IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 10, Issue 2 Ver. I (Mar-Apr. 2014), PP 56-59 www.iosrjournals.org

Some Results Concerning Memomorphic Matrix Valued Functions. 1

1

Department of Mathematics, Karnatak University, Dharwad-580003-INDIA Associate Professor, Department of Mathematics, Karnatak Arts College, Dharwad-580001, INDIA

2

Abstract: In this paper we have extended some basic results of Nevanlinna theory to

Matrix valued

meromorphic functions. Key words:Nevanlinna theory,Matrix valued meromorphic functions.

I.

Preliminaries:

By a meromorphic function we shall always mean a transcendental meromorphic function in the plane.

a  C and r > 0, we use the following notations of frequent use in Nevanlinna 1   1   theory with their usual meaning: m(r, a , f )  m r  , n (r, a , f )  n r, ,  f a   f a  n(r, a, f ), N(r, a, f), N(r, a, f), T(r, f) ,  (a, f ), (a, f ), (a, f ) etc..as in [2] If f is a meromorphic function,

As usual, if a = , then by a zero of f-a, we mean a pole of f. We define a meromorphic matrix valued function as in [1]. 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

  1

A(z) if it is a pole of A z . For a meromorphic m  m matrix valued function A(z), let mr, A  

2

 

1 log A re i  2 0

d

(1)

where A has no poles on the circle z  r .

Here,

Az   M ax x 1

Az x

x  Cn n t, A dt t 0 r

Set

Nr, A  

(2)

where n(t, A) denotes the number of poles of A in the disk z : z  t , counting multiplicities. Let T(r, A) = m(r, A) + N(r, A) Definition : Let A = Let

And

a 

m ij i , j1

be a memomorphic matrix valued function of finite order. Let a

 C.

m( r , a ) N( r , a )  lim r  T ( r , a ) r  T ( r , a ) ij ij

(a ij )  (a , a ij )  lim

(a ij )  (a , a ij )  1  lim

r 

N( r , a ) T(r, a ij ) www.iosrjournals.org

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Some Results Concerning Memomorphic Matrix Valued Functions. Also, Let

(a, A)  Max (a ij )  Max (a, a ij )

and

(a, A)  Max (a ij )  Max (a, a ij )

1i , j m

1i , j m

1i , j m

1i , j m

We wish to prove the following results. Theorem 1 : Let A =

a 

distinct numbers, then

1  i, j  m.

q

 m(r, a k 1

m ij i , j1

be a meromorphic matrix valued functions of finite order. If a1, a2, a3, ….. a q are

 1 a )  N r,   T(r, a ij' )  S(r, a k , a ij )  a   ij 

k , ij

Where S(r, ak, aij)=S(r, aij )= 0 { T(r, aij) } r

a 

m ij i , j1

Theorem 2 :Let A =

  through all values of aij.

be a meromorphic matrix valued function of finite order. Then,

T(r, A ' )   (a , A) r  T ( r , A ) aC

lim

Theorem 3 Let A =

a 

m ij i , j1

be meromorphoic matrix valued function of finite order.

T(r, A ' ) Then, lim  2  (, A) r  T ( r , A ) Proof of theorem 1: Without loss of generality, let us assume that for q  2, q

F(z) =

a k 1

1 , 1 < i, j < m. ij  a k

Then by a known result [Hayman, p.33], we have q

M(r, F) >

 m( r , a k 1

q

Therefore,

 m( r , a k 1

a )  q log 

k , ij

3q  log 2 , where   min a k1  a k 2 k1  k 2 

a )  m(r, F)  O(1)

k , ij

 1   m r, ' .F.a ij'   O(1)  a   ij   1  m r, '   m(r, F.a ij' )  O(1)  a   ij   1  q a ij'    O(1)  m r, '   m r,   a   k 1 a a  ij ij  k     ' q  1  a ij    O(1)  m r, '    m r,  a  k 1  a  a  ij ij k      1 = m r, '   S(r, a ij )  a   ij  www.iosrjournals.org

57 | Page

Some Results Concerning Memomorphic Matrix Valued Functions.

 1  1 a )  N r, '   T r, '   S(r, a ij )  a   a   ij   ij   1  T r, '   S(r, a ij )  a   ij 

q

Thus,

 m( r , a

k , ij

k 1

Hence, the result. Proof of theorem 2: Let

{a ij}i1 be an infinite sequence of distinct elements of C which includes for each a  C,  (a, aij) > 0. q

Then,

 (a a

i , ij

ac

)   (a ,a ij ) ac

Let q be a positive integer. Then by previous theorem, q

 m(r, a k 1

 1 a )  N r,   T(r, a ij' )  S(r, a k , a ij )  a   ij 

k , ij

q

Adding  N ( r , a k , aij) both sides, we get k 1

q  1   1 '   T  T ( r , a )  N(r, a k , a ij )  N r, '   S(r, a ij )   ij    a  k 1  a ij  a k  k 1  ij  q  1 ' = T(r, a ij )   N(r, a k , a ij )  N 0  r, '   S(r, a ij )  a  k 1  ij   1 ' where N 0  r, '  is formed with the zeros of a ij which are not zeros of any of aij – ak.  a   ij   1 Since N 0  r, '  > 0, We have  a   ij  q q  1  '  (*) T r ,  T ( r , a )  N(r, a k , a ij )  S(r, a ij )   ij  a a  k 1  k 1 ij k   1  Now, T r,  T(r, aij) + O (Log r)  a a  ij k   = T(r, aij) + o{T(r, aij)} as r   q

Thus (*) takes the form q

' ij

q T(r, aij) < T(r, a ) +

Hence, q<

Therefore,

lim

r 

T(r, a ij' ) T(r, a ij )

q<

lim

r 

 N(r, a k 1

q

+

 lim k 1

T(r, a ij' ) T(r, a ij )

a ) + S( r, aij)

k , ij

N(r, a k , a ij )

r 

r 

T(r, a ij ) q

+

{1  (a k 1

 lim

k

S(r, a ij ) T(r, a ij )

, a ij )}  lim

r 

S(r, a ij ) T(r, a ij )

Taking maximum over 1 < i, j < m, we get q

T(r, A ' ) r  T ( r , A )

 (a k , A) < lim k 1

Hence the result. www.iosrjournals.org

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Some Results Concerning Memomorphic Matrix Valued Functions. Proof of theorem 3:

 f'  From [1],We know that m r,  = S(r, f)  f  ' Therefore, m(r, a ij ) < m(r, a ij ) + S(r, a ij ) Also, N(r,

a ij' ) = N(r, aij) + N (r, aij)

Hence, T(r, Therefore,

Therefore,

T(r, a ij' ) T(r, a ij ) lim

r 

a ij' ) = T(r, aij) + N (r, aij) + S(r, aij) <1+

T(r, a ij' ) T(r, a ij )

N (r, aij) + O(1)

 2  (, a ij )

T(r, A ' ) 2  (, A) Taking maximum over 1 < i, j < m, we get lim r  T (r, A ) References [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.

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