On the order and type of integral functions of several complex variables

C OMPOSITIO M ATHEMATICA

R. K. S RIVASTAVA

V INOD K UMAR

On the order and type of integral functions of several complex variables

Compositio Mathematica, tome 17 (1965-1966), p. 161-166

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161

of several complex ^variables

by

R. K. Srivastava and Vinod Kumar

1.

Let1)

be a function of two

complex

variables z, and Z2’ where the coeffi- cients _amn are

complex

numbers. The series

(1.1) represents

^an

integral

function of two variables

z1, z2 , if it

converges

absolutely

for all values of

|z1|

^{oo and}

|z2|

^{oo. M. M.}

Dzrbasyan (1,

p.

1)

has shown that the necessary and sufficient condition for the series

(1.1)

^to

represent

^an

integral

function of

variables z,

and z2, is

Let Gr

^{be the}

family

of closed

polycircular

^{domains in space}

(z1, Z2) dependent

^on

parameter

^r &#x3E; 0 and possess the

property

that

(z1, Z2) ~ Gr,

^{if and}

only

^if

(zl/r, z2/r) ~ G1.

The maximum modulus of the

intégral

^function

f(z1, Z2)

^{is denoted}

by

and the function will be called G - order and G -

type

respec-

tively,

^if

Denote

A. A.

Goldberg (2,

p.

146)

^has

proved

^the

following

^theorems.

1 For simplicity ^{we consider} only ^two variables, though the results can easily

be extended to several complex ^variables.

162

THEOREM A. All orders _pg be

equal

^and

THEOREM B.

G-type TG

^{satisfies the} correlation

In this _paper we shall obtain the relations between two or more

integral

^{functions and}

study

^the relations between the coefficients in the

Taylor expansion

^of

integral

functions and their orders and

types.

2.

THEOREM

1. Let

f1(z1, z2)

⁼

03A3~m,n=0 ^amnzm1zn2

^and

f2(z1, z2)

⁼

03A3~m,n=0 bmnzm1zn2

^{be two}

integral functions o f

^non-zero

tinite

^orders

pl and _P2

respectively.

^{Then the}

function

where 2

is an

integral function

^{such that}

where _p is the order

of f(zl’ z2).

PROOF: Since

/1 (Zl Z2)

^and

t2 (Zl Z2)

^are

integral ^functions, therefore, using (1.2),

^{we have}

Hence

t(Zl , z2)

^{is an}

integral

^function.

Now

using (1.5)

^{for the functions}

/1 (zl, z2)

^and

f2(z1, z2),

^{we have}

and

Therefore,

^{for an}

arbitrary

^e &#x3E; 0, ^we

get

and

Thus,

^for

or,

Therefore,

^if

Hence

COROLLARY. Let

fs(z1, Z2)

⁼

03A3~m,n=0 a(s)mnzm1zn2,

^{where s = 1, 2, ...,}

p be _p

intégral

functions of non-zéro f inite orders _{03C11, P2, ...,} PlI

respectively.

^{Then the function}

is an

integral

function such that

where _p is the order of

f(z1, z2).

THEOREM 2. Let

f1(z1, z2) = 03A3~m,n=0 amnzm1zn2

^and

f2(z1, z2) =

03A3~m,n=0bmnzm1zn2 be

^two

integral functions o f finite

^{non-zero orders}

03C11 and _P2

respectively.

^{Then the}

function f(z1, Z2)

⁼

03A3~m,n=0 cmn zm1zn2,

where

is an

integral function,

^{such that}

where _p is the order

o f f(zl’ z2).

164

PROOF : Since

/1 (Zl z2)

^and

12(ZI’ Z2)

^are

integral funetions, therefore, using (1.2),

^{we have for an}

arbitrary e

&#x3E; 0 and

large

^R

and

Therefore,

^for

Thus,

^if

then,

^for

large

m + n

or,

Hence

f(z1, x2)

^{is an}

integral

^function.

Now,

^from

(2.1)

^and

(2.2),

^{we have for}

sufficiently large (m+n)

or,

Thus,

^if

log

or,

COROLLARY: Let

fs(z1, z2)

⁼

03A3~m,n=0 a(s)mnzm1zn2,

^{where s =} 1, 2, ..., p be _p

integral

functions of finite non-zéro orders _pi? 03C12,..., Pv

respectively.

^Then the function

where

is an

integral

function such that

where _p is the order of

f(z1, z2).

THEOREM 3. Let

f1(z1, z2)

⁼

03A3~m,n=0amnzm1zn2

^and

^{t2 (Zl Z2)}

⁼

03A3~m,n=0 ^bmnzm1zn2

^{be two}

^integral functions o f finite

^{non-zero orders}

03C11, P2 and

finite

^non-zero

types3 T1, T2 respectively.

^{Then the}

function

where

is an

integral function

^{such that}

where _p and T are the order and

type O!!(ZI’ Z2) respectively

^and

PROOF: ^{We can} _prove, ^{as in the}

proof

of Theorem ¹ that

f(z1, z2)

^{is an}

integral function,

^when

Further, using (1.6)

^{for the functions}

fl(zl, z2)

^and

t2(Zll Z2)’

^we

have

From

(2.3)

^and

(2.4),

^we

get

^{for an}

arbitrary -

&#x3E; 0

for m+n &#x3E;

kl

^and

Thus,

^for

’ The types Ti ^and TI correspond ^{to the same} family ^{of closed} polycircular

domains Gr.

166

Therefore,

^{if we obtain}

or,

where _p, Tare

respectively

^{the order and}

type

^of

f(z1, %2). Hence,

COROLLARY 1. Let

fs(z1, Z2)

⁼

03A3~m,n=0a(s)mnzm1zn2,

^{where s = 1,}

2, ..., p be p

intégral

^functions of f inite non-zéro orders _03C11, 03C12, ... , 03C1p and f inite non-zéro

types T1, T2, ... , Tp respectively.

Then the function

f(z1, Z2)

⁼

03A3~m,n=0 amnzm1zn2,

^where

^{|cmn| ~}

|(03A0ps=1|a(s)mn|)1/p|

^{is an}

^intégral

function such that

where _p and T are the order and

type

^of

f(z1, xz) respectively

^and

lP

⁼

03A3ps=1 ^1/P..

COROLLARY 2. If in the above theorem the functions

fi(xl, z.)

and

f2(z1) z2)

are of the same finite non-zero

order,

^then

COROLLARY 3. The result of the

corollary

^{2 can} be extended to p

integral

^functions.

We are

grateful

^to Dr. S. K. Bose for the

suggestion

^and

guidance

in the

preparation

of this paper.

REFERENCES M. M. DZRBASYAN,

[1] ^{On the} theory ^{of some class of} integral functions. Izv. Akademi. Nauk Ar-

myanskoi. ^S.S.R. VIII, No. 4, 1955, pp. ^1-23.

A. A. GOLDBERG,

[2] Elementary ^{remarks on} the formulae defining ^{order and} type ^of functions of several variables. Doklady Akademi Nauk Armyanskoi S.S.R., ²⁹ (1959),

pp. ^145-151.

Department of Mathematics and Astronomy, (Oblatum 21-5-64) ^Lucknow University, ^Lucknow (India)