Since PI(t) = Pot + PI(O) (Po(t) = Po), equality (7.5) may be represented in
the form
J JP~(t)h'(t)dt.
I I
I
By successive integration by parts of J PHt)h'(t)dt (n-I)-times, we obtain
o
Po J
o
I n
J
I
+ (_l)n Pn(t)h(n)(t)dt,
o
from which (7.4) follows.
COROLLARY 1. Let function f(x) satisfy tbe conditions as in Theorem 2 and let
f(k)(b) = (_l)k-I f(k)(a) (k = 0,1, ... , n - 1). Tben
J
b
M(b - a)n+l
(7.6) f(x)dx ::S 2n(n+1)! .
a
INEQUALITIES INVOLVING FUNCTIONS WITH BOUNDED DERIVATIVES 483
va
(8.1) J F( x )dx ~ 115 v'3.
-va
This is a simple consequence of the well-known two-point Gauss-Legendre
quadrature formula
va
J F(x)dx = (F( -1) + F(l))· v'3 + 115 v'3. F(4)(~), I~I < v'3.
-va
REFERENCES
(n:
+1
(1.1) J
-1
(ao + a1 x + ... + anx n )2dx ~ 1)2'
Except for an obvious error, this relation has been obtained by S. N. Bernstein
[1].
A related result is obtained by O. Bottema in [2]:
b
Let J(ao + a1x + '" + a n x n )2dx = J, where a, b, n and J are fixed numbers.
a
The maximum value of an appears to be
J
1
(1.3) J
o
00
485
486 CHAPTER XVI
F. Smithies [5] has proved the same result using the orthogonality properties of
the Laguerre polynomials. At the same time he has derived the explicit expression
for minimizing polynomial. In [6] L. J. Mordell has solved, in some cases, the
problem of finding the minimum value of integrals of the form
J J
b b
b
where p(x) ~ 0 is such that the integrals J p(x)xrdx (r > 0) exist and the
a
coefficient ak of the term xk in the bracket is given as 1.
L. Mirsky [7] proved
J
b
b
where M(p) = J w(t)tPdt, and where the minimum on the left-hand side is taken
a
with respect to all real numbers AI, . .. ,An-
2. In [8]-[10] the minimum values of (1.4) were found by using a different kind
of normalization of fn(x) which is especially appropriate in solving the approxi-
mation problem in synthesis of filtering networks in communication engineering.
We shall give results from [9] due to Rakovich and Vasic.
Let w(x) be a nonnegative function on the interval [a, b] such that all the
b
moments J w(x)xrdx are finite for r ~ o.
a
Consider the integral of the form
J
b
If Qo, Q1, Q2, ... is a set of orthogonal polynomials associated with the weight
function w( x) on [a, b], the polynomial f n can be expanded into a finite series of
the Q;(x) so that
(2.2) In = J b
w(x)
(
~ a;Q;(x)
n )2 dx.
a
(2.3) ,,(a., a" ... ,an, ,1) ~ j w{x) (~a;Q;{x»)' dx + ,1 (~a;Q;(p) -1)
a
8c.p == 2
8a; J b
w(x)a;Q;(x)2dx + (3Q;(p) = 0,
(2.4) a
n
L a;Qi(p) = l.
;=0
Let
J
b
(2.5) b; = w(x)Q;(x)2dx.
a
Then-
Qi(p) 1
(2.6) a;= - - n
bi " Q j (p)2
L..J b-
j=O J
so that the minimum value M of the integral (2.1) under the constraint is
M= 1
(2.7)
f: Qi(p)2
;=0 bi
488 CHAPTER XVI
3.
THEOREM 1. If c is finite and c ~ a or c ~ b, the integral
J
b
(3.1) In = w(x)ln(x)2dx,
11
where In is any real polynomial of degree n such that In (c) = 1, reaches its
minimum value if and only if 1o, 11, h, . .. form a set of orthogonal polynomials
on [a,b] with respect to the weight function (x - c)w(x) for c ~ a, and with
respect to the weight function (c - x )w( x) for c ~ b.
PROOF: Suppose c(~ a) is finite. From (2.2) and (2.6) the polynomial In that
minimizes the definite integral (2.1), subject to the condition In(c) = 1, is
(3.3)
(3.4)
The Christoffel formula [27, p. 28] states that if Qo, Ql, Q2, ... form a set of
orthogonal polynomials associated with the weight function w on [a, b], then the
polynomials Ro, R 1 , R 2 , ••• , where
(3.5)
are orthogonal on the same segment [a, b] with respect to the weight function
(x - c)w(x). This evidently completes the proof of Theorem 1 for c ~ a. A
similar results holds if c(~ b) is finite.
An immediate consequence of Theorem 1 is the following, more general result:
INEQUALITIES OF BERNSTEIN-MORDELL TYPE 489
J
b
w(x)g(xrdx (r ~ 0)
a
exists. Then the sequence of functions fo (g(x», h (g(x», h (g(x», ... that
minimizes the integrals
J
b
J
g(b)
where
w (g-l(t»)
(3.8) pet) = g' (g-l(t»
According to Theorem 1, the functions that minimize (3.6) form an orthogonal
sysem on [g(a),g(b)] with respect to (t - c)p(t) for c::; g(a), that is
J
g(b)
J
b
J (
b 2
Rn(X»)
= w(x) Rn(C) dx,
a
(4.2) t
.
1=1
Q;(c)M Qi(X)
bi
= Rn(x),
Rn(c)
(4.3)
(4.4)
]'cl - (t,
x')" aox ) , dx
> 7rn!r(2v + 3) .
- (2v + 2)n r (v + ~)2 (v + n + 1) 22+2v
INEQUALITIES OF BERNSTEIN-MORDELL TYPE 491
The last result is obtained for In(l) = 1 but it still holds if the polynomial In
reaches the value of 1 at any point in the interval [-1, + 1].
30 In [9] was proved that (1.2) can be obtained from (1.1), i.e. the following
results are obtained from (1.1):
and
J
c
n 2 l+c
(ao+a1 x +· .. + a n x ) dx~ (n+1)2'
-1
1= J
1
o
(1 - x 2 )P-q x 2q - 1
(
?= )2 dx
n
.=0
ai x2i (p - q > -1, c> 0).
492 CHAPTER XVI
. JOO x" e- x (1
mm + alx + ... + anxn) 2 dx = rev + 1)/ (n + nv+ 1) .
o
where Pn,k runs through the set of all polynomials with real coefficients
and degree at most n such that the coefficient of xk is 1. Janous proved
that
6. Finally, we shall give some related results which were obtained as applications
of quadrature formulae. The following text was communicated to us by D. Acu,
[13]-[16]:
INEQUALITIES OF BERNSTEIN-MORDELL TYPE 493
If [a,b] is a finite interval we shall <;all L[a,b] the class of functions f(x)
Lebesgue-integrable (summable) in [a, b] and ACk[a, b] the class offunctions f(x)
whose k-th derivative f{k}(x) is absolutely continuous in [a, b] (k = 0,1,2, ... ).
We shall call quadrature formula relative to the function f E L[a, b] and to the
nodes Xl, X2,"" x n , any formula of the type
(6.1) J
a
b n
The constants Ai are the coefficients and Rn(f) is the remainder of the quadra-
ture formula (6.1).
The number 'Y E N with the property that Rn(f) = 0 for all f E Pr and
there exists agE PrH so that R(g) ~ 0 is named the degree of exactness of the
quadrature formula.
The problem which arises regarding to a quadrature formula is to determine
the parameters Ai, i = 1, ... , m, and Xi, i = 1, ... , k, in some given conditions
and to study the remainder term R(f) fo the obtained values of Ai and Xi.
For a broader information on quadrature formulae the works [17]-[22] can be
consulted.
FUrther on we shall present two procedures for using the quadrature formulae
in obtaining inequalities.
1. If in a quadrature formula (6.1) all coefficients are nonnegative, then for
f E L[a, b], removing the quadrature sums, we obtain inequalities of the
type
J
b
J
b
for X E [a,b] in the case of the inequalities (6.2) and hex) ~ 0 for X E [a,b] in the
case of the inequalities (6.3).
494 CHAPTER XVI
This method has been used by F. Locher [23] to obtain certain inequalities
with polynomials, inequalities used in solving some extremal problems for the
quadrature formulae.
Consider Gauss-Jacobi's quadrature formula [20]:
1 m
a> -1, f3 > -1, in which the nodes x~iP), i = 1, ... ,m, are the zeros of Jacobi's
. 1 p(a,p)()
po1ynonua m X , an C1ent s A(a,p).
d the coeffi' mi ,Z = 1, ... , mare POS1't'1ve.
For a function f E AC - [_1, 1] the remainder is written in the form
2m 1
R2m - 1 (P2m).
-1
J
1
/(1-
1
x)"'(l + X)PP2m+1(X)dx
-1
/(1-
1
x)"'(l + X)PP2m+1(X)dx
-1
/(1 -
1
x)"'(l + X)pP2m(x)dx
-1
J
00
J
00
J
00
o
xcxe- x f(x)dx = ?: A;J(xj) + R 2m - 1(J)
m
.=1
in which the coefficients Aj, i = 1, ... ,m, are positive and the remainder is given
by
R 2m-1 (f) =m!r(a+m+l)f(2m)(t) t ( )
(2m)! .", ." E 0,00.
J
o
00
where
Ai > 0, i = 1, ... , m, B >0
and
R (f) = mIre a + m + 2) f(2m+I)(C).
2m (2m+1)! \,
The generalizations for the results from the Propositions 1-6 were presented in
[16].
7. Examples given in the previous text of D. Acu are special cases of general
results which follow from Gauss quadrature formula (see [25, pp. 86-91]):
(7.1) J a
b
w(x)f(x)dx =
i=l
n
L Ad(xi) + Rn,
where w( x) 2: °is an integable function and
(7.2)
f(2m)(o
Rn = (2m)!
J b
w(x) (1l'n(x)) dx
2
(a < ~ < b),
a
J
b
In this case we have Ai 2: 0 (i = 1, ... ,n), so for a positive function f(x) we get
(7.5) J
b
w(x)f(x)dx 2:
f(2n)(~)
(2n)!
J b
For an arbitrary polynomial P2n of the degree 2n, P2n(X) 2: 0 and with the
dominant coefficient equal to 1, we get from (7.5):
J J
b b
REFERENCES
J
b
provided u E AC[O, rr] with u' E L 2 [0, rr]. Moreover, equality holds in (1.2)
precisely for u( x) == C sin x, for some constant C.
The first six methods we discuss are from the unpublished manuscript [1], with
few editorial changes.
2. Use of calculus of variations. The inequality (1.2) is the special case of (1.1)
with F(x, y, y') = y,2 - y2. The Euler equation in this case is
500
METHODS OF PROOFS FOR INTEGRAL INEQUALITIES 501
J
b
is, of course, the main problem of the calculus of variations. Sufficient conditions
depend on the existence of an appropriate field (or flow) of solutions of the Euler
equation
d
(2.1) dXFyl - Fy = O.
S = {(x,y): a::; x::; b, ~(x,a)::; y::; ~(x,a') for some a,a' E (al,a2))'
That is we assume that through each point of S there passes a unique curve
y = ~(x,a). In addition it is assumed that the equation y - ~(x,a) = 0 defines
a = a( x, y) uniquely for (x, y) E S as a twice-continuously differentiable function.
The slope-function p: S --+ R of the field :F is defined by p( x, y) == ~x [x, a( x, y)].
The Weierstrass E-function (E for excess) for J is defined by
Under appropriate smoothness conditions on F (cf. [2, pp. 55-59], [3, pp. 26-27,
30]) a sufficient condition for J(yo) to be a minimum value of J is that Yo be an
extremal (so y = Yo is a solution of (2.1) and yo(a) = A, yo(b) = B) embedded
in a field :F as above, such that
holds for all admissible functions u (so u(a) = A, u(b) = B in particular) whose
graphs lie in S. In this case the conclusion is only that J(yo) ::; J(u) for all such
u. If strict inequality holds in (2.3) provided u' (x) :j. p (x, u( x)), the conclusion
is that J(yo) < J( u) for all such u.
We will not be more precise concerning the class of admissible functions for
J(u) except to note that in the classical theory it is usually assumed that u
is piecewise smooth on [a, b]. (In [2, Ch. 9] the theory is extended to include
502 CHAPTER XVII
which can be written as a line integral, and is independent of the path in 5. That
is, if a ~ Xl < X2 ~ band Ul, U2 are two piecewise smooth functions satisfying
Ul(Xl) = U2(Xl), Ul(X2) = U2(X2) and whose graphs lie in 5, then
J;l,x2(Ul) = J;l,X2(U2).
The integral identity in question now is
X2 X2
valid for all piecewise smooth functions u whose graphs lie in 5 and which satisfy
u(x;) = Yo(x;) for i = 1,2. The proof of (2.5) depends on the fact that the left
side of (2.5) is equal to J;1,X2(U) - J;1,X2(YO), using the invariance of J*.
By taking Xl = a, X2 = b the proof of the sufficiency of the Weirstrass condition
(2.3) follows directly from (2.5) - when the embedding field :F exists. However,
for a < Xl < X2 < b, the identity (2.5) remains valid provided only the one-
parameter family y = cp(x, a) of solutions of (2.1) simply covers the subset 51 of
5 for which Xl ~ X ~ X2. In our example, with
F( X,y,y ') = Y,2 -
Y2, E( X,y,p,q ) = ( q - p)2 , Yo = csinx,
cp(x, a) = asinx, a(x,y) = y/sinx, p(x,y) = ycotx,
and °< Xl < X2 < 7r, u(x;) = csinx;, the identity (2.5) becomes
X2 X2 X2
valid for piecewise smooth functions u on [Xl, X2] for which U(Xi) = csin Xi (i =
1,2). Letting Xl --+ 0+ and X2 --+ 7r-, this suggests, but does not prove, that
the inequality (1.2) is valid for all piecewise smooth functions U on [0,7r] having
u(O) = u(7r) = O.
Instead of trying to justify this procedure, we return to the identity (2.6), let
Xl --+ 0+, X2 --+ 7r- to "obtain" the conjectured identity
,.. ,..
(2.7) l(u,2 - u2)dt = I(u' - ucott)2dt.
o 0
We now show directly that (2.7) is valid for all u E AC[O,7r] with u' E L 2 [0,7r]
and u(O) = u(7r) = O. In fact, if 0 < a < f3 < 7r, then
p p p p
t ,..
Now u(t) = J u'ds = - J u'ds, so that
o t
t ,..
0$ u2(t) $ t I u,2ds , 0$ u 2(t) $ (7r - t) I u,2 ds
o t
follows from the Cauchy-Schwarz inequality. It now follows that u 2 (f3) cot f3 --+ 0
as f3 --+ 7r-, and u 2 ( a) cot a --+ 0 as a --+ 0+, so that (2.7) follows. This proves
(1.2), and by (2.7) we see that equality holds in (1.2) precisely when u' = u cot t
a.e. on [0,7r], that is precisely when u = csint for some constant c.
We have dealt with this example in some detail to show that although the
classical calculus of variations may not, by itself, completely handle an integral
inequality such as (1.1), nevertheless an appropriate use of the Euler equation
(2.1) and the Hilbert identity (2.5) may lead to a direct solution. For further
discussion and examples of this approach, see also Hardy, Littlewood, and P6lya
[4, Ch. 7].
Before leaving this subsection, we note that the Hilbert identity (2.6) can be
used to obtain other integral inequalities to which the standard theory of the
504 CHAPTER XVII
calculus of variations does apply. For example, taking 0 < Xl < 7r /2 < X2 =
7r - Xl < 7r, and U(XI) = u(7r - xI) = Uo, where Uo = csinxI, (2.6) becomes
so that
J J
11'-201 11'- 20 1
holds for all such u, with equality if and only if U = Uo sin t / sin Xl. A simple
change of variable gives: if a < b with b - a < 7r, and v(a) = v(b) = Uo, then
(2.9)
a a
holds for all piecewise smooth functions v on [a, b], with equality if and only if
v = Uo sec (b;a ) cos (~ - t) . Although this result requires only the classical
theory of the calculus of variations, a direct proof which shows that (2.9) is valid
for all v E AC [a, bj with v' E L 2[a, bj and v(a) = v(b) = Uo, can also be given by
expanding the integral
J
x
This proves that lim v{ x) = A exists, and similarly lim v{ x) = B exists. Since
x .....o+ x ..... ,..-
x
vex) - vee:) = J v'dt and v' ELI, on letting e: -+ 0+ we obtain
e
J
x
h = y~/yo,
then h is a solution of the associated Riccati equation
p p p p
obtained before.
5. Integral identities and Jacobi multipliers. Let Yo = sin t and for v E AC(O, 11")
define z by
or using (5.1),
(5.2)
a a a a
This problem has the simple eigenvalues An = n 2 (n = 1,2, ... ) and the corre-
sponding eigenfunctions Yn = sin nx. By known results from the spectral theory
of linear differential equations ([8, p. 268] or [9, p. 273]) we have
(6.2)
(6.3)
J :2 J
". ".
J u(t) sin k t dt = °
".
7. Use of Fourier series. If u E AC[0,1I"] with u(O) = u( 11") = 0 and u' E L2 [0,11"]
then we have the Fourier series expansions (for the odd periodic extension of u
to [-11",11"])
00 00
11"
where bk = 211"-1 J u(t)sinktdt, as well as the Parseval relations
o
(7.2) j 1l"
U
2
dt
11"~2
= '2 L...Jbk ,
o 1
and that equality holds precisely when bk = 0 for k ~ 2, that is for u = b1 sin x
for some constant b1 • Similary, if u satisfies the above conditions and, in addition
11"
(7.3) n2
o 0
with equality precisely for u = bn sin n x. This is just the inequality (6.4) obtained
in the last subsection.
The same technique leads to the following inequality, usually referred to as
Wirtinger's inequality: if u E AC[0,211"] with u' E L 2[0,211"] and u(O) = u(211"),
211"
J udt = 0, then
o
(7.4)
METHODS OF PROOFS FOR INTEGRAL INEQUALITIES 509
f
(7.5)
u'(x) rv !Ao + (Ak cos kx + Bk sin kx),
2 1
2,..
then ao = 11"-1 J udt = 0, and
o
J
2,..
Ak = ~ u'(t) cos kt dt
o
J
2,.. 00
J an·
2,.. 00
The inequality (7.4) follows at once, together with the conditions for equality.
For further details of Wirtinger's inequality and related results, see Chapter II.
As in (7.3) if to the hypotheses for (7.4) we add the orthogonality conditions
J J
2,.. 2,..
then
(7.8)
with equality precisely for u = A cos nx + B sin nx. This extension of Wirtinger's
inequality was given by Everitt [10].
There are other extensions of (7.4) which follow by the arguments used above.
211'
For example, we may delete the condition J u dt =0 and obtain the following:
o
if u E AC[0,27r] with u(O) = u(27r) and u' E L2[0, 27r], then
(7.9)
with equality precisely for u = C + Acosx + B sinx. This follows from the (full)
Parseval relation
Similarly, if u' E AC[0,27r] with utI E L2[0, 27r] and u(O) = u(27r), u'(O) = u'(27r),
then
(7.10)
with equality precisely for u = C + A cos x + B sin x. The inequlaity (7.10) was
proved as long ago as 1915 by N. M. Krylov [11], who proved a number of
inequalities of this kind using the Fourier series method. We note that this same
method can be applied to any complete orthonormal sequence of functions using
the corresponding Parseval relations.
0, if aa = 0, { 0 if ab = 0,
(8.2) A(y) = { ~y2(a), if aa -:f. 0, , B(y) = ~y2(b), if ab -:f. o.
J J
b b
(If aa -:f. 0, 'Do includes no boundary condition at a, and similarly for b; only
"essential" boundary condition of (8.1') remain.)
For n 2: 1, set
wbere A, B are defined by (8.2). Tbe minimum in (8.3) is attained if and only if
u = CYn for some constant c -:f. O.
We shall give some applications of this theorem given in [1].
First, we note that the above theorem remains valid, with only minor mod-
ifications to the proof, if it is only supposed that the nth eigenfunction Yn has
kn ::; n zeros on (a, b). Moreove, corresponding results hold for certain singular
512 CHAPTER XVII
systems, that is cases where a or b or both are infinite, or where one or more of
the coefficients p, q, r are singular at the end-points, or r( x) = 0 for x = a or b.
See [12] for a general theorem of this type. Examples of singular cases given in
[12] are listed below:
1 1
1 1
for all u E AC[-I, 1) 8uch that J (1 - X2)U l2 dx < 00, and J Pj(x)u(x)dx = 0
-1 -1
for 0 '5:. j '5:. n -1 there P k i8 the Legendre polynomial of degree k. Equality hold8
in (8.4) if and only if u = CPn for 80me con8tant c.
-00 -00
for all u E AC(R)) 8uch that u(x) = 0 (Ixlk) a8 Ixl -+ 00 for 80me k > 0, the
right 8ide of (8.5) i8 finite, and J e- x2 Hj(x)u(x)dx = 0 for 0 '5:. j '5:. n -1 where
R
Hk i8 the Hermite polynomial of degree k. Equality hold8 in (8.5) if and only if
u = cHn for 80me con8tant c.
!
1 1 1
for m ~ ~, and all u E AC[O,I] 8uch that u(O) = 0, both integra18 on the right
1
8ide of (8.6) are finite, and J
x 1 / 2 J m (k j x)u(x)dx = 0 for 1'5:. j '5:. n -1, where
o
k n i8 the nth p08itive zero of the Be88el function Jm(x). Equality hold8 in (8.6)
if and only if u = cx 1 / 2 Jm(knx) for 80me con8tant c.
For further examples of singular eigenvalue problems, see [15, pp. 324-331,
400,415]. In the proof of Theorem 1 the following lemma is used:
LEMMA. Let n ~ 1, so that the n-th eigenfunction Yn has consecutive zeros at
Xk E (a, b), 1'5:. k '5:. n, where
J J
b b
Then
(8.9)
PROOF: Since Yn has n zeros on (a, b), it follows from Lemma 2 that
To prove the opposite inequality, take any u E 'Hn and suppose the zeros of u on
( a, b) are Xl, ... , X k where k ~ n. We now construct a function v E 'Hn (u) such
that R(v) ~ An- This will prove dn(u) ~ An for all u E 'Hn whence (8.9) will
n
follow. In fact it suffices to take v = I: CiYi. We have v E 'Hn( u) if the Ci (not
o
all 0) can be chosen so that
n
L CiYi(Xj) = 0, 1 ~ j ~ k (~ n).
i=l
514 CHAPTER XVII
Such a nontrivial solution always exists for this homogeneous system. But then,
as in the proof of Theorem 1 (see [12]), we have:
b n b
t
b b
An j pv 2 dx = An cr j Pyr dx.
a 0 a
b b
Thus, An J pv 2 dx 2:: J(rv l2 + qv 2 )dx + A( v) + B( v) is clear so that R( v) ::; An
a a
holds, completing the proof of (8.9).
The above result is given in Courant-Hilbert [15, p. 463] and attributed there
to K. Hohenemser. A second maximum-minimum characterization of An is due
to R. Courant [15, p. 406] and can be formulated as follows.
THEOREM 3. Let C n be the class of all n-tuples (vo, . .. , Vn-l) of functions such
b
that 0 < J pv;dx < 00 for 0 ::; i ::; n - 1, and set
i
a
u(x) = L CiYi(X)
°
where the Ci are any nontrivial solution of the n homogeneous equations (in n + 1
unknowns)
J
b b
Then u E Cn(vo, ... ,vn-d and, as in the proof of Theorem 2, An > R(u) ~
dn(vo, ... , vn-t}, completing the proof of (8.10).
which is true for every real x and n = 1,2, ... , equality holds if and only if x = 1,
and he got the following:
THEOREM 1. Let u( x), P( u, x), G( u, x) be continuously differentiable functions
and let P( u, x)> O. Then, the following inequality holds
b
j (pu,2n + (2n - l)p-l/(2n-1)G~n/(2n-l) + 2nG x ) dx
(9.2)
a
~ 2n(G(u(b),b) - G(u(a),a)),
J
u
J
u
I
X2
(u/ 2 + (M - u)(u - m)) dx 2: -2
M
j ((M - u)(u _ m))1/2 du = ?r(M ~ m/
and
J J
X2+211" m
211"
If additionally we assume J u dx = 0" then we have
o
J(u/
211"
2 - u2) dx 2: ?r(M; m)2
o
METHODS OF PROOFS FOR INTEGRAL INEQUALITIES 517
Since the right-hand side of this inequality is always positive, as a special case
we obtain Wirtinger's inequality (see Chapter II; for a generalization of the above
result see especially 12. from Chapter II).
Let us now state a useful special case of (9.2) which can be applied to prove
Weyl's inequality ((6.8) from Chapter I.).
Let u and 9 be continuously differentiable. Then
J
b
Since this inequality holds for every real A, the discriminant of the above
(i i i
quadratic polynomial in A must be nonpositive, i.e.
J
L
J(
L
,2n + ( 2n - rrn(2n )!- ( M - m )2n ,
1)( u - m )n(M - u )n) d x ~ (n!)22
U 2n 2
o
J b
(p(u(n»)2m + (2m - l)p-l/(2m-l)G!~~(;)m-l) + 2mu(m-l)G u (n_2) + ... +
a
Here, sand t are nonnegative (positive if p < 0), and in both cases strict in-
equality holds unless s = t. (Note also that when p = 0 or p = 1, the left sides
of both (9.5) and (9.6) become identically zero for all s and t.)
THEOREM 4. Let vex) be absolutely continuous on [a,p] with v'(x) ~ 0 a.e.
Also, suppose that Q(x) is nonnegative a.e. and measurable on [a, P], and G(v, x)
METHODS OF PROOFS FOR INTEGRAL INEQUALITIES 519
is continuously differentiable for x E [a,,8] and v in the range of the function v( x),
with Gv(v,x);::: 0 (or Gv(v,x) > 0 in case P < 1). Then if the integrals exist
J
(3
(9.8) J f3
{Qv'P + (p -l)(G v Y/(P-l)Q-I/(P-I) + pG x } dx
'"
:::; p{G(v(,8),,8) - G(v(a),a)} (O<p<l),
with the same abbreviations as in Theorem 1. Equality in both (9.7) and (9.8)
holds if and only if the differential equation
That is,
proving (9.7) by integrating both sides of the above inequality from a to ,8.
The proof of (9.8) follows from the above argument, but using (9.6) instead of
(9.5).
Using these results, Shum obtained various generalizations of Opial's inequal-
ities (for some of his generalizations see Chapter III. and generalizations of
Bomieri's inequality (27.2) from Chapter IV.
D. C. Benson [22] extended a method of his own to comparison theory. So, he
proved:
THEOREM 5. Let p(x), PI(X), q(x) and ql(X) be continuous on [c,d] with p > 0,
PI > o. On [c,d], let y(x) and YI(X) be solutions of
520 CHAPTER XVII
and let a and b satisfy Xl (a) = c, Xl (b) = d, with a, b > o. Then the following
inequality holds:
REFERENCES
PARTICULAR INEQUALITIES
-00
holds on the real x-axis, then for all positive integers k the inequality
-00
2° I f(x)cosxdx = 0 and
°
3° there exists a number>., with 0 -:; >. -:; 27r /3, such that
7r
f(x) = 0 for>. -:; x -:; >. + 3.
Then we have
J
7!"
f ( x ) sin x dx -:; 1,
11"
f(x)=o for ). :s: x :s: ). + 3"'
f(x) = ° for I). + ~6 - x I -> ~2'
f(x) = 1 otherwise
4. Let f(t) be a real-valued integrable and 211"- periodic function. Consider the
n- th
Fourier sum
fn(t)=.!..
11"
J f(t+2z)sin(2~+1)zdz.
n:/2
SIn z
-n:/2
Let, further, M[g] denote upper bound of Igl and A, B, C, AI, BI absolute
constants.
Then we have the following Lebesgue inequality [3]:
P(x) = Jv'f=U2 x
=J
x
QI(X) dQ(u) with dQ(u) ;:::: 0,
v'f=U2
-1
mk-
-J 1
u k p(u) d -
~ u-
v1-u 2
J1
u k dQ(u)
~,
v1-u 2
-1 -1
(4.3) j ukdQ(u)-fn(t)du+M(Jn]du=M[fn]juk du .
~ ~
-1 -1
x:::;
l
From (4.2) and (4.3) we obtain, for -1 :::; 1,
l
1
l
Inequalities (4.4) and (4.5) yield
X dQ(u) _ jX p(u)du
j
~ v'1-u 2
1) ]
1 -1
Al
< CM[Jn] = -;-M[P(x)] + B1log
n nM [p ( x +~ - p(x) .
REMARK: The Lebesgue inequality (4.1) can be found in [3]' pp. 19-23. The
above results are due to M. Krawtchouk [4].
j exp[if(t)]dt <~.
v
u
PARTICULAR INEQUALITIES 525
This lemma of Van der Corput is used in [6] in the proof of the following result:
Let al < a2 < ... < an be fixed nonnegative real numbers and let bl , ... , bn
+00
be real numbers. Then f exp{i(bdx]a 1 + ... + bn[x]a n }
-00
d:
~ K(al,"" an),
where K does not depend on bl, ... , bn , and [x] is the integer part of x.
J J
U n -2 Un-l
8. Some inequalities are ofthe Tauberian type, for example a result ofV. Avaku-
movie and J. Karamata [8].
Let f(t) be a nonnegative nondecreasing function for t 2:: O. Further let the
integral
J«()) = J
+00
o
e- 9t f(t)dt be convergent for () > O.
10. Inequalities for the incomplete gamma function are given by H. R. van der
Vaart [11, pp. 442-443].
x
Consider the incomplete gamma function ,(a, x) = J e-tta-1dt. As e-aa a >
o
a+l
J e-ttadt, we obtain
a
a J
a
e-tta-1dt = e-aa a + J a
e-ttadt >
o 0
whence
a,( a, a) > ,(a + 1, a + 1).
If(x)1 :::; L (L is a positive constant) for -00 < x < +00, then
12. Here is a probem posed in the SIAM Review, April 1973. The proof is by
the proposer, Robert Shafer.
If a, b, c, m, n denote real numbers with a, b > 0, 12: min > 0, show that
J
00
where
PROOF: Let a = min. Note that for z > 0 and 0 < a ~ 1, the maximum of
exp{ -az - bz-a} occurs at z = p where p satisfies bp-a = aPia. We shall show
that for z > 0 and 0 < a ~ 1,
from which the stated result will readily follow. Letting u = log(z/p), we trans-
form inequality (12.1) to the equivalent inequality
For -00 <u ::; 0, (12.2) can be verified by expanding both sides in powers of u
and using the fact that 0 < a ::; 1. For 0 < u < 00, the inequality is somewhat
more difficult to prove. First, we note that for all real s,
or equivalently,
which follows because the exponential function is convex. It then follows that
JJ JJ
.. t .. t
J
(12.3) 00
~pc-qexp{(c-q)/a} zq-1exp{-Az-B/z}dz,
o
where A = a(2 - a)(1 + a)/2, B = ap2a(1 + a)/2, and q = c + ap(1 - ( 2). From
an integral representation of the modified Bessel function /{q, [14], we obtain
J
00
J
1
-x xp-1(x - I)P
Sp= e (p-l)! dx,
o
PARTICULAR INEQUALITIES 529
then
(13.1)
(13.2)
1
REMARK: In order to prove (13.2), consider J xm(1- x )ndx, where m and n are
o
natural numbers. See Ostrowski [15, p. 96].
n;j
o < -.,fi - V. bt
a
then
b b
q q u
and u
min f(q)g(q)
u~q<b
~ jf(t)y'(t)dt.
a
530 CHAPTER XVIII
Hence,
b b u
b b
The above results are due to J. G. van der Corput [16]. Paper [16] contains
also a number of similar inequalities.
o < a ~ b, to ~ x ~ y ~z ~w
such that
z w
dt dt< b
/ get) = a, / -
get) - ,
y x
and
Ig'(t) I ~ c for t E [to,+oo).
If h is defined by
PARTICULAR INEQUALITIES 531
then we have
J J
w z
J J
A
hg:S
A
iIg·
J
00
J J
00 00
h(x)dg(x) :S h(x)dh(x)
o 0
J J
00 00
KI(X,t)dg(x):S K2(X"t)dh(x).
o 0
19.
1. If a twice differentiable real function f satisfies the inequalitity f( x)1" (x) <
o for x E (0,1), then f satisfies at least one of the following inequalities
If(x)l> _P_xr+I
r+1
or If(x)l> -P-(l -
r+1
xr+ l
0, and if
then
b- a < Am-lin,
20. Let a function f(x) be defined on [a,bj. We shall say that f belongs to the
class Lip(a,p) if
( !
b ) lip
If(x + h) - f(x)IPdx ~ MhCX,
JJI =~(y) I J
1 1 ~ 1
is valid if the righthand integral exists as a Lebesgue integral, and the constant
log 4 cannot be improved for A = 1.
22. A. F. Timan [24] determines explicitly the best bound for the difference
between a continuous function of period 271" and its Poisson integral
1
f(r, h) = 271"
J".
1- r2
f(x+t)I_2rcost+r2dt,
-".
when
If(x + h) - f(x)1 ~ Mlhl
or when f( x) has a pth derivative with a prescribed bound. In the first case the
result is, for 0 ~ r < 1,
2M
If(x)-f(r,h)I~-
71"
(
1
(l-r)log-+
l-r
J
l-r
o
1 2-t )
( - l o g - t +1)dt .
I-t t
23. Let S( k, n) denote the class of functions f which are continuous on [0,1]
such that:
e~k)(g) = min
/ES(k,n)
IIf - gllC[a,bj'
Starting from these assumptions the following results was proved by G. Freud
and V. Popov [26].
Let f E C[O, 1] be absolutely continuous, and for a function Fn E S(k,n) let
J
1
J
1
Let it, fz ... be an infinite sequence of real measurable functions on [0,1] with
the properties
and
J
1
J
1
25. Let f(x) and f'(x) be continuous functions on [a,b] and let t' exist and be
absolutely bounded, i.e. It' (x) I :S m. Then, for any x E [a, b] we have
26. Let f(x) be a differentiable function on [a, b]. Let a denote the oscillation
of f on [a, b], and let, for a positive constant m,
- f'(y) I
Ifl(x)x-y :S m if x,y E [a,b], x i= y.
536 CHAPTER XVIII
b-a
max (f(x) - fey)) ~ (b - a) max f(x) - m -2- ,
z~y a~z~b
when a ~ y ~ x ~ b.
If b - a ~ V2u/m, then
1f'(x)1 ~ V2mu.
See A. M. Ostrowski [29].
28. Let f(x) be a real 271"-periodic function, absolutely continuous in [0,271"] and
let f' E L2(0,271"). Then for all x and y,
J J
2,.. 2,..
This inequality is due to S. E. Warschawski [31], p. 18, and is used and proved
also in [32], pp. 68-69.
PROOF: We may take x,y E [0,271"] and y < x. Since f(x)2 is also absolutely
continuous, we have
J J
z y+2,..
J
x
and
J
y+2,..
If(x)2 - J(y)21 ~~ (
2! X
IJ(t)IIJ'(t)ldt + 2! y+2,..
IJ(t)IIJ'(t)ldt
)
J
2,..
= If(t)IIf'(t)ldt
o
s; (1 (1)'dl),,' (7f'(t)'dl),,'
1
J J
2,.. 2,..
i JIJ(t)la-11f'(t)ldt
2,..
If(xW::;
o
538 CHAPTER XVIII
For a = p = q = 2 we have
J J
271" 271"
The constant 2'1r in (28.4) is best possible. In order to prove this fact consider
J
tn
tnJ(tn) ~q J(t)dt.
o
30. (a) Let \II be positive and monotone increasing and let w be positive on (0, A)
where 0 <A ~ +00 with w( x) ~ hx for 0 < x < A and some constant h > o. If
0< q < p < 00, then there exists an absolute constant M = M(h,p, q) > 0 such
that for all positive, measurable" subadditive functions cp on (0, A) we have
(b) Moreover, if \II is positive and monotone decreasing on (0, A) and there are
constants e, k such that 0 < e < 1/2, k > 0, and \II(ex) ~ k\Il(x) for 0 < x < A,
then the inequality (30.1) is still valid for some M = M(c, k, h,p, q) > o.
A special case of these results is:
PARTICULAR INEQUALITIES 539
Let w be positive on (O,A), where 0 < A < 00 and satisfy w(x) $ hx b for
some constants h > 0, b > 1. Let 0 < q < p < 00 and \II, cP satisfy the previous
hypotheses. Then the inequality (30.1) holds with
31. Let M be a real continuously differentiable function of [a, b] with M' > O.
If z is an absolutely continuous real function on [a,b] with z(a) = z(b) = 0 and
z' E L2( a, b), then
b b b
32. If a < b are real numbers and f is a continuous complex valued function on
[a,b], with continuous first derivative in (a, b), then, for any u in [a,b],
b b u
hence
b b
then
(34.1)
(34.2)
with the condition that the assumptions made above for (34.1) remain valid for
(34.2).
35. Let J(t) = eitj(e it - a)" with t real and a complex (lal-=ll). Then J is
infinitely differentiable in t and we have
Suppose that there is a sequence (Yn) with Yn --+ +00 as n --+ +00, and also an
integer m < n such that the following limits
exist.
Consider further the expressions
holds for any numbers Xo, Xl, ••. , Xm greater than a. These results appear in N.
Obrechkoff, [46].
37.
J
x+1
The bound l/x is the best possible, see M. Landau, J. Gillis and M. Shimshoni,
[47]. For generalizations of this result see E. K. Godunova and V. I. Levin, [48].
offunctions on [a, b] and let us suppose that <l>n(x) can be expressed in the form
<l>n(X) = w(x)d~nnFn(x), where Fn(x) and w(x) satisfy the following conditions:
(i) F~p) (x) = 0, p = 0,1,2" ... ,n - 1, (for x = a and x = b),
(ii) Fn(x) does not change sign in [a,b],
(iii) f(x)/w(x) is continuous in [a,b].
542 CHAPTER XVIII
A. ~ ( ! ;\:j:dX / I! F.(x)dxl
b ) 1/2 b
An ~
(2n)!
2n+ 1 / 2 r(n+l/2)
(
f1
(I-x)
2 -1/2 2
J(x) dx
) 1/2
L. S. Bosanquet [51] has replaced integrals by sums and has proved the fol-
lowing:
Let A~ = (0" + v)!/(O"!v!). If 0 ~ ,\ ~ 1 and 0 ~ m ~ n (m and n are integers),
then
PARTICULAR INEQUALITIES 543
J J J
00 00 00
If 0 < p ~ q we have
J
q 2
and
J
q
-p
Jf
-p
P
(t )dt 2: 1 - ;2'
2
For the above results, see [52]. In [53] the following is proved.
If there exists xo such that the function f, given above, is nondecreasing for
x < xo and nonincreasing for x > xo, then
J
p
a2 3 a2
f( t)dt >
-
1 - 0- > 1 - - -
p2 5p2'
-p
42. Let I, g be two integrable functions on [0,1) such that 0 :s: g( x) :s: 1 for all
1
x E [0,1) and J I(x)dx = O. Then
o
! 1
l(x)2dx 2: 4
(
!
1
g(x)/(x)dx
) 2
then
J
z
R(t)Q(t)dt ~ R(;;'~~O(~O) (R(x) - R(xo)) (a ~ xo < x ~ b).
zo
See [56].
00
44. A moment inequality is given by G. Aumann [57]. Let>. > 0, J W(t)dt = 1
o
00
(2)' + 2)A
(2)' + l)A+1 ~
J00
A
t W(t)dt
o
or if
2>' + 2
W( x) =0 for x > 2>' + 1 .
As a corollary we get
J
00
for W(t)dt = 1.
o
45. Let the function f be defined and integrable on each compact subinterval of
[0,+00), with f(x) -+ 0 as x -+ +00. Let 9 be a continuous function for x ~ 0
z 00
More generally, let J(Xl, ... , xn) be defined and integrable on each compact
subset of the domain {(xl, ... ,xn)lxl ~ O""xn ~ O} and let J -t 0 as any
Xi - t +00 (i = l, ... ,n). Let (-1)n{rJ/8xl ... fJx n ~ O. Let gl(X), ... ,gn(x)
x x
be continuous on x ~ 0 and let h1(x) = J gl(t)dt, ... , hn(x) = J gn(t)dt be
o 0
bounded and nonnegative. Then the repeated integral
f··· JJ(Xl,""
00 00
46. Let J(x) and g(x) be complex-valued functions of the real variable x. Con-
sider the following two conditions.
A. IJ(x)1 S Ig(x)l;
B. For any real number t the function arg (eitg(x) . J(x») is nondecreasing in
some neighbourhood of x.
is strictly increasing in any subinterval of (a, b), where J(x)/g(x) is not constant,
and if
then
Ih(X2) - h(Xl)1 S 2 sin ~ (B(X2) - B(Xl»'
1f'(x)1 S 19'(x)l;
PARTICULAR INEQUALITIES 547
If'(x)1 ~ 19'(x)lcos(,8 - a) or
(46.1)
1f'(x)1 ~ -1g'(x)1 cos(,8 + a),
If(x)1 Ig(x)I'
cos a = Ig(x)1 and cos,8 = Ig'(x)l;
4. If f is a real-valued function on [a, b], where the conditions A and B are
satisfied, then
J
1
J J
1 1
e- f(x)
Z = 1
o
Z
we get
(47.2)
and similarly
(47.3) e~-Zlf(xW
smh(l- x)
$; 1
1
The function sinh x sinh(1 - x)/ sinh 1 takes its maximal value ! tanh! at
x = !. This proves (47.1).
48. Let the function f( x, y) vanish on the boundary of the square S = {(x, y) 10 $;
x$;1 and 0 $; y $; I} and let 104 f(x, y)/8x 28y21 $; A (A is a positive constant).
Then
11
B
f(x,y)dxdy $; 1!4 A .
upon-Wabash" [61]. The authors ofthis book have learned this pseudonym from
Flanders himself.
PARTICULAR INEQUALITIES 549
49. If an = - E(-ll(~)kr for 0 < r < 1 and n 1,2, ... , then an > 0
(n = 1,2, ... ) and n--++oo
lim an = O.
PROOF: From
J(1-
00
h = e- yk ) y-r-1dy = krII
o
follows that
J(1 -
00
J J
00 M
0< anII < yn-r-1dy + (1- e-M)n y-r-1dy
o 1
J
00
+ y -r-Id y,
M
with M > 1.
The right side of the above inequality becomes smaller than any c; > 0 if we
take first M and then n large enough. It therefore follows that an > 0 and
lim an = o.
n--++oo
J tne-te-atk dt,
00
1 1 3a(n+2)(n+3)
n + 1 < Zn < n + 1 + n +1 , n = 0,1,2, ....
550 CHAPTER XVIII
51. Let J(t) be a nonnegative summable function on [0, a] such that it is either
nondecreasing or twice differentiable with f'2(t) - J(t)f"(t) ~ 0 for every t E
(0, a). Then the sequence
jtkJ(t)dt}
{ .!:.
k I
0
t k- 1 J(t)dt
(k=1,2,3,)
is nonincreasing.
This result is due to W. Sciamplicotti [63]. Some generalizations are obtained
by D. S. Mitrinovic and J. E. Pecaric [64].
52. A necessary and sufficient condition for the existence of three linearly inde-
pendent periodic solutions of
, ,
XIII - !!....x"
p
+ (1 + p2)X' - !!....x
p
= 0, pet + w) == pet), pet) =fo 0,
is that
J'" V1+p
o
2 dt>27r.
Let u and v be positive increasing functions defined for t E [1, +00), such that
u"(t) V"(t)
(54.1) -->--
u(t) - v(t)·
Then
(54.2)
u'(t)
--> ---c
u(t) - vet) ,
v'(t) .h
WIt c-
-lu'(l) _ v'(l)
u(l) v(l)·
I
PROOF: Put Vet) = v'(t)fv(t), U(t) = u'(t)fu(t), wet) = Vet) - U(t). Then
U(t) > 0 and Vet) > 0, and in view of (54.1) also
From (54.3) follows that w is a decreasing function in all its points of positivity.
IO Let w(l) > O. Then wet) ~ w(l) for all t 2: 1 such that wet) > O. If
wet) ~ 0, then a fortiori wet) < w(I).
2 0 Let w(l) ~ o. Then wet) ~ 0 for all t 2: 1, and therefore wet) ~ Iw(l)1
for all t E [1,+00). Namely, in the opposite case for some b > 1 we would have
w(b) > 0, and the zeros of w on [1, b] would form a nonempty set. Let a be
the supremum of that set. Since the function w is nonnegative on [a, b], it is
nonincreasing on that interval. Hence web) ~ w(a), and thus web) ~ 0, which
contradicts the assumption.
55. If L: akeikx and L: Ake ikx are trigonometric polynomials and lak I ~ Ak for
all k, then for all even integers p
JII: JII:
2tt 2tt
akeikxlPdx ~ AkeikxlPdx.
o 0
Then
J
v
J
±v'2
e- z Jk(x)dx
o
PARTICULAR INEQUALITIES 553
3x 3x
3+x
--2 < tanh x < - ---.
3 + x2 _ .:4 ,
15
where the upper bound is valid for 0 ~ x a(15 + V405)) 1/2. This is a result of
A. Ronveaux [72J.
60. Let x(t) = (Xl(t), ... , xn(t» be a vector, whose components Xl, ... , xn are
real-valued and continuous functions on [a, bJ. Let A(t) = (aij(t» be a square
n X n matrix whose entries are also real-valued and continuous functions on [a, b].
Let further
n
where Ax = (AI' ... ' An) with Ai = L: aijXj.
j=l
Then, any solution x(t) of the system of linear differential equations
dx;(t) ~
---;u- = L..Ja;j(t)xj(t) (i = 1, ... ,n)
j=l
554 CHAPTER XVIII
A very short and elegant proof of the above result [73], due to H. Toyama,
was given by E. Kamke [74].
(61.1) dx dY )
d ( p(x) dx + q(x)y = 0,
where p and q are real continuous functions for x > 0 such that
d
p(x) > 0 and dx (p(x)q(x)) = 0,
(61.2)
Thus it follows that the neither g(x) and g'(x) can vanish at Xl or X2' Whence
(61.3)
PARTICULAR INEQUALITIES 555
J
X2
Hence,
(61.4)
L JJfn(t)dt
+00 b X 2
dx ~ ~ (a - C)2 + (b - C)2) ,
n=l a c
63. Let ao, al , ... , an be nonnegative real numbers and let f (t) be an increasing
function for t ~ 0 such that f(O) = O. If
L ai ~ f-l(al-
n
r) (k=O,l, ... ,n)
i=k
n
where b = L: ai.
y
For x = {xii} E D let J(x) = J{xii} denote the point of D, whose (i,j)-
coordinate is
ap
xii ""liXii I(x)
J(X)ii =
qi ap
2: Xij ax"
j=1 OJ
(x)
Then
:S maxlf(x,y,z)l·
X,y,Z
This result is connected with the problem of determining the majorants for
solutions of the equation
U xx + U yy + U zz - Ut = o.
PARTICULAR INEQUALITIES 557
we have
with p;::: 1.
When p ---+ +00, the last inequality reduces to
67. A function f is called cyclically monotone on (a, b), if f and its derivatives
are each of constant sign on (a, b) and
J<kl(x)f(k+ 2 )(x) :::; 0 for a:::; x:::; b and for every k;::: O.
For a cyclically monotone function the following inequality, for any r = 0,1, ... ,
is valid.
This result, due to Bernstein [82], is dated from 1928.
68. Let f be defined and bounded on the rectangle D = [a, b] x [c, dJ. Then the
lower and upper iterated and double Riemann integrals of f over D satisfy the
inequalities
J- -J f(x, y)dA:::; J- -J
f(x, y)dx dy:::; {JJ
J J- f(x, y)dx dy
-=-
}
(68.1) f(x,y)dxdy
The same inequalities hold with the opposite order of integration for the iterated
integrals. A proof of this is given in T. Apostol [83, p. 260] together with
examples where strict inequality holds. See also the examples (and partial proof)
given in Hobson [84, pp. 514-518]. As noted in Hobson, the existence of both
b d
the Riemann integrals J I(x,y)dx for all y E [c,d) and J I(x,y)dy for all x E
A C
[a, b) implies the existence and equality of the two iterated integrals. However,
examples [84, Ex. 4, p. 518], [83, Ex. 10-9, p. 299] show that this does not
imply the existence of the double Riemann integral.
If I is not bounded on D, W. H. Young [85] has shown that the inequalities
(68.1) may not remain valid. In this case the integrals are improper lower and
upper integrals in the sense of de la Vallee Poussin, defined as follows. For
unbounded functions I on an interval I, one sets
Then by definition,
i l dx = N,M-co
lim iIN,M dx, i l dx = N,M-+co
lim iIN,M dx,
- -
provided these limits exist. The function I is said to be imporperly integrable
precisely when its lower and upper integrals exist and are equal. In this case,
it is clear that the common value is just the Lebesgue integral of I. Similar
definitions apply to unbounded functions I defined on the rectangle D.
For such improper integrals, Young [85, Th. 3] proves that the first inequality
of (68.1) remains valid if I is bounded below on D, while the last inequality of
(68.1) remains valid if I is bounded above on D. Moreover, in the first case
equality holds if I is lower semi continuous on D, and in the second case if I is
upper semi continuous on D. An explicit example is given of a function I which
PARTICULAR INEQUALITIES 559
1(1 f(x, Y)dX) dy = 11 f(x, y)dA < 1(1 f(x, Y)dX) dy.
Here, the inner integrals and the double integral are imporperly integrable.
Young also gives two examples of bounded functions f, g for which
which shows that no general inequality can hold for these repeated integrals.
1 1
~ ~
O. Then
o 0
1 1
~ ~
70. If f and 9 are such that all the integrals exist, then
Js~p J
00 00
PROOF:
Js~p J
00 00
J [S~Pf(X)]
J
00 00
J [S~Pf(x-t)]9(t)dt= Js~p[J(x-t)g(t))dt
00 00
=
-00 -00
s~p J s~p J
00 00
72. Tagamlicki gave related results in three papers [87], [88], and [89]. We shall
give only the following two results from [87] and [89] respectively.
(i) Let f(x) have derivatives of all order for x < a and let
Then
f(x)=Be x ,
then g(x) == 0 on [.
This is a Gronwall type lemma. It implies the usual Gronwall lemma since if
g(a) = 0 and Ig'(x)1 ~ Ig(x}1 on [ then 9 == 0 (the case f == 0 and), = 1). See
[90].
J
R
(74.2) - = 4~
S(u) (u'',). + U·),'.) (u'.,). + U·),'.) dV,
R
where Latin subscripts have the range 1 to n, and n is the number of dimensions.
Summation over repeated subscripts is implied, and subscripts preceded by a
comma indicate differentiation with respect to the corresponding coordinate.
A. Korn [91]' [92] proved the existence of a number ]{l > 0, depending only
on the domain R, such that
(74.3)
for all vector fields 1J satisfying certain side conditions. Since equality holds in
(74.3) for any ]{l if 1J is a constant vector, we will agree henceforth to identify
vector fields differing only by a constant.
The necessity for imposing some side conditions on 1J In order that (74.3)
should hold can be seen from the fact that if 1J is a pure rotation, then S( u) == 0,
D(u) > 0.
Let
- = ~J(u
R(u) 4 .. -u'),'·)(u··
I,) ',) -u'
),'·)dV.
R
Then
D(1J) = S(1J) + R(1J),
so that ]{l cannot be less than unity, and the inequality (74.3) is equivalent to
either of the following inequalities:
J( Ui,j - uj,i)dV = 0,
Main Case: { R
C 2 (R), Uj,jj + Uj,ij = 0 in R.
1J E
Either of the conditions in First and Second cases serves to eliminate pure rota-
tions. The class of admissible domains R is specified in detail in [93] (see also
[94]).
Extension of Main case is considered in [94] and [95]. For other references
about this and related problems see [94], [95].
(75.3)
In 1966, the editor of the Amer. Math. Monthly pointed out [97]: From a
reexamination of the original proposal there is reason to think that the inequality
has been stated as
(75.4)
564 CHAPTER XVIII
a2 = 1 - al ~ 0.8175121,
and n = 2. Then the inequality (75.4) is true for al (X2 - xo) < Xl - Xo < a2 (X2 -
xo), and it is untrue for Xl - Xo < al (X2 - xo) or Xl - Xo > a2( X2 - xo). Similarly,
for n > 2, the validity of the inequality (75.4) depends on the distribution of
Xo < Xl < ... < Xn ·
THEOREM 2. Let Xo < Xl < ... < Xn be n + 1 uniformly spaced points of the
closed interval [xo, xn], then for n ~ 6, form (75.4) is true, and for large n (75.4)
is untrue.
THEOREM 3. Let a < Xo < Xl < ... < Xn < b be the roots of the (n + l)-th
orthogonal polynomial P n+ 1 (X), associated with the weight function p( x) in the
interval [a, b], where
J
b
76. Montgomery [99] discusses certain results of Gallagher [100] and improves
upon them.
PARTICULAR INEQUALITIES 565
k I
b b
b b
The result (76.1) is in [100] while the second is a special case of a general
Sobolev inequality.
b b x
t-b jt-a
f(x) = (b - a)-l j f(x) + j b _ af'(t)dt + b _ af'(t)dt.
a x a
The fractions :=! and :=: are at most 1 so (76.2) follows. If x = ~, these
fractions are at most 1/2 so (76.1) is obtained.
The inequality (76.2) may be iterated.
N.,(x) = L 1.
tEP
It-xl<.,
H f E C1(To, To + 1) then
566 CHAPTER XVIII
PROOF: By Lemma 1
x+! x+!
If(x)1 ::; 8- 1 J If(t)dt + ~ J If'(t)ldt.
x-! x-!
Thus
J J
~+T ~+T
LEMMA 3. With the hypotheses of Lemma 2, let IX1 - x21 ~ 8 for any two points
X1,X2EP. Then
J J
To+T To+T
L
xEP
N6(x)-1If(x)1 2 ::; 8- 1 J
To+T
To
If(t)12+ J
( To+T
To
If(tWdt
) 1/2 (TO+T
J
To
IS'(t)1 2dt
) 1/2
PROOFS: Lemma 3 follows from Lemma 2 by observing that N6(X) == 1 for each
x E P. To get Lemma 4 we apply Lemma 3 to P and use the Cauchy-Schwarz
inequality.
77. G. Freud and G. P. Nevai [101, pp. 68-69] proved the following lemma. If
= 0,1,2, ... , then
x > 0, a ::; 0, (3 ~
Ij(x)=
1 and for j
J00
Io(x) = j
x
ya e - yP dy ::; /3::-1 j /3yP-1
x
e - YP dy
1 a -xP
= /3xP-1X e
(1 41 1
then
(j o
Xf'(X)2 dX) ' S; k J J
0
xf(x)'dx
0
x!,,(x)'dx.
REFERENCES
GABRlEL, R. M.: 262, 273, 295, 311. GREENE, D. E.: 416, 426.
GABUSIN, V. N.: 12, 13,33,56. GREENHALL, Ch. A.: 551, 571.
GAIER, D.: 536, 569. GRONWALL, T.: 345, 349, 353, 392.
GALANIS, E.: 51, 64. GROSS, L.: 93, 110.
GALBRAITH, A.: 230,238. GRUDO, E. I.: 391, 397.
GALJAUTDINOV, I. G.: 12,56. GUANCHU, H.: 564, 573.
GALLAGHER, P. X.: 564, 565, 566, GUGGENHEIMER, H.: 550, 571.
573. GUILLIANO, L.: 391, 397.
GAMIDOV, S. T.: 357, 361, 393. GUINAND, A. P.: 567, 573.
GANELIUS, T.: 88, 107. GUSTAFSON, G.: 233,239.
GARSIA, A.: 551, 571. GUTOVSKI, R.: 345, 350, 354.
GAUSS, C. F.: 260, 273. GUTOWSKI, R.: 345, 349, 350, 425,
GEISBERG, S. P.: 11, 12, 55, 56. 429.
GEL'FAND, A. V.: 336,.347. GYORI, I.: 353, 363, 369, 391, 392,
GERAGHTY, M.: 459, 462. 397.
GERARD, F. A.: 485, 498. HADAMARD, J.: 2,53,68,102.
GHIZZETTI, A.: 493, 494, 498. HADWIGER, H.: 544, 570.
GIERTZ, M.: 16, 17, 22, 58, 91, 109, HAJNOSZ, A.: 391, 396.
175,184. HALBERSTAM, H.: 202, 214.
GILLIS, J.: 541, 570. HALL, R. R.: 89, 108.
GILLMAN, D.: 51, 65. HALPERlN, I.: 8, 54, 84, 106.
GINDLER, H. A.: 32, 33, 61. HAMAD, G.: 391, 399.
GLUK, L.: 161, 181. HARDY, G. H.: 2, 3, 5, 6, 19, 53, 54,
GODLEVSKII, V. S.: 391, 399. 66, 70, 72, 79, 102, 103, 143, 144, 145,
GODUNOVA, E. K: 120, 139, 152, 146, 147, 148, 150, 153, 158, 177, 178,
153, 154, 180, 202, 214, 267, 274, 541, 179, 180, 187, 188, 189, 190, 192, 194,
570. 200, 203, 205, 207, 212, 213, 240, 247,
GOLAB,: 231. 257, 259, 262, 273, 285, 290, 295, 296,
GOLDBERG, S.: 12,56. 297, 298, 300, 302, 303, 307, 308, 310,
GOLDSTEIN, J. A.: 24,25,32,33,37, 311, 312, 313, 503, 518, 520, 533, 551,
59,61, 62. 569,571.
GOLLWITZER, H.: 370, 382, 394. HARLAMOV, P.: 356, 393.
GOODMAN, T. N. T.: 28,60. HARRIS, B. J.: 233,239.
GORNY, A.: 7, 54. HARTMAN, P. H.: 217, 218, 226, 228,
GOSHAL, A.: 425, 429. 235, 237, 511, 521.
GOSSELIN, R. P.: 539,569. HE, T. X.: 131, 140.
GRACE, S. R.: 334,347,434,435,460. HEADLEY, V.: 415,426.
GRAMMATIKOPOULIS, M. K: 345, HEDBERG, L. I.: 307, 312.
349. HElL, E.: 544, 570.
GRANDJOT, K: 143, 177.
GRAVES, R. E.: 555, 571.
INDEX 579
HEINIG, H. P.: 163, 164, 165, 166, IL'IN, V. P.: 201, 202, 213, 214, 533,
167, 168, 175, 177, 181, 186,304,312. 569.
HEJHOL, D. A.: 564. IMORU, C. 0.: 134, 141, 165, 173,
HELTON, B.: 360, 394. 181, 183, 288, 304, 310, 312, 391, 396.
HELTON, J. C.: 391, 397, 425, 429. INGHAM, A. E.: 192,213.
HEROD, J.: 360, 393. IONESCU, D. V.: 250, 253, 258, 493,
HEROLD, H.: 125, 140. 498.
HERZBERGER, J.: 460,463. IONESCU, N. M.: 406,425,429.
HEUER, G. A.: 550, 571. ISIHARA, A.: 257, 258.
HILBERT, D.: 96, 98, 112, 508, 512, IYENGAR, K. S. K.: 471,483.
514,521. IZUMI, M.: 158, 159, 180.
HILLE, E.: 14, 15, 28, 57, 315, 345, IZUMI, S. I.: 158, 159, 180.
353,354,392,418,427,453,454,462. JACKSON, D.: 523,524,568.
HINTON, D. B.: 39, 51, 63, 65, 175, JACKSON, L.: 345,350.
176, 183, 184, 186, 539, 570. JACOBI, C. G.: 68,507.
HO, T. K.: 361, 394. JAGERS, A. A.: 540.
HOA, N. T. T.: 53,65. JAKIMOVSKI, A.: 174,183.
HOBSON, E. W.: 558, 572. JAKUBOV, A.: 364,366,394.
HOCHSTADT, H.: 229,237. JAN, Tan Men: 345, 348.
HOHENEMSER, K.: 514. JANCUK, L. F.: 391, 397.
HOHRJAKOV, A. Ja.: 345,349. JANET, M.: 69, 70, 88, 98, 103, 107,
HOLBROOK, J. A. R.: 15,57. 175, 185.
HOLT, J. M.: 118, 132, 138. JANOUS, W.: 486, 492, 498.
HONG, Y.: 132,140. JENSEN, J. L. W. V.: 208.
HORGAN, C. 0.: 98, 112, 563, 572, JIANKANG, T.: 564,573.
573. JODEIT, Jr., Max: 260, 273.
HORMANDER, L.: 71, 104. JOHNSON, P. D.: 177,186.
HORST, H.: 96, 112. JONES, G. S.: 360, 393, 436, 445, 460.
HOU, M. S.: 129, 140. JONES, P. S.: 101, 113.
HRISTOVA, S. G.: 391,397,399,425, JU-DA, L.: 136, 141.
429,430. JURKAT, W. B.: 291, 292, 310.
HRYPTUN, V. G.: 89, 108. KADLEC, J.: 152,174,180.
HSIANG, F. C.: 201, 213, 539, 569. KAHANE, J. P.: 308,313.
HUA, L. K.: 118,138. KALF, H.: 68 .
. HUANG, D.: 52,65. KALLIO NIEMI, H.: 24, 59.
HUGHES, 1. M.: 275, 283. KALLMAN, R. R.: 13,57.
HUKUHARA, M.: 80, 105. KALUZA, Th.: 143,177.
HULL, T.: 437, 461. KAMKE, E.: 354,392,507,511,520,
HUNT, R. A.: 175,184,267,274. 554, 57l.
HURODZE, T. A.: 345,351. KAMZOLOV, A. 1.: 94,111.
580 INDEX
KANNAN, R.: 459,463. KRALJEVIC, H.: 14, 40, 45, 53, 57,
KAPER, H.: 51, 65. 64,65.
KARAMATA, J.: 525, 568. KRASNOSEL'SKIf, M. A.: 345,350.
KARLIN, S. J.: 252, 253, 258, 332, KRAWTCHOUK, M.: 524,568.
347, 561, 572. KRECMETOV, G. S.: 391, 397.
KARTSATOS, A. G.: 335, 347. KREIN, M. G.: 233,239.
KASCEEV, N. A.: 331,346. KREITH, K.: 93, 101, 110, 113.
KASATKINA, N. V.: 425, 430. KRILOFF (KRYLOV), N. M.: 68, 103,
KASATURE, D.: 425,430. 510,521.
KASTURE, D. Y.: 345, 348, 425, 43l. KRZYWICKI, A.: 85, 106, 175, 183.
KATO, T.: 15, 51, 57, 64. KUDRJAVCEV, L. D.: 53, 65.
KAUFFMAN, R. M.: 175, 184. KUFNER, A.: 94, 110, 152, 168, 172,
KECKIC, J. D.: 242, 257. 174, 175, 176, 180, 181, 182, 184, 186,
KELLNER, Richard: 559. 202,214.
KENNEBECK, D.: 360,393. KUPCOV, N. P.: 22,58.
KEOGH, F. R.: 308, 313. KUREPA, S.: 14, 57.
KERIMOV, Dz. S.: 425, 430. KURPEL', N. S.: 345, 350, 453, 462.
KHRISTOVA, S. G.: 425,428. KURTZ, D. S.: 304, 312.
KIGURADZE,1. T.: 334,347. KURTZ, T. G.: 27, 59.
KIM, W. J.: 75, 104, 175, 183, 227, KVAPIS, M.: 460, 464.
237. KWONG, M. K.: 30, 33, 37, 51, 61,
KINGMAN, J. F. C.: 276, 283, 284. 62,65, 233, 239, 567,573.
KIRSANOVA, G. V.: 24, 58. LACKOVIC, 1. B.: 15, 16, 57.
KJELLBERG, B.: 266, 274. LAHARIEV, A. J.: 391,396.
KLASI, M.: 360,393. LAKSHMIKANTHAM, V.: 315,345,
KLEFSJO, B.: 269, 274. 350, 353, 378, 381,391, 392, 397,460,
KLOOSTERMAN, H. D.: 9,55. 463,464.
KNESER, A.: 67, 102. LALLI, B. S.: 334,347,434,435,460.
KNOBLOCH, H. W.: 345, 350. LANDAU, E.: 1, 2, 3, 47, 53, 54, 143,
KNOPP, K.: 143, 145, 148, 178. 177.
KOHN, R.: 39,63. LANDAU, M.: 541,570.
KOKIASVILI, V. M.: 172, 175, 182. LANDBERG, L.: 88, 107.
KOLMOGOROV, A.: 7, 54, 305. LANGENHOP, C. E.: 365, 394.
KNOWLES, J. K.: 563, 572. LAPTINSKII, V. N.: 391, 397.
KOMAROFF, N.: 309, 313. LASALLE, J. P.: 363,394.
KONOVALOV, V. N.: 29, 33, 60, 61. LASOTA, A.: 223, 236.
KONSTANTINOV, M.: 391,397. LAVRENT'EV, M. A.: 240.
KONYUSKOV, A. A.: 150,179. LEE, Cheng-Ming: 123, 139.
KORN, A.: 562, 572. LEE, C. S.: 130, 140, 174, 183.
KOVACEC, A.: 294,311. LEE, Kin-Chun: 177, 186.
KRAFT, M.: 523. LEE, S. L.: 28, 60.
INDEX 581
LEELA, S.: 315, 345, 350, 353, 392, LIVINGSTON, A. E.: 201, 213.
460,463. LOCHER, P.: 494, 495, 499.
LEES, M.: 353, 392, 441, 461. LOEWNER, C.: 561, 572.
LEHMAN, A. L.: 296,311. LONDON, D.: 293,310.
LEHMER, D. H.: 308, 313. LORENTZ, G. G.: 291, 297, 308, 310,
LEIGHTON, W.: 93, 110, 229, 237, 313.
239. LOSONCZI, L.: 457, 458, 462.
LEINDLER, L.: 159, 160, 180, 181. LOSSERS, O. P.: 544,559,560,570.
LEPIN, A. Ya.: 34,62. LOVE, E. R.: 172, 174, 175, 182, 183,
LETTENMEYER, F.: 223, 236. 185.
LEVI, E. E.: 66,67,68,98, 102, 103. LU, W.: 345,348.
LUPA~, A.: 27, 59, 88, 107, 124, 139,
LEVIN, A. Ju.: 80,105,345,350.
LEVIN, A. Yu: 79, 105} 218, 223, 229, 222,247,258.
235, 236, 237. LUTTINGER, J. M.: 287, 298, 310,
LEVIN, V. 1.: 72, 86, 104, 106, 120, 31l.
139, 148, 149, 151, 153, 179, 192, 210, LUXEMBURG, W. A. J.: 90,308,312,
213, 231, 238, 263, 266, 267, 273, 274, 437, 461, 540, 570.
541,470. LUZIN, N. N.: 325,345,346,350.
LEVINE, H. A.: 93, 110. LYAPUNOV, A. M.: 216,235.
LEVINSON, N.: 114, 138, 150, 151, MADSEN, K.: 460,463.
179,507,511,521. MAGARIL-IL'JAEV, G. G.: 25, 33,
LEWIS, R. T.: 175,183,539,570. 59,6l.
LIANG, Z. J.: 131, 140. MAHAJANI, G. S.: 474, 484.
LIEB, E. H.: 203,214,287,298,310. MAHMUDOV, A. N.: 425,430.
LIEBERMAN, G. J.: 291,310. MAKAI, E.: 98, 113, 227, 237.
LIGUN, A. A.: 25, 36, 53, 59, 62, 65. MAKHMUDOV, A. P.: 460, 464.
LIN, C. S.: 39, 63. MALLOWS, C. L.: 114, 138.
LIN, C. T.: 133, 134, 135, 138, 141, MAMEDOV, Ja. D.: 318, 319, 320,
142, 391, 397, 425, 430. 332, 336, 337, 338, 339, 341, 345, 346,
LINENKO, V.: 460, 464. 350, 376, 377, 378, 395, 406, 407, 409,
LIU, W. B.: 391, 397. 410,413,416,417,426,450,453,460,
LIZORKIN, P. I.: 176, 186. 462, 463, 464, 465.
LJUBIC, Ju.: 10, 11, 15, 55. MAMEDOV, G. K.: 425,429.
LITTLEWOOD, J. E.: 2,3,5,6,19, MAMIKONYAN, F. U.: 391, 397.
53, 54, 66, 70, 72, 79, 102, 103, 143, MAMMANA, G.: 332, 346.
144, 147, 148, 150, 153, 178, 179, 188, MANDEL, S. P. H.: 275, 283.
189, 190, 192, 194, 200, 203, 212, 213, MANDELBROJT, S.: 89, 108.
240, 247, 257, 259, 285, 290, 295, 296, MANGERON, D.: 86,98,106,112.
297, 298, 300, 302, 303, 307, 308, 310, MANOUGIAN, M. N.: 391, 397.
311, 312, 313, 503, 518, 520, 533, 551, MAO, X. R.: 425, 430.
569,571. MARCINKIEWICZ,: 303.
582 INDEX
MARGOLIS, B.: 345,348. 475, 484, 499, 500, 511, 518, 520, 521,
MARONI, P. M.: 121, 139, 363, 394. 550,57l.
MARTELLI, M.: 227,237. MLAK, W.: 345,350.
MARTIN-REYES, F. J.: 175, 177, 185. MOHAPATRA, R N.: 172,173,175,
MARTYNJUK, A.: 354, 393. 177,182, 185, 186.
MARTYNYUK, A. A.: 345, 350, 460, MOORE, R A.: 175, 183.
463. MOORE, R E.: 460, 463.
MASANI, P.: 524, 568. MONTGOMERY, H. L.: 308,313,564,
MASOOD, M. S.: 425, 428. 565, 573.
MATTHEWS, K. R: 202, 214. MORAN, P. A. P.: 275,276,283.
MATORIN, A. P.: 10, 12, 55. MORDELL, L. J.: 2,53,485,486,498,
MAUKEEV, B. L: 545, 570. 546,570.
MAZ'YA, V. G.: 175, 184,218. MOROZOV, S. F.: 86, 106.
McKEE, S.: 460, 465. MORREY, Jr., C. B.: 96, 97, 112.
MElMAN, N.: 546,547, 57l. MOVLJANKULOV, H.: 358, 393.
MELENCOVA, Ju. A.: 92, 110. MUCKENHOUPT, B.: 157, 158, 180,
METCALF, F. T.: 83, 84, 106, 295, 304,312.
311, 359, 393. MULDOWNEY, J. S.: 91, 109, 335,
MEYERS, N. C.: 91, 109. 345, 347, 351, 391, 397.
MULHOLLAND, H. P.: 148, 178, 179,
MICHAEL, J. H.: 96,112.
188, 190, 191, 192, 212, 277, 284.
MIKHLIN, S. G.: 98, 112.
MULLER, W.: 11, 55, 84, 106.
MIKUSINSKI, L G.: 532,569.
MURDESHWAR, M.: 370, 373, 391,
MILLER, J. C. P.: 202,214.
394,396.
MILNE, W. E.: 526, 568.
MUSAEV, V. M.: 425, 430.
MILOVANOVIC, G. V.: 16,27,30,45, MUSIELAK, H.: 165, 181.
46, 57, 59, 60, 87, 107, 131, 133, 140, MYSKIS, A. D.: 345, 351.
141, 233, 239, 241, 244, 246, 247, 257,
258, 468, 472, 474, 475, 476, 481, 483, NADENIK, Z.: 94, 111.
484. NARSIMHA REDDY, K.: 391, 397.
MILOVANOVIC, L Z.: 30,60,87,107, NATANSON, L P.: 533,569.
131, 133, 140, 14l. NECAEV, L D.: 98, 101, 113, 126, 140.
MINC, H.: 292, 310. NEDER, L.: 2, 53.
MINGARELLI, A. B.: 446, 462. NEHARI, Z.: 75, 104, 175, 183, 185,
MIRANDA, M.: 83, 105. 226, 232, 239.
MIRSKY, L.: 486, 498. NEMETH, J.: 160, 173, 180.
MITRINOVIC, D. S.: 3,8, 17, 39,40, NERSESYAN, A. B.: 391,397.
45, 52, 63, 64, 88, 95, 107, 111, 136, NETA, B.: 30, 60.
137, 142, 170, 171, 177, 185, 186, 203, NEUMAN, E.: 52, 65, 257, 258.
207, 208, 214, 215, 243, 257, 269, 274, NEUMANN, E.: 257,258.
283, 284, 299, 311, 324, 332, 353, 392, NEVAI, G. P.: 566,573.
INDEX 583
NEWMAN, D. J.: 544. 393, 391, 395, 398, 406, 409, 417, 421,
NIKOL'SKIi, S. M.: 53, 65, 493, 499. 425, 426, 427, 428, 431, 435, 441, 442,
NIRENBERG, L.: 10, 39, 55, 63, 84, 444,446,460,461,462,465,466.
88, 106, 107. PAGANI, C. D.: 52.
NORMURADOV, H.: 425,430. PAK, S. A.: 345,35l.
NORTHCOTT, D. G.: 72, 104. PALEY, R. E. A. C.: 308,313.
NOVOTNA., J.: 87, 107,446,462. PARTINGTON, J. R.: 51,64.
NOVRUZOV, G. M.: 450. PEANO, G.: 316, 346, 354, 392.
NUIJ, W.: 547,548. PEASLEE, D. C.: 165, 18l.
NUMIROW, T.: 460, 465. PECARI<\ J. E.: 27, 35, 36, 38, 39,
NURIMOV, T.: 402, 403, 405, 406, 40, 45, 52, 53, 59, 62, 63, 64, 65, 95,
425, 426, 427, 430. lll, 136, 137, 142,173, 177, 182, 186,
202, 203, 207, 214, 215, 244, 246, 252,
OBRECHKOFF, N.: 541, 570.
256, 257, 258, 261, 269, 273, 274, 277,
OHRONCUK, V. I.: 345, 350, 35L 278, 281, 283, 284, 288, 289, 291, 299,
OLECH, C.: ll4, 138, 425, 430. 310, 311, 332, 347, 369, 394, 468, 470,
OL'HOVSKIi, Yu. G.: 555,572. 472, 474, 475, 476, 481, 483, 484, 550,
OLOVYANISNIKOV, V. M.: ll, 12, 57l.
35,55. PECH, Pavel: 425, 432.
O'NEIL, R.: 267, 274, 306, 308, 312, PEDERSEN, R. N.: 138.
313. PEETRE, J.: 89,108.
ONESTI, N. B.: 425,430. PELCZAR, A.: 460,466.
OPREA, A.: 86, 106. PENG, G.: 425, 43l.
OPIAL, Z.: ll4, 138, 221, 223, 228, PERESTYUK, N. A.: 391, 399.
235, 236, 416, 426. PEROV, A. I.: 320,337,338,346,347,
OPIC, B.: 176, 186. 360, 394,460,464, 466.
OPOfCEV, V. I.: 345,35L PETERSEN, G. M.: 158, 159, 173,
OPPENHEIM, A.: 294,31L 180.
ORNELAS, A.: 425, 430. PETROV, B. N.: 345,35l.
OSSICINI, A.: 493, 494, 498. PETROVITCH, M.: 316, 346.
OSTASZEWSKI, K.: 391,397. PFEFFER, A. M.: 12, 55, 84, 87, 106.
OSTROGORSKI, T.: 175, 185. PH6NG, V. Q.: 33, 6l.
OSTROWSKI, A. M.: 8, 54, 143, 178, PICARD, E.: 67,102.
468, 483, 529, 533, 535, 536, 537, 552,
PICHORIDES, S. K.: 305, 312.
568,569.
PICONE, M.: 68, 103.
OWEN, P. M.: 188,212.
PIGOLKIN, G. M.: 267,274.
OZEKI, N.: 95, lll.
PINTER, L.: 391,397.
PACHPATTE, B. G.: 38, 63, 95, 96, PITT, H. R.: 8,54,84, 106, 194, 213.
101, lll, ll2, 134, 136, 137, 138, 141, PITTENGER, A. P.: 175, 185.
142, 174, 175, 177, 183, 185, 186, 342, PLEIJEL, A.: 69, 103.
344, 345, 347, 348, 357, 383, 386, 387, PLOTNIKOV, V. I.: 86, 106.
584 INDEX
POINCARE, H.: 67, 96, 98, 102, 112. RIESZ, F.: 188,212.
POLYA, G.: 3,5,6, 19,54,66,70,98, RIESZ, M.: 187, 305, 542, 570.
102, 113, 126, 140, 143, 144, 148, 177, RINOTT, Y.: 291, 292, 293, 310.
178, 188, 189, 190, 192, 200, 203, 212, RIPIANU, D.: 86,221,235.
240, 247, 257, 259, 273, 285, 290, 295, RISHEL, R.: 82, 105.
296, 297, 298, 300, 302, 310, 311, 332, RODOV, A. M.: 9, 24, 55.
346, 503, 518, 520. ROGERS, T.: 369, 391, 394, 399.
POPENDA, J.: 425, 428, 444, 461, RONVEAUX, A.: 553, 571.
466. ROSEN, G.: 88,91,94, 108, 109, 110.
POPOV, V.: 534, 569. ROSS, S. M.: 291, 310.
PORTNOY, V. R.: 175, 184. ROTA, G. C.: 13, 57.
PROTTER, M.: 51, 64. ROZANOVA, G. 1.: 125, 129, 140.
PUTNAM, C. R.: 175,184. RUDERMAN, H. D.: 290, 294, 310.
RUDNICK, K.: 155, 161, 180.
QI, Z.: 134, 14l. RUSSELL, A.: 24,58.
QUADE, W.: 325, 346. RUSSELL, A. M.: 95,111.
RAB, M.: 345, 348. RUSSELL, D. C.: 173, 175, 182, 185.
RABCZUK, R.: 321, 325, 327, 332, RYBARSKI, A.: 85, 89, 94, 96, 106,
346, 433, 435, 460. 108, 110, 160, 175, 181, 183, 184, 505,
RABINOWITZ, B.: 493, 498. 520.
RADEMACHER, H.: 523, 568. RYFF, J. V.: 121,298,311.
RADISIN, Z.: 331. RYSER, H. J.: 291,292,310.
RADZISZEWSKI, B.: 345, 350. SABIROV, T.: 320,346.
RAFAL'SON, S. Z.: 36,62,526,568. SADIKOVA, R. H.: 87,107.
RAGHAVENDRA,: 391,399. SADOVSKII, B. N.: 345,351.
RAHMAN, Q. 1.: 552,571. SADRIN, G. A.: 89, 108.
RAHMATULLINA, L. F.: 345, 351. ST. MARY, D. F.: 229, 231, 237, 238,
RAKOVICH, B. D.: 486,487,490,491, 239.
492, 496, 498. SAITAH, S.: 95, 113.
RALL, L. L.: 460, 463. SAITHOH, S.: 309, 313.
RASMUSSEN, D.: 425, 431. SALPAGAROV, H. M.: 383, 390,391,
RASSIAS, Th. M.: 96,97,112. 395, 405, 425.
REDHEFFER, R. M.: 11, 29, 51, 55, SANSONE, G.: 218, 228, 231, 237.
60, 65, 84, 85, 88, 106, 107, 121, 124, SARDAR'LY,S. M.: 359, 393, 460,
139, 175, 184, 202, 203, 214, 425, 431, 465.
435,460. SAROVA, L.: 354,355,357,370,383,
REID, W.: 315, 345, 353, 354, 392, 392, 437, 46l.
507, 511, 520. SATAKE, 1.: 51, 64..
RHOADES, B. E.: 174, 183. SATO, M.: 34, 62.
RICHARD, U.: 88, 107, 223, 236. SATO, R.: 34,62.
RICE, N. M.: 308, 313.
INDEX 585