some growth properties on integer translation of ... - RS Publication

International journal of advanced scientific and technical research Available online on http://www.rspublication.com/ijst/index.html

Issue 2 volume 6, December 2012 ISSN 2249-9954

SOME GROWTH PROPERTIES ON INTEGER TRANSLATION OF ENTIRE AND MEROMORPHIC FUNCTIONS SANJIB KUMAR DATTA1 AND SAMTEN TAMANG2 1&2 Department of Mathematics, University of Kalyani, Kalyani, Dist. - Nadia, Pin - 741235, West Bengal, India.

Abstract In the paper we study some comparative growth properties of composite entire and meromorphic fucntions on the basis of integer translation applied upon them. AMS Subject Classi…cation (2010) : 30D35; 30D30: Keywords and phrases: Compartive growth property, entire function, meromorphic function, integer translation.

1

Introduction, De…nitions and Notations.

Let f (z) be a meromorphic fucntion de…ned in the open complex plane C: For n 2 N; the translation of f (z) be denoted by f (z + n) : We now describe or investigate the changes to Nevanlinna’s Charateristic function of the translated meromorphic functions. In the paper we investigate the changes to Nevanlinna’s Characteristic function of the translated meromorphic functions and establish some new results on the comparative growth properties related to the translated entire and meromorphic functions. We do not explain the standard de…nitions and notations in the theory of entire and meromorphic functions as those are available in [4] and [3]. For each n 2 N we may obtain a function with some properties. Let us denote this family by fn (z) i.e., fn (z) = ff (z + n) : n 2 N g :

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The Nevanlinna’s Characteristics function of a meromorphic function f denoted by T (r; f ) is de…ned as T (r; f ) = N (r; f ) + m (r; f ) where Zr n (t; f ) n (0; f ) dt + n (0; f ) log r N (r; f ) = t 0

1 and m (r; f ) = 2

Z2

log+ f rei

d

fcf: [1]g :

0

It is clear that the number of zeros of f may be changed in a …nite region after translation but it remains unaltered in the open complex plane C i.e., N (r; f (z + n)) = N (r; f ) + en 1 Also m (r; f (z + n)) = 2

Z2

where en ! 0 as n ! 1:

log+ f rei + n d

0

= m (r; f ) + e0n where e0n ! 0 as r ! 1: Therefore on adding we get that N (r; f (z + n)) + m (r; f (z + n)) = N (r; f ) + en + m (r; f ) + e0n i.e., T (r; f (z + n)) = T (r; f ) + en + e0n : Now if n varies then the Nevanlinna’s Characteristic function for the family fn is X T (r; fn ) = nT (r; f ) + (en + e0n ) n

i.e., log T (r; fn ) = log T (r; f ) + log n:

(A)

The following de…nitions are well known: De…nition 1 The order de…ned as f

= lim sup r!1

f

and lower order

log[2] M (r; f ) and log r

f

f

of an entire function f is

= lim inf r!1

log[2] M (r; f ) log r

where log[k] x = log log[k 1] x for k = 1; 2; :::; n and log[0] x = x: When f is meromorphic, one can easily verify that f

= lim sup r!1

log T (r; f ) and log r

f

= lim inf r!1

log T (r; f ) : log r

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De…nition 2 The type f

f


of a meromorphic function f is de…ned as

= lim sup r!1

log T (r; f ) ;0 < rf

f

< 1:

If f is entire then f

= lim sup r!1

log[2] M (r; f ) ;0 < rf

f

< 1:

Applying (A) on De…nition 1 and De…nition 2 we respectively get that fn

=

f

and

fn

=n

f

and the relations can easily be veri…ed on considering f = exp z: In this paper we develop some remarkable results in connection with the comparative growth properties of composite entire and meromorphic functions by using integer translation upon them.

2

Lemmas.

In this section we present some lemmas which will be needed in the sequel. Lemma 1 f[1]gIf f be a meromorphic function and g be an entire fucntion then for all su¢ ciently large values of r; T (r; f

g)

f1 + o (1)g

T (r; g) T (M (r; g) ; f ) : log M (r; g)

Lemma 2 f[2]g Let f be a meromorphic function and g be an entire fucntion and suppose that 0 < < g 1: Then for a sequence of values of r tending to in…nity, T (r; f g) T (exp (r ) ; f ) :

3

Theorems.

In this section we present the main results of the paper. Theorem 1 Let f be meromorphic and g be entire such that 0 < 1: If fn = f (z + n) and gn = g (z + n) for n 2 N then lim sup r!1

f

0) is arbitrary it follows from (1) and (4) that lim sup r!1

log T (r; fn gn ) = 1: T (r; fn )

Thus the theorem is established. Theorem 2 Let f be a meromorphic function and g be an entire function such that 0 < g g < 1 and 0 < f f < 1. If fn = f (z + n) and gn = g (z + n) for n 2 N then for every positive constant A and for every real number ; log[2] T (r; fn gn ) r!1 flog T (r A ; f )g1+ n

= 1

lim

= 1:

(i) lim and (ii)

log[2] T (r; fn gn ) r!1 flog T (r A ; g )g1+ n

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Proof. In view of fn = f and fn = f let us choose 0 < " < min fn ; gn = min f ; g : Since T (r; gn ) log+ M (r; gn ) we obtain in view of Lemma 2, for a sequence of values of r tending to in…nity,

i.e.,

log T (r; fn gn ) log T (r; fn gn )

Again from the de…nition of of r;

fn

=

f

( (

f

(5)

it follows for all su¢ ciently large values

log T rA ; fn

fn

i.e., log T rA ; fn If 1 +

") r gn " ") r g " :

fn

+ " A log r

+ " A log r:

f

> 0 then we get from above for all su¢ ciently large values of r;

i.e.,

log T rA ; fn

1+

log T rA ; fn

1+

fn

+"

1+

+"

f

1+

A1+ (log r)1+ A1+ (log r)1+ :

(6)

So from (5) and (6) we obtain for a sequence of values of r tending to in…nity that log T (r; fn gn ) flog T (rA ; fn )g1+ i.e., As

r g " (log r)1+

log T (r; fn gn ) flog T (rA ; fn )g1+

( fn

f

+"

f 1+

") r

gn

"

A1+ (log r)1+ ") r g " A1+ (log r)1+

:

(7)

! 1 as r ! 1; we get from (7) that lim sup r!1

log T (r; fn gn ) = 1: flog T (rA ; fn )g1+

On the other hand from the de…nition of large values of r;

gn

log T rA ; gn

=

gn

i.e., log T rA ; gn If 1 +

+" (

fn 1+

g

g

it follows for all su¢ ciently

+ " A log r

+ " A log r:

> 0 then we obtain from above for all su¢ ciently large values of r,

i.e.,

log T rA ; gn

1+

log T rA ; gn

1+

gn g

+"

+"

1+ 1+

A1+ (log r)1+ A1+ (log r)1+ :

(8)

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Since

r g " (log r)1+


! 0 as r ! 1 we get from (5) and (8) that lim sup r!1

log T (r; fn gn ) = 1: flog T (rA ; gn )g1+

When 1 + 0; Theorem 2 is trivial. Thus the theorem is established. Theorem 3 Let fn be meromorphic and gn be entire such that 0 < f < 1 and 0 < g < 1. Then g

lim sup

f

r!1

log[2] T (r; fn gn ) log T

(k) r; fn

f

g f

where k = 0; 1; 2; ::: Proof. In view of fn = f and fn = f let us choose that 0 < " < min gn ; fn = min g ; f : Since T (r; gn ) log+ M (r; gn ) by Lemma 1 we obtain for all su¢ ciently large values of r;

i.e., i.e.,

f1 + o (1)g T (M (r; gn ) ; fn ) log T (M (r; gn ) ; fn ) + O (1) fn + " log M (r; gn ) + O (1)

T (r; fn gn ) log T (r; fn gn ) log T (r; fn gn )

i.e., log[2] T (r; fn gn ) i.e., log[2] T (r; fn gn )

log[2] M (r; gn ) + O (1) gn + " log r + O (1)

i.e., log[2] T (r; fn gn )

g

+ " log r + O (1) :

(9)

Again from (2) we get for a sequence of values of r tending to in…nity that log[2] T (r; fn gn ) i.e., log[2] T (r; fn gn ) Also from the de…nition of large values of r;

fn

=

f

and

gn

" log r + O (1)

g

" log r + O (1) :

fn

log T r; fn(k) i.e., log T

=

f

fn

r; fn(k)

f

(10)

we have for all su¢ ciently

+ " log r

+ " log r:

(11)

and log T r; fn(k)

(

fn

i.e., log T r; fn(k)

(

f

") log r ") log r:

(12)

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From (9) and (12) it follows for all su¢ ciently large values of r; log[2] T (r; fn gn )

g

(k)

log T r; fn

+ " log r + O (1) : ( f ") log r

As " (> 0) is arbitrary, it follows from above that lim sup

log[2] T (r; fn gn )

r!1

log T

g

(k) r; fn

:

(13)

f

Again from (10) and (11) it follows for a sequence of values of r tending to in…nity, log[2] T (r; fn gn )

" log r + O (1) fn + " log r

gn

(k)

log T r; fn i.e.,


" log r + O (1) : f + " log r

g

(k)

log T r; fn

As " (> 0) is arbitrary, it follows from above that lim sup r!1

log[2] T (r; fn gn ) log T

(k) r; fn

g

:

(14)

f

Thus the theorem follows from (13) and (14) : Remark 1 In addition to the conditions of Theorem 3, if fn is of regular growth i.e., fn = fn equivalently f = f then lim sup


r!1

log T

(k) r; fn

=

g

for k = 0; 1; 2; :::

f

Remark 2 Under the same conditions of Theorem 3, lim sup r!1

log[2] T (r; fn gn ) (k)

1 for k = 0; 1; 2; :::

log T r; gn

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References [1] Bergweiler, W. : On the Nevanlinna characteristic of a composite function, Complex Variables, Vol. 10(1988);pp.225 236: [2] Bergweiler, W.: On the growth rate of composite meromorphic functions, Complex Variables, Vol. 14 (1990) ; pp. 187 196: [3] Hayman, W.K.: Meromorphic Functions, The Clarendon Press, Oxford (1964): [4] Valiron, G.: Lectures on the General Theory of Integral Functions, Chelsea Publishing Company (1949):

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