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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
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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)
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The Growth Properties of Composite Entire and Meromorphic Functions of Several Complex Variables 3
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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∗ −
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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
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/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 /]
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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
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# /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.
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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)
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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
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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)
8/17/2019 1. IJMCAR - The Growth Properties of Composite Entire and Meromorphic
http://slidepdf.com/reader/full/1-ijmcar-the-growth-properties-of-composite-entire-and-meromorphic 11/12
The Growth Properties of Composite Entire and Meromorphic Functions of Several Complex Variables 11
www.tjprc.org [email protected]
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'
8/17/2019 1. IJMCAR - The Growth Properties of Composite Entire and Meromorphic
http://slidepdf.com/reader/full/1-ijmcar-the-growth-properties-of-composite-entire-and-meromorphic 12/12
12 Rakesh Kumar, Anupma Rastogi & Balram Prajapati
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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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Liao, L., and yong, C.C., On the growth properties composition of entire functions, Yokohama Math.J.tŒ(+),™™.F−+PF.
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Song, G.D., and yong, C.C., Further growth properties composition of entire and meromorphic functions, Indian J. Pure Appl.
Math. +Š(+nt),–T.+,™™.ŒF−n. 9. Valiron, G., Lectures On the General theory of integral functions, Chelsea Publishing, Company,+t. 10.
Fuks, B. A., theory of analytic functions of several complex variables, (1963) Moscow.