International Journal of Mathematics and Statistics Invention (IJMSI)

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 4767 || P-ISSN: 2321 - 4759

www.ijmsi.org || Volume 2 || Issue 2 || February - 2014 || PP-28-32
www.ijmsi.org 28 | P a g e
A Short Note On The Order Of a Meromorphic Matrix Valued
Function.
1
Subhas S. Bhoosnurmath,
2
K.S.L.N.Prasad
1
Department of Mathematics, Karnatak University, Dharwad-580003-INDIA
2
Associate Professor, Department Of Mathematics, Karnatak Arts College, Dharwad-580001, INDIA

ABSTRCT: In this paper we have compared the orders of two meromorphic matrix valued functions A(z) and
B(z) whose elements satisfy a similar condition as in Nevanlinna-Polya theorem on a complex domain D.

KEYWORDS: Nevanlinna theory, Matrix valued meromorphic functions.

Preliminaries: We define a meromorphic matrix valued function as in [2].
By a matrix valued meromorphic function A(z) we mean a matrix all of whose entries are meromorphic on the
whole (finite) complex plane.
A complex number z is called a pole of A(z) if it is a pole of one of the entries of A(z), and z is called a zero of
A(z) if it is a pole of ( ) | |
1
z A

.
For a meromorphic m m matrix valued function A(z),
let ( ) ( )
}
t
u
u
t
=
2
0
i
d re A log
2
1
A , r m (1)
where A has no poles on the circle r z = .
Here, ( ) ( ) x z A
C x
1 x
Max z A
n
e
=
=
Set ( )
( )
dt
t
A , t n
A , r N
r
0
}
= (2)
where n(t, A) denotes the number of poles of A in the disk { } t z : z s , counting multiplicities.
Let T(r, A) = m(r, A) + N(r, A)
The order of A is defined by
( )
r log
A , r T log
sup lim
r
=
We wish to prove the following result
Theorem: Let
(
=
) z ( f
) z ( f
A
2
1
and
(
=
) z ( g
) z ( g
B
2
1
be two meromorphic matrix valued functions. If f
k
and g
k

(k=1, 2) satisfy

2
2
2
1
2
2
2
1
) z ( g ) z ( g ) z ( f ) z ( f + = + (1)
on a complex domain D, the n
B A
= , where
A
and
B
are the orders of A and B respectively.
A Short Note on The Order of a Meromorphic Matrix Valued Function
We use the following lemmas to prove our result.
Lemma 1[3] [ The Nevanlinna Polya theorem]
Let n be an arbitrary fixed positive integer and for each k (k = 1, 2, n) let f
k
and g
k
be analytic
functions of a complex variable z on a non empty domain D.
If f
k
and g
k
(k = 1,2,n) satisfy

2
k
n
1 k
2
k
n
1 k
) z ( g ) z ( f

= =
=
on D and if f
1
,f
2
,f
n
are linearly independent on D, then there exists an n n unitary matrix C, where each of
the entries of C is a complex constant such that
(
(
(
(
(
=
(
(
(
(
(

) z ( f
.
) z ( f
) z ( f
C
) z ( g
) z ( g
) z ( g

n
2
1
n
2
1

holds on D.
Lemma 2 Let f
k
and g
k
be as defined in our theorem (k = 1,2). Then there exists a 2 2 unitary matrix C where
each of the entries of C is a complex constant such that B = CA
(2)
where A and B are as defined in the theorem.
Proof of Lemma 2
We consider the following two cases.
Case A: If f
1
, f
2
are linearly independent on D, then the proof follows from the Nevanlinna Polya theorem.
Case B: If f
1
and f
2
are linearly dependent on D, then there exists two complex constants c
1
, c
2
, not both zero
such that
c
1
f
1
(z) + c
2
f
2
(z) = 0 (3)

We discuss two subcases
Case B
1
: If c
2
= 0, then by (3) we get ) z ( f
c
c
) z ( f
1
2
1
2
= (4)
holds on D.
If we set b = ,
c
c
2
1
then by (4) we have f

2
(z) = b f
1
(z) on D. (5)
Hence (1) takes the form

2
2
2
1
2
1
2
(z) g (z) g (z) f ) b 1 ( + = + . (6)
We may assure that f
1
/ 0 on D. Otherwise the proof is trivial.
Hence by (6), we get

2
2
1
2
2
1
1
b 1
) z ( f
) z ( g
) z ( f
) z ( g
+ = + (7)
Taking the Laplacians
2
2
2
2
z x c
c
+
c
c
= A of both sides of (7) with respect to z = x + iy (x, y real),
we get
0
) z ( f
) z ( g

) z ( f
) z ( g

2
1
2
2
1
1
=
'
|
|
.
|
\
|
+
'
|
|
.
|
\
|
(8)
Since
2 2
(z) P 4 P(z) ' = A , [14] where P is an analytic function of z, by (8), we get
0
) z ( f
) z ( g
, 0
) z ( f
) z ( g
1
2
1
1
=
'
|
|
.
|
\
|
=
'
|
|
.
|
\
|

Hence, g
1
(z) = c f
1
(z) and g
2
(z) = d f
1
(z) (9)
where c, d are complex constants.
Substituting (9) in (7), we get
2 2 2
b 1 d c + = + (10)
Let us define

(
(
(
(
(
(
(
+ +
+
+
=
b 1
1
b 1
b
b 1
b

b 1
1
: U
2 2
2 2
(11)
and
(
(
(
(
(
(
(
+ +
+
+
=
b 1
c
b 1
d
b 1
d

b 1
c
: V
2 2
2 2
(12)
Then, it is easy to prove that

(
=
(
(
+
b
1

0
b 1
U
2
(13)
and
(
=
(
(
+
d
c

0
b 1
V
2
(14)
Set C = V U
-1
(15)

Since all 2 2 unitarly matrices form a group under the standard multiplication of matrices, by (15), C is a 2 2
unitary matrix.
Now, by (13), we have
(
(
+
=
|
|
.
|
\
|
0
b 1
b
1
U
2
1
(16)
Thus, we have on D,

b
1
C ) z ( f
) z ( f
) z ( f
C
1
2
1
|
|
.
|
\
|
=
(
, by (5)

|
|
.
|
\
|
|
|
.
|
\
|
=
b
1
U V ) z ( f
1 -
1
, by (15)

|
|
.
|
\
|
+
=
0
b 1
V ) z ( f
2
1
, by (16)

|
|
.
|
\
|
=
d
c
) z ( f
1
, by (14)

(
=
) z ( g
) z ( g
2
1
, by (9)
Hence, the result.
Case B
2
. Let c
2
= 0 and c
1
= 0.
Then by (3), we get f
1
= 0.
Hence, by (1),
2
2
2
1
2
2
(z) g (z) g (z) f + = (17)
holds on D.
By (17) and by a similar argument as in case B
1
, we get the result.
Proof of the theorem
By Lemma 2, we have B = CA where A and B are as defined in the theorem.
Therefore, T(r, B) = T(r, CA)
using the basics of Nevanlinna theory[2], we can show that,
T(r, B) s T(r, A) as T(r,C)=o{T(r,f}
On further simpler simplifications, we get

A B
s
.
(18)
By inter changing f
k
and g
k
(k=1, 2) in Lemma 2,
we get A =CB , which implies
T(r, A) s T(r, B) and hence
B A
s (19)
By (18) and (19), we have
B A
=
Hence the result.

REFERRENCES
[1.] C. L. PRATHER and A.C.M. RAN (1987) : Factorization of a class of mermorphic matrix valued functions, Jl. of Math.
Analysis, 127, 413-422.
[2.] HAYMAN W. K. (1964) : Meromorphic functions, Oxford Univ. Press, London.
[3.] HIROSHI HARUKI (1996) : A remark on the Nevanlinna-polya theorem in alanytic function theory, Jl. of Math. Ana. and Appl.
200, 382-387.

International Journal of Mathematics and Statistics Invention (IJMSI)

Uploaded by

Copyright:

Available Formats

International Journal of Mathematics and Statistics Invention (IJMSI)

Uploaded by

Document Information

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

International Journal of Mathematics and Statistics Invention (IJMSI)

Uploaded by

Copyright:

Available Formats

International Journal of Mathematics and Statistics Invention (IJMSI)

E-ISSN: 2321 4767 || P-ISSN: 2321 - 4759

then by (4) we have f

You might also like