1. ijmcar - the growth properties of composite entire and meromorphic

8/17/2019 1. IJMCAR - The Growth Properties of Composite Entire and Meromorphic

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THE GROWTH PROPERTIES OF COMPOSITE ENTIRE AND MEROMORPHIC

FUNCTIONS OF SEVERAL COMPLEX VARIABLES

RAKESH KUMAR, ANUPMA RASTOGI & BALRAM PRAJAPATI

Department of Mathematics & Astronomy, University of Luck now, India

ABSTRACT

D. Somasundaram and Thamizharasi have proved growth the properties of entire functions of − index

and − . We will discuss some growth properties of composite entire and meromorphic functions based on relative order

due to slowly changing function ≡ (, , , … … . . , ) . In this paper we obtain the some improved results based on slowly

changing function.

KEYWORDS: Order of Entire and Meromorphic Functions, Slowly Changing Function, ∗ − , ∗ − Lower Order, ∗ − .

Received: Feb 06, 2016; Accepted: Feb 24, 2016; Published: Mar 02, 2016; Paper Id.: IJMCARAPR20161

INTRODUCTION

Definition and Notation

Let be a meromorphic function and be an entire function defined on ℂ where ℂ is the set of complex

number. We use the standard notations and definitions in the theory of entire and meromorphic functions which are

available in [9], [6] and [10].

We use the following notation.

log! " # log$log%&! "' , o * # +,,-, … …

And

log! " # "

Some useful following definitions.

Definition 1. The ordernv

/0 and lowernv

10 of a meromorphic function are defined as

nv /0 # l2345,46,…..,4789 :;< =>?@A(45,46,….,47)=>?(4546....47)

And

nv 10 # l2345,46,…..,4789 BC =>?@A(45,46,….,47)=>?(4546....47)

If is an entire function, then

nv /0 # l2345,46,…..,4789 :;< =>?6! DA(45,46,....,47)

=>?(4546....47)

Or i gi n al Ar t i c l e

International Journal of Mathematics and Computer

Applications Research (IJMCAR)

ISSN(P): 2249-6955; ISSN(E): 2249-8060

Vol. 6, Issue 2, Apr 2016, 1-12

© TJPRC Pvt. Ltd



2 Rakesh Kumar, Anupma Rastogi & Balram Prajapati

Impact Factor (JCC): 4.6257 NAAS Rating: 3.80

And

nv 10 # l2345,46,…..,4789 BC =>?6! DA(45,46,….,47)=>?(4546....47)

Definition 2. The hyper order nv /0and hyper lower order nv 10 of a meromorphic function are defined as

follows

nv /0 # l2345,46,…..,4789 :;< =>?6! @A(45,46,….,47)=>?(4546....47)

And

nv 10 # l2345,46,…..,4789 BC =>?6! @A(45,46,....,47)=>?(4546….47)

If

is an entire function then

nv /0 # l2345,46,…..,4789 :;< =>?E! DA(45,46,….,47)=>?(4546....47)

And

nv 10 # l2345,46,…..,4789 BC =>?E! DA(45,46,….,47)=>?(4546....47)

Definition 3. F! Let be meromorphic function of order zero. Then nv /0∗ , nv 10∗ and /0∗ , 10

∗ are defined as

following

nv /0∗ # l2345,46,…..,4789 :;< =>?@A(45,46,….,47)=>?6!(4546....47)

nv 10∗ # l2345,46,…..,4789 BC =>?@A(45,46,....,47)=>?6!(4546....47)

And

nv /0∗ # l2345,46,…..,4789 :;< =>?6! @A(45,46,....,47)=>?6!(4546....47)

nv 10∗ # l2345,46,…..,4789 BC =>?6! @A(45,46,….,47)=>?6!(4546....47)

If is an entire, then clearly,

nv /0∗ # l2345,46,…..,4789 :;< =>?6! DA(45,46,….,47)=>?6!(4546....47)

nv G H ∗ # l23I5,I6,…..,IJ89 2K =>?6! LM (I5,I6,....,IJ)=>?6!(I5I6....IJ)

And

nv /0∗

# l2345,46,…..,4789 :;< =>?E! DA(45,46,....,47)

=>?6!(

4546....47)



The Growth Properties of Composite Entire and Meromorphic Functions of Several Complex Variables 3


nv 10

∗ # l2345,46,…..,4789 BC =>?E! DA(45,46,....,47)=>?6!(4546....47)

Definition 4. The type nv N0 of a meromorphic function is defined as

nv N0 # l2345,46,…..,4789 :;< @A(45,46,....,47)

(4546....47) nv OA, P Q nv /0 Q R

If is an entire, then

nv N0 # l2345,46,…..,4789 :;< =>?DA(45,46,....,47)

(4546....47) nv OA, P Q nv /0 Q R

D. Somasundaram and Thamizharasi ! introduced the notions of S − TUVWU and S − XY<W for entire functions

which S ≡ S(U&, UZ, . . . . , U) is a positive continuous function increasing slowly

B . W . , S([U&, [UZ, . . . . , [ U)\S(U&, UZ, . . . . , U) [: U&, UZ, … . , U 8 R for every positive constant [ their definitions are as

follows.

Definition 5. ! The S − TUVWU nv /0] and the S − ^T_WU TUVWU n

v 10] of an entire function are defined as

follows.

nv /0] # l2345,46,…..,4789 :;< =>?6! DA(45,46,....,47)=>?(4546.…47)](45,46,....,47)!

And

nv 10] # l2345,46,…..,4789 BC =>?6!

DA(45,46,....,47)=>?(45464E.…47)](45,46,....,47)!

When is meromorphic, then

nv /0] # l2345,46,…..,4789 :;< =>?@A(45,46,….,47)=>?(4546.…47)](45,46,....,47)!

And

nv 10] # l2345,46,…..,4789 BC =>?@A(45,46,....,47)=>?(45464E….47)](45,46,....,47)!

Definition 6. ! S − XY<Wnv

N0]

of an entire function with S − TUVWUnv

/0] is defined as

nv N0] # l2345,46,…..,4789 :;< =>?DA(45,46,....,47)

(4546.…47)](45,46,....,47)!nv OA

, P Q nv /0] Q R

For meromorphic , the S −XY<W nv N0] becomes

nv N0] # l2345,46,…..,4789 :;< =>?@A(45,46,….,47)

(4546.…47)](45,46,....,47)! nv OA

, P Q nv /0] Q R

Similarly we can define

S − aY<WUand

S − aY<WU ^T_WUorder of entire and meromorphic

.

The same generalized concept of S − TUVWU and S − XY<W of entire and meromorphic functions are S∗ −





TUVWU [CV S∗ − XY<W. their definitions are as follows.

Definition 7. The S∗ − TUVWU [CV S∗ − ^T_WU TUVWU [CV S∗ − XY<W of a meromorphic function are as defined

by

nv /0]∗ # l2345,46,…..,4789 :;< =>?@A(45,46,....,47)

=>?b(4546.…47)c`(d5,d6,…..,d7)e

nv 10]∗ # l2345,46,…..,4789 BC =>?@A(45,46,....,47)

=>?b(4546.…47)c`(d5,d6,…..,d7)e

And

nv N0]∗ # l2345,46,…..,4789 :;< @A(45,46,….,47)

b(4546.…47)c`(d5,d6,…..,d7)env OA

∗, P Q nv /0]

∗ Q R

Where

is entire then,

nv /0]

∗ # l2345,46,…..,4789 :;< =>?6! DA(45,46,….,47)=>?b(4546.…47)c`(d5,d6,…..,d7)e

nv 10]∗ # l2345,46,…..,4789 BC =>?6! DA(45,46,....,47)

=>?b(4546.…47)c`(d5,d6,…..,d7)e

And

nv N0]

∗ # l2345,46,…..,4789 :;< =>?DA(45,46,....,47)

b(4546.…47)c`(d5,d6,…..,d7)env OA

∗, P Q n

v /0]∗ Q R

In this paper we intend to establish some results relating to the growth properties of composite entire and

meromorphic functions on the basis of their relative order which improving some earlier results.

In this section we present some lemmas

Definition 7. Let be a meromorphic function and be transcendental entire. If nv 10f] Q R then nv 10] # P.

Definition 8. Let be meromorphic and be transcendental entire. If nv /0f Q R then nv /0 # P.

Definition 9. Let be meromorphic and be transcendental entire. If nv /0f] Q R then nv /0f] # P.

Lemma 1. -! If and two entire functions, then for all sufficiently large values of U&, UZ, … … . . , U

h0 i&j h i45

Z , 46Z , … . , 47

Z k − (P)k m h0f(U&, UZ, … . , U) m h0 ih(U&, UZ, … . , U)k.

Lemma 2. n! Let be entire and be a transcendental entire function of finite lower order. then for any p P

h0f$U&&qr , UZ&qr , … . , U&qr' p h0 ih(U&, UZ, … . , U)k , (U&, UZ, . . . . , U s U).

Lemma 3.t! Let be a meromorphic function and be transcendental entire. If nv 10f Q R then nv 10 # P.

Lemma 4 Let be entire and be transcendental entire with nv 1] Q R Also let nv /0f] # P. Then





/u7∗ 0 1u7

∗ ] m /u7∗ 0f] m /u7

∗ 0 /u7∗ ] .

Proof by lemma 2

.

/u7∗

0f]

# l2345,46,…..,4789 :;<=>?6! DAvwi455xy,465xy,….,475xyk

=>?6!(4546.…47)](45,46,....,47)!

s l2345,46,…..,4789:;< logZ! h0 ih(U&, UZ, … . , U)k

logZ! h(U&, UZ, . . . . , U) . l2345,46,…..,4789BC logZ! h(U&, UZ, . . . . , U)

logZ!(U&UZ. … U)S(U&, UZ, . . . . , U)!

# nv /0∗ 1u7∗ ]

Again by lemma 1.

/u7∗ 0f] # l2345,46,…..,4789 :;< =>?6! DAvw(45,46,....,47)

=>?6!(4546.…47)](45,46,....,47)!

m l2345,46,…..,4789 :;< =>?6! DAiDw(45,46,....,47)k=>?6! Dw(45,46,....,47) .l2345,46,…..,4789 :;< =>?6! Dw(45,46,….,47)

=>?6!(4546.…47)](45,46,....,47)!

# nv /0∗ /u7∗ ]

Above from two inequalities we get that

nv /0∗ 1u7∗ ] m /u7

∗ 0f] m nv /0∗ /u7∗ ]

Lemma 5. Let and be two entire functions such that nv /0] # P and nv 1] Q R. Also let be transcendental

entire. Then

1u7∗ 0] nv /] m nv /0f] m nv /0∗ nv /] .

z{| Lemma 1, we get

nv /0f] # l2345,46,…..,4789 :;< =>?6! DAvw(45,46,....,47)=>?(4546.…47)](45,46,....,47)!

m l2345,46 ,…..,4789:;< logZ! h0 ih(U&, UZ, . . . . , U)k

logZ! h(U&, UZ, . . . . , U) . l2345,46,…..,4789:;< logZ! h(U&, UZ, . . . . , U)

log(U&UZ. … U)S(U&, UZ, . . . . , U)!

# nv /0∗ nv /]

Also from lemma, it follows that

nv /0f] # l2345,46,…..,4789 :;< =>?6! DAvwi455xy,465xy,….,475xyk=>?(4546.…47)](45,46,....,47)!5xy

s l2345,46,…..,4789BC logZ! h0 ih(U&, UZ, . . . . , U)k

logZ! h(U&, UZ, . . . . , U) . l2345,46,…..,4789:;< logZ! h(U&, UZ, . . . . , U)

log(U&UZ. … U)S(U&, UZ, . . . . , U)!&qr

# 1u7∗ 0] nv /]





Continuing the above two inequalities we obtain that

1u7∗ 0] nv /] m nv /0f] m nv /0∗ nv /] .

Lemma 6: If be an entire function and be transcendental entire with

nv 10f] # P, nv 1] Q R

Then

1u7∗ 0f] s 1u7

∗ 0] 1u7∗ ]

Proof: Lemma

1u7∗ 0f] # l2345,46,…..,4789 BC =>?6! DAvwi455xy,465xy,….,475xyk

=>?6!(4546.…47)](45,46,….,47)!5xy s

l2345,46,…..,4789 BC =>?6! DAiDw(45,46,….,47)k=>?6! Dw(45,46,....,47) .l2345,46,…..,4789 BC =>?6! Dw(45,46,....,47)

=>?6!(4546.…47)](45,46,....,47)!

# nv 10∗ 1u7∗ ]

This proof the theorem.

Lemma 7: Let be an entire and be transcendental entire with nv 1]∗ Q R. Also let

nv /0f]∗ # P. Then

/u7∗ 0 1u7

∗ ] m /u7∗ 0f] m /u7

∗ 0 nv /0]

Proof. By lemma ,

/u7∗ 0f]∗ # l2345,46,…..,4789 :;< =>?6! DAvwi455xy,465xy,….,475xyk

=>?6!b(4546.…47)c`(d5,d6,…..,d7)e

s l2345,46,…..,4789 :;< =>?6! DAiDw(45,46,....,47)k=>?6! Dw(45,46,....,47) .l2345,46,…..,4789 BC =>?6! Dw(45,46,....,47)

=>?6!b(4546.…47)c`(d5,d6,…..,d7)e

#nv

/0

∗∗

nv

1

]∗

Again by lemma +,

/u7∗ 0f]∗ # l2345,46,…..,4789 :;< =>?6! DAvw(45,46,....,47)

=>?6!b(4546.…47)c`(d5,d6,…..,d7)e

m l2345,46,…..,4789 :;< =>?6! DAiDw(45,46,....,47)k=>?6! Dw(45,46,....,47) .l2345,46,…..,4789 :;< =>?6! Dw(45,46,....,47)

=>?6!b(4546.…47)c`(d5,d6,…..,d7)e

# nv /0∗∗

nv /]∗

From the above two inequalities we get that





# /u7∗∗ 0]

∗ nv 1]∗ m /u7

∗ 0f]∗ m nv /0∗∗ nv /]∗

This proof the lemma.

Lemma 8. Let and be two entire functions such that nv /0]∗ # P [CV P Qnv 10]∗ Q R, also let be

transcendental entire. That

nv 10∗ nv /]

∗ m nv /0f]∗ m n

v /0∗ nv /]

∗

Proof. From lemma1, we get

nv /0f]∗ # l2345,46,…..,4789 :;< =>?6! DAvw(45,46,....,47)=>?b(4546.…47)c`(d5,d6,…..,d7)e

m l2345

,46

,……..,4789

:;< =>?6! DAiDw(45,46,….,47)k

=>?6!

Dw(45,46,....,47) .l234

5,4

6,…..,4

789 :;< =>?6! Dw(45,46,….,47)

=>?b(4546.…47)c`(d

5,d

6,…..,d

7)e

# nv /0∗ nv /]

∗

Again from lemma it follows that

nv /0f]∗ # l2345,46,…..,4789 :;< =>?6! DAvwi455xy,465xy,….,475xyk=>?b(4546.…47)c`(d5,d6,…..,d7)e5xy

s l2345,46,…..,4789 BC =>?6! DAiDw(45,46,....,47)k=>?6! Dw(45,46,....,47) .l2345,46,…..,4789 :;< =>?6! Dw(45,46,....,47)

=>?b(4546.…47)c`(d5,d6,…..,d7)e5xy

# nv 10∗ nv /]∗

Now combining the above two inequalities we obtain.

nv 10∗ nv /]∗ m nv /0f]∗ m nv /0∗ nv /]

∗

Lemma 8 If is an entire function and be transcendental entire with

nv 10f]∗ # P, P Q nv 1]∗ Q R

Then

1u7∗ 0f]∗ s 1u7

∗ 0]∗ 1u7

∗ ]∗

Proof. By lemma ,

1u7∗ 0f]∗ # l2345,46,…..,4789 BC =>?6! DAvwi455xy,465xy,….,475xyk

=>?6!b(4546.…47)c`(d5,d6,…..,d7)e5xy

s l2345,46,…..,4789 BC =>?6! DAiDw(45,46,....,47)k=>?6! Dw(45,46,....,47) .l2345,46,…..,4789 :;< =>?6! Dw(45,46,....,47)

=>?b(4546.…47)c`(d5,d6,…..,d7)e5xy

#nv

10∗ 1u7∗ ]∗

This proof the lemma.





Theorems

In this section we present the main results of this paper.

}~• . Let be meromorphic and transcendental entire such thatnv

10f]

p P. Then for every positiveno.

€,l2345,46,…..,4789 :;< =>?@Avw(45,46,....,47)=>?@A$45,46,….,47' m ‚ƒ7∗ Avw`

„ ‚ƒ7∗ A

z{. Case ([). If nv 10f] # R, the theorem is obvious event line case (), If nv 10f] Q R, then by lemma t,

nv 10] # P and the theorem follows.

}~• . Let be meromorphic and be transcendental entire such that nv /0f] # P, Also let P Q 1u7∗ 0f] m

/u7∗ 0f] Q R and P Q 1u7∗ 0] m /u7∗ 0] Q R.

Then for any positive number €, ‚ƒ7∗ Avw`

„ †ƒ7∗ A m l2345,46,…..,4789 BC =>?@Avw(45,46,….,47)

=>?@A$45,46,.…,47' m ‚ƒ7∗ Avw`

„ ‚ƒ7∗ A,

m l2345,46,…..,4789:;< =>?@Avw(45,46,....,47)

=>?@A$45,46,.…,47' m ‚ƒ7∗ Avw`

„ ‚ƒ7∗ A,

z{. Since nv /0f] =0 by definition 9, nv /0] # P. From the definition of /u7∗ 0] and 1u7

∗ 0] we have for arbitrary

positive ‡ and for all large values U&, UZ, . . . . , U

log 0f(U&, UZ, . . . . , U) s $ 1u7∗ 0f] −‡'logZ!(U&UZ. … U)S(U&, UZ, . . . . , U)! (+)

And

log 0(U&„ , UZ„ , . … , U„) m €$ /u7∗ 0] ‰ ‡'logZ!(U&UZ. … U)S(U&, UZ, . . . . , U)! ()

From (1) and (2) it follows for all large values of U&, UZ, . . . . , U,

log 0f(U&, UZ, . . . . , U)

log 0(U&

„

, UZ

„

, . … , U

„

)

s $ 1u7∗ 0f] − ‡'

€$ /u7

∗0

]

‰ ‡'

As ‡(p P) is arbitrary, we obtain that

l2345,46,…..,4789 BC =>?@Avw(45,46,…..,47)=>?@A$45,46,.…,47' s ‚ƒ7∗ Avw`

„ †ƒ7∗ A (-)

Again for a sequence values of U&, UZ, … . , U,tending to infinity.

log 0f(U&, UZ, . . . . , U) m $ 1u7∗ 0f] ‰‡'logZ!(U&UZ. … U)S(U&, UZ, . . . . , U)! (t)

and for all large values of U&, UZ, . . . . , U,

log 0(U&„ , UZ„ , . … , U„) s €$ 1u7∗ 0] − ‡'logZ!(U&UZ. … U)S(U&, UZ, … . , U)! (Š)





Combining (4) and (5) we get for a sequence of values of U&, UZ, … . , U, tending to infinity,

=>?@Avw(45,46,….,47)=>?@A$45,46,.…,47' m i ‚ƒ7∗ Avw` q‹k

„i ‚ƒ7∗ A%‹k

Since ‡(p P) is arbitrary it follows that

l2345,46,…..,4789 BC =>?@Avw(45,46,…..,47)=>?@A$45,46,.…,47' m ‚ƒ7∗ Avw`

„ ‚ƒ7∗ A (Œ)

Also, for a sequence of U&, UZ, . . . . , U, tending to infinity,

log 0(U&„ , UZ„ , . … , U„) m €$ 1u7∗ 0] ‰ ‡'logZ!(U&UZ. … U)S(U&, UZ, . . . . , U)! (F)

Now from (1) and (7) we obtain for a sequence of values of U&, UZ, . . . . , U, tending to infinity,

=>?@Avw

(45,46,....,47)=>?@A$45,46,.…,47' s

i ‚ƒ7

∗ Avw` %‹k„i ‚ƒ7∗ Aq‹k

As ‡(p P) is arbitrary, we get

l2345,46,…..,4789 :;< =>?@Avw(45,46,…..,47)=>?@A$45,46,.…,47' s ‚ƒ7∗ Avw`

„ ‚ƒ7∗ A (n)

Also for all large value of U&, UZ, . . . . , U,

log 0f(U&, UZ, . . . . , U) m $ /u7∗ 0f] ‰‡'logZ!(U&UZ. … U)S(U&, UZ, . . . . , U)! ()

From (5) and (9) it follows for all large value of U&, UZ, . . . . , U,

=>?@Avw(45,46,….,47)=>?@A$45,46,.…,47' m i †ƒ7∗ Avw` q‹k

„i ‚ƒ7∗ A%‹k

Since ‡(p P) is arbitrary, we obtain that

l2345,46,…..,4789 :;< =>?@Avw(45,46,…..,47)=>?@A$45,46,.…,47' m †ƒ7∗ Avw`

„ ‚ƒ7∗ A (+P)

Thus the theorem follows from (3), (6), (8),and (10).

}~• . Let be entire and be transcendental entire satisfying the following conditions.

• nv /0f] # P [CV nv 1] Q R

• P Q 1u7∗ 0f] m /u7

∗ 0f] Q R

And

• P Q 1u7∗ 0] m /u7

∗ 0] Q R

Then

‚ƒ7

∗ A ‚ƒ7

∗ w„ †ƒ7∗ A m

‚ƒ7

∗ Avw`

„ †ƒ7∗ A





m l2345,46,…..,4789 BC =>?6! DAvw(45,46,….,47)=>?6! DA$45,46,.…,47'

m ‚ƒ7∗ Avw`

„ ‚ƒ7

∗

A

`

m l2345,46,…..,4789 :;< =>?6! DAvw(45,46,….,47)=>?6! DA$45,46,.…,47'

m nv †Avw`

„ ‚ƒ7∗ A m †ƒ7∗ A †ƒ7∗ w

„ ‚ƒ7∗ A .

z{. From in lemma Œ and the second part of lemma 4, Theorem - follows from Theorem.

}~• Ž. Let be meromorphic and be entire such that nv /0f] # P. Also let

P Q 1u7∗ 0f] m /u7∗ 0f] Q R [CV P Q /u7∗ 0] Q R. Then for any positive number €.

2345,46,…..,4789 BC =>?@Avw(45,46,….,47)=>?@A$45,46,.…,47' m †ƒ7∗ Avw`

„ †ƒ7∗ A

m l2345,46,…..,4789 :;< =>?@Avw(45,46,....,47)=>?@A$45,46,.…,47'

z{. From definition 9, nv /0f] # P implies that nv /0] # P from the definition of S − TUVWU we get for a

sequence of values of

U&, UZ, . . . . , U, tending to infinity,

log 0(U&„ , UZ„ , . … , U„) s €$ /u7∗ 0] − ‡'logZ!(U&UZ. … U)S(U&, UZ, . . . . , U)! (++)

Now from (9) and (11) it follows for a sequence of values of U&, UZ, . . . . , U, tending to infinity,

=>?@Avw(45,46,....,47)=>?@A$45,46,.…,47' m i †ƒ7∗ Avw` q‹k

„i †ƒ7∗ A%‹k

As (‡ p P) is arbitrary, we obtain,

l2345,46,…..,4789 BC =>?@Avw(45,46,…..,47)=>?@

A$4

5

,46

,.…,47

' m †ƒ7∗ Avw`

„ †ƒ7

∗

A

` (+)

Again for a sequence of values of U&, UZ, … … . . , U, tending to infinity,

log 0f(U&, UZ, . . . . , U) s $ /u7∗ 0f] −‡'logZ!(U&UZ. … U)S(U&, UZ, … . , U)! (+-)

For combining (2) and (13) we get for a sequence of values of U&, UZ, . . . . , U, tending to infinity,

=>?@Avw(45,46,....,47)=>?@A$45,46,.…,47' s i †ƒ7∗ Avw` %‹k

„i †ƒ7∗ Aq‹k

Since ‡(p P) is arbitrary, it follows that

l2345,46,…..,4789 :;< =>?@Avw(45,46,…..,47)=>?@A$45,46,.…,47' s †ƒ7∗ Avw`

„ †ƒ7∗ A (+t)





Thus the theorem follows (12) and (14).

}~• . Let be an entire function and be a transcendental entire function satisfying.

• nv

/0f]

# P [CVnv

1] Q R

• P Q 1u7∗ 0f] m /u7

∗ 0f] Q R

And

• P Q /u7∗ 0] Q R

Then

l2345,46,…..,4789 :;< =>?6! DAvw(45,46,....,47)=>?6! DA(45,46,....,47) s 1u7

∗ ]

And

l2345,46,…..,4789 BC =>?6! DAvw(45,46,....,47)=>?6! DA(45,46,....,47) m /u7

∗ ]

z{. In view of lemma 4 we obtain from theorem 4 for € # +,

l2345,46,…..,4789 :;< =>?6! DAvw(45,46,....,47)=>?6! DA(45,46,....,47) s †ƒ7∗ A ‚ƒ7∗ w

†ƒ7∗ A # 1u7

∗ ]

And

l2345,46,…..,4789 BC =>?6! DAvw(45,46,....,47)=>?6! DA(45,46,....,47) m †ƒ7∗ A †ƒ7∗ w†ƒ7∗ A # /u7∗ ]

Thus the theorem follows from theorem 2 inequalities.

The following theorem is a natural consequence of theorem 2 and theorem 4.

}~• ‘. Let be meromorphic and be entire such that nv /0f] # P. Also let

P Q 1u7∗ 0f] m /u7

∗ 0f] Q R. and P Q 1u7∗ 0] m /u7

∗ 0] Q R. Then for any positive number €,

2345,46,…..,4789 BC =>?@Avw(45,46,….,47)=>?@A$45,46,.…,47'

m ’BC “ ‚ƒ7∗ Avw`

„ ‚ƒ7∗ A , †ƒ7∗ Avw`

„ †ƒ7∗ A”

m ’[" “ ‚ƒ7∗ Avw`

„ ‚ƒ7∗ A , †ƒ7∗ Avw`

„ †ƒ7∗ A”

m l2345,46,…..,4789 :;<=>?@

Avw(4

5,4

6,….,4

7)

=>?@A$45,46,.…,47'





The proof is omitted.

ACKNOWLEDGEMENTS

The author are thankful to the referee for valuable suggestion towards the improvement of the paper.

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1. ijmcar - the growth properties of composite entire and meromorphic

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