ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA

Volume 20, Number 1, June 2016

Available online at http://acutm.math.ut.ee

Some generalizations of the Eneström–Kakeya

theorem

Eze R. Nwaeze

Abstract. Let p(z) = a0 + a1 z + a2 z 2 + · · · + an z n be a polynomial of

degree n, where the coefficients aj , j = 0, 1, 2, . . . , n, are real numbers.
We impose some restriction on the coefficients and then prove some
extensions and generalizations of the Eneström–Kakeya theorem.

1. Introduction
A classical result due to Eneström [5] and Kakeya [7] concerning the
bounds for the moduli of zeros of polynomials having positive coefficients
is often stated as follows.
Xn
Theorem A. Let p(z) = aj z j be a polynomial with real coefficients
j=0
satisfying
0 < a0 ≤ a1 ≤ a2 ≤ · · · ≤ an .
Then all the zeros of p(z) lie in |z| ≤ 1.
In the literature there exist several extensions and generalizations of this
result (see [1], [2], [6] and [8]). Joyal et al. [6] extended Theorem A to the
polynomials whose coefficients are monotonic but not necessarily nonnega-
tive. In fact, they proved the following result.
n
X
Theorem B. Let p(z) = aj z j be a polynomial of degree n, with real
j=0
coefficients satisfying
a0 ≤ a1 ≤ a2 ≤ · · · ≤ an .

Received January 8, 2015.

2010 Mathematics Subject Classification. 30C10, 30C15.
Key words and phrases. Real polynomials, location of zeros, MATLAB.
http://dx.doi.org/10.12697/ACUTM.2016.20.02
15
16 EZE R. NWAEZE

Then all the zeros of p(z) lie in the disk

1
|z| ≤ (an − a0 + |a0 |).
|an |
Aziz and Zargar [3] relaxed the hypothesis in several ways and among
other things proved the following result.
n
X
Theorem C. Let p(z) = aj z j be a polynomial of degree n such that
j=0
for some k ≥ 1,
0 < a0 ≤ a1 ≤ a2 ≤ · · · ≤ kan .
Then all the zeros of p(z) lie in the disk
|z + k − 1| ≤ k.
In 2012, they further generalized Theorem C which is an interesting ex-
tension of Theorem A. In particular, the following theorems are proved in
[4].
n
X
Theorem D. Let p(z) = aj z j be a polynomial of degree n. If for some
j=0
positive numbers k and ρ with k ≥ 1, 0 < ρ ≤ 1,
0 ≤ ρa0 ≤ a1 ≤ a2 ≤ · · · ≤ kan ,
then all the zeros of p(z) lie in the disk
2a0
|z + k − 1| ≤ k + (1 − ρ).
an
n
X
Theorem E. Let p(z) = aj z j be a polynomial of degree n. If for
j=0
some positive number ρ, 0 < ρ ≤ 1, and for some nonnegative integer λ,
0 ≤ λ < n,
ρa0 ≤ a1 ≤ a2 ≤ · · · ≤ aλ−1 ≤ aλ ≥ aλ+1 ≥ · · · ≥ an−1 ≥ an ,
then all the zeros of p(z) lie in the disk
an−1 1 h i
z+ −1 ≤ 2aλ − an−1 + (2 − ρ)|a0 | − ρa0 .
an |an |
.
Looking at Theorem D, one might want to know what happens if ρa0 is
NOT nonnegative. In this paper we prove some extensions and generaliza-
tions of Theorems D and E which in turn give an answer to our enquiry.
 SOME GENERALIZATIONS OF THE ENESTRÖM–KAKEYA THEOREM 17

2. Main results
n
X
Theorem 1. Let p(z) = aj z j be a polynomial of degree n. If for some
j=0
real numbers α and β,
a0 − β ≤ a1 ≤ a2 ≤ · · · ≤ an + α,
then all the zeros of p(z) lie in the disk
α 1 h i
z+ ≤ an + α − a0 + β + |β| + |a0 | .
an |an |
If α = (k − 1)an and β = (1 − ρ)a0 with k ≥ 1, 0 < ρ ≤ 1, then we get
the following corollary.
n
X
Corollary 1. Let p(z) = aj z j be a polynomial of degree n. If for some
j=0
postive numbers k ≥ 1 and ρ with 0 < ρ ≤ 1,
ρa0 ≤ a1 ≤ a2 ≤ · · · ≤ kan ,
then all the zeros of p(z) lie in the disk
1 h i
|z + k − 1| ≤ (kan − ρa0 ) + |a0 |(2 − ρ) .
|an |
If a0 > 0, then Corollary 1 amounts to Theorem D.
n
X
Theorem 2. Let p(z) = aj z j be a polynomial of degree n. If for some
j=0
real number s and for some integer λ, 0 < λ < n,
a0 − s ≤ a1 ≤ a2 ≤ · · · ≤ aλ−1 ≤ aλ ≥ aλ+1 ≥ · · · ≥ an−1 ≥ an ,
then all the zeros of p(z) lie in the disk
an−1 1 h i
z+ −1 ≤ 2aλ − an−1 + s − a0 + |s| + |a0 | .
an |an |
If we take s = (1 − ρ)a0 , with 0 < ρ ≤ 1, then Theorem 2 becomes
Theorem E. Instead of proving Theorem 2, we shall prove a more general
case. In fact, we prove the following result.
n
X
Theorem 3. Let p(z) = aj z j be a polynomial of degree n. If for some
j=0
real numbers t, s and for some integer λ, 0 < λ < n,
a0 − s ≤ a1 ≤ a2 ≤ · · · ≤ aλ−1 ≤ aλ ≥ aλ+1 ≥ · · · ≥ an−1 ≥ an + t,
18 EZE R. NWAEZE

then all the zeros of p(z) lie in the disk

an−1  t  1 h i
z+ − 1+ ≤ 2aλ − an−1 + s − a0 + |s| + |a0 | + |t| .
an an |an |

3. Proofs of the theorems

Proof of Theorem 1. Consider the polynomial
g(z) = (1 − z)p(z)
= −an z n+1 + (an − an−1 )z n + (an−1 − an−2 )z n−1 + · · · + (a1 −a0 )z + a0
= −an z n+1 − αz n + (an + α − an−1 )z n + (an−1 − an−2 )z n−1 + . . .
+ (a1 − a0 + β)z − βz + a0
= −z n (an z + α) + (an + α − an−1 )z n + (an−1 − an−2 )z n−1 + . . .
+ (a1 − a0 + β)z − βz + a0
= −z n (an z + α) + φ(z),
where
φ(z) = (an + α − an−1 )z n + (an−1 − an−2 )z n−1 + · · · + (a1 − a0 + β)z − βz + a0 .
Now for |z| = 1, we have
|φ(z)| ≤ |an + α − an−1 | + |an−1 − an−2 | + · · · + |a1 − a0 + β| + |β| + |a0 |
= an + α − an−1 + an−1 − an−2 + · · · + a1 − a0 + β + |β| + |a0 |
= an + α − a0 + β + |β| + |a0 |.
Since this is true for all complex numbers with a unit modulus, it must also
be true for 1/z. With this in mind, we have, for all z with |z| = 1,
|z n φ(1/z)| ≤ an + α − a0 + β + |β| + |a0 |. (1)
Also, the function Φ(z) = z n φ(1/z)
is analytic in |z| ≤ 1, hence, inequality
(1) holds inside the unit circle by the maximum modulus theorem. That is,
for all z with |z| ≤ 1,
an + α − a0 + β + |β| + |a0 |
|φ(1/z)| ≤ .
|z|n
Replacing z by 1/z, we get
h i
|φ(z)| ≤ an + α − a0 + β + |β| + |a0 | |z|n
if |z| ≥ 1. Now, for |z| ≥ 1, we obtain that
|g(z)| = | − z n (an z + α) + φ(z)|
≥ |z n ||an z + α| − |φ(z)|
h i
≥ |z n ||an z + α| − an + α − a0 + β + |β| + |a0 | |z|n
 SOME GENERALIZATIONS OF THE ENESTRÖM–KAKEYA THEOREM 19
 h i
= |z n | |an z + α| − an + α − a0 + β + |β| + |a0 |
>0
if and only if
h i
|an z + α| > an + α − a0 + β + |β| + |a0 |

or, equivalently, if and only if

α 1 h i
z+ > an + α − a0 + β + |β| + |a0 | .
an |an |
Thus, all the zeros of g(z) whose modulus is greater than or equal to 1 lie in
α 1 h i
z+ ≤ an + α − a0 + β + |β| + |a0 | . (2)
an |an |
But those zeros of p(z) whose modulus is less than 1 already satisfy (2),
because |φ(z)| ≤ an + α − a0 + β + |β| + |a0 | for |z| = 1 and φ(z) = g(z) +
z n (an z + α). Also, all the zeros of p(z) are zeros of g(z). That completes
the proof of Theorem 1. 
Proof of Theorem 3. Consider the polynomial
g(z) = (1 − z)p(z)
= −an z n+1 + (an − an−1 )z n + (an−1 − an−2 )z n−1 + · · · + (a1 − a0 )z + a0
= −an z n+1 + (an − an−1 )z n + (an−1 − an−2 )z n−1 + . . .
+ (aλ+1 − aλ )z λ+1 + (aλ − aλ−1 )z λ + · · · + (a1 − a0 )z + a0
= −z n [an z − an + an−1 − t] − tz n + (an−1 − an−2 )z n−1 + . . .
+ (aλ+1 − aλ )z λ+1 + (aλ − aλ−1 )z λ + · · · + (a1 − a0 + s)z − sz + a0
= −z n [an z − an + an−1 − t] + ψ(z),
where
ψ(z) = −tz n + (an−1 − an−2 )z n−1 + · · · + (aλ+1 − aλ )z λ+1
+ (aλ − aλ−1 )z λ + · · · + (a1 − a0 + s)z − sz + a0 .
For |z| = 1 we get
|ψ(z)| ≤ |t| + |an−1 − an−2 | + · · · + |aλ+1 − aλ | + |aλ − aλ−1 | + . . .
+ |a1 − a0 + s| + |s| + |a0 |
= |t| + an−2 − an−1 + · · · + aλ − aλ+1 + aλ − aλ−1 + . . .
+ a1 − a0 + s + |s| + |a0 |
= |t| − an−1 + 2aλ − a0 + s + |s| + |a0 |.
20 EZE R. NWAEZE

It is clear that
|z n ψ(1/z)| ≤ |t| − an−1 + 2aλ − a0 + s + |s| + |a0 | (3)
on the unit circle. Since the function Ψ(z) = z n ψ(1/z)
is analytic in |z| ≤ 1,
inequality (3) holds inside the unit circle by the maximum modulus theorem.
That is,
|t| − an−1 + 2aλ − a0 + s + |s| + |a0 |
|ψ(1/z)| ≤
|z|n
for |z| ≤ 1. Replacing z by 1/z we get
h i
|ψ(z)| ≤ |t| − an−1 + 2aλ − a0 + s + |s| + |a0 | |z|n
for |z| ≥ 1. Now for |z| ≥ 1, we have
|g(z)| ≥ |z n ||an z − an + an−1 − t| − |ψ(z)|
≥ |z n ||an z − an + an−1 − t|
h i
− |t| − an−1 + 2aλ − a0 + s + |s| + |a0 | |z|n
 h i
= |z n | |an z− an + an−1 − t| − |t| − an−1 + 2aλ − a0 + s + |s| + |a0 |
>0
if and only if
h i
|an z − an + an−1 − t| > |t| − an−1 + 2aλ − a0 + s + |s| + |a0 | .
But this holds if and only if
an−1  t  1 h i
z+ − 1+ > |t| − an−1 + 2aλ − a0 + s + |s| + |a0 | .
an an |an |
Hence, the zeros of p(z) with modulus greater or equal to 1 are in the
closed disk
an−1  t  1 h i
z+ − 1+ ≤ |t| − an−1 + 2aλ − a0 + s + |s| + |a0 | .
an an |an |
Also, those zeros of p(z) whose modulus is less than 1 already satisfy the
above inequality since ψ(z) = g(z) + z n [an z − an + an−1 − t] and, for |z| = 1,
|ψ(z)| ≤ |t| − an−1 + 2aλ − a0 + s + |s| + |a0 |. That completes the proof. 

4. Demonstrating examples
Example 1. Let us consider the polynomial
p(z) = 3z 5 + 4z 4 + 3z 3 + 2z 2 + z − 1.
The coefficients here are a5 = 3, a4 = 4, a3 = 3, a2 = 2, a1 = 1 and a0 = −1.
We cannot apply Theorems A, B, C and D. But we can apply Theorem 1
to determine where all the zeros of the polynomial lie. Using MATLAB, we
obtain the following zeros : −0.9154 + 0.4962i, −0.9154 − 0.4962i, 0.0530 +
 SOME GENERALIZATIONS OF THE ENESTRÖM–KAKEYA THEOREM 21

0.8845i, 0.0530 − 0.8845i, 0.3916. Taking α = 2 and β = 0, Theorem 1 gives

that all the zeros of the polynomial lie in the closed disk |3z + 2| ≤ 7.
Example 2. Next, consider
q(z) = −z 6 + 2z 5 + 2z 4 + 3z 3 + z 2 − 2.
The coefficients of q(z) are a6 = −1, a5 = 2, a4 = 2, a3 = 3, a2 = 1,
a1 = 0 and a0 = −2. Using MATLAB, we obtain the following zeros:
3.0197, −0.7682+0.5814i, −0.7682−0.5814i, −0.0803+1.0233i, −0.0803−
1.0233i, 0.6773. Taking λ = 3, t = 1 and s = 0, Theorem 3 gives that the
zeros lie in |z − 2| ≤ 9.

5. Acknowledgement
The author is greatly indebted to the referee for his/her several useful
suggestions and valuable comments.

References
[1] N. Anderson, E. B. Saff, and R. S. Varga, On the Eneström-Kakeya theorem and its
sharpness, Linear Algebra Appl. 28 (1979), 5–16.
[2] N. Anderson, E. B. Saff, and R. S. Varga, An extension of the Eneström-Kakeya theo-
rem and its sharpness, SIAM J. Math. Anal. 12 (1981), 10–22.
[3] A. Aziz and B. A. Zargar, Some extensions of Eneström-Kakeya theorem, Glas. Mat.
Ser. III 31 (51) (1996), 239–244.
[4] A. Aziz and B. A. Zargar, Bounds for the zeros of a polynomial with restricted coeffi-
cients, Appl. Math. (Irvine) 3 (2012), 30–33.
[5] G. Eneström, Härledning af en allmän formel för antalet pensionärer, som vid en
godtycklig tidpunkt förefinnas inom en sluten pensionskassa, Stockh. Öfv. L. 6 (1893),
405–415. (Swedish)
[6] A. Joyal, G. Labelle, and Q. I. Rahman, On the location of zeros of a polynomial,
Canad. Math. Bull. 10 (1967), 55–63.
[7] S. Kakeya, On the limits of the roots of an algebraic equation with positive coefficients,
Tôhoku Math. J. 2 (1912), 140–142.
[8] M. Kovac̆ević and I. Milovanović, On a generalization of the Eneström-Kakeya theorem,
Pure Math. Appl. Ser. A 3 (1992), 43–47.

Department of Mathematics and Statistics, Auburn University, Auburn, AL

36849, USA
E-mail address: ern0002@auburn.edu