# On Some Growth Properties of Entire Functions Using Their Maximum Moduli Focusingp

Authors: Luis Manuel Sanchez Ruiz; Sanjib Kumar Datta; Tanmay Biswas; Golok Kumar Mondal
Journal: The Scientific World Journal
 (2014)

---

## Abstract

We discuss some growth rates of composite entire functions
on the basis of the definition of relativep
,
qth order (relative p
,
qth lower order) with respect to another entire function which improve some
earlier results of Roy (2010) where p and q are any two positive integers.


---

## Body

## 1. Introduction, Definitions, and Notations

Letf be an entire function defined in the open complex plane and let
(1)
M
f
r
=
max
⁡
f
z
:
z
=
r
be its maximum modulus function. If f is nonconstant then M
f
(
r
) is strictly increasing and continuous and its inverse M
f
-
1
(
r
)
:
f
0
,
∞
→
0
,
∞ exists and is such that
(2)
lim
⁡
s
→
∞
M
f
-
1
s
=
∞
.
We use the standard notations and definitions in the theory of entire functions which are available in [1]. In the sequel we use the following notation:
(3)
log
⁡
[
0
]
x
=
x
,
log
⁡
[
k
]
x
=
log
⁡
log
⁡
[
k
-
1
]
x
for
k
=
1,2
,
3
,
…
,
exp
⁡
[
0
]
x
=
x
,
exp
⁡
[
k
]
x
=
exp
⁡
exp
⁡
[
k
-
1
]
x
for
k
=
1,2
,
3
,
…
.The following definitions are well known.Definition 1.
The orderρ
f and the lower order λ
f of an entire function f are defined as
(4)
ρ
f
=
lim
⁡
sup
⁡
r
→
∞
log
⁡
2
M
f
r
log
⁡
r
,
λ
f
=
lim
⁡
inf
⁡
r
→
∞
log
⁡
2
M
f
r
log
⁡
r
.Juneja et al. [2] defined the (
p
,
q
)th order and (
p
,
q
)th lower order of an entire function f, respectively, as follows:
(5)
ρ
f
p
,
q
=
lim
⁡
sup
⁡
r
→
∞
log
⁡
[
p
]
M
f
(
r
)
log
⁡
q
r
,
λ
f
p
,
q
=
lim
⁡
inf
⁡
r
→
∞
log
⁡
[
p
]
M
f
(
r
)
log
⁡
q
r
,
where p, q are any two positive integers with p
≥
q.Ifp
=
l and q
=
1 then we write ρ
f
l
,
1
=
ρ
f
l and λ
f
l
,
1
=
λ
f
l.Also forp
=
2 and q
=
1 we, respectively, denote ρ
f
2,1 and λ
f
2,1 by ρ
f and λ
f.In this connection we just recall the following definition.Definition 2 (see [2]).
An entire functionf is said to have index-pair p
,
q, p
≥
q
≥
1, if b
<
ρ
f
p
,
q
<
∞ and ρ
f
p
-
1
,
q
-
1 is not a nonzero finite number, where b
=
1 if p
=
q and b
=
0 if p
>
q. Moreover if 0
<
ρ
f
p
,
q
<
∞, then
(6)
ρ
f
p
-
n
,
q
=
∞
for
n
<
p
,
ρ
f
p
,
q
-
n
=
0
for
n
<
q
,
ρ
f
p
+
n
,
q
+
n
=
1
for
n
=
1,2
,
…
.
Similarly for 0
<
λ
f
p
,
q
<
∞, one can easily verify that
(7)
λ
f
p
-
n
,
q
=
∞
for
n
<
p
,
λ
f
p
,
q
-
n
=
0
for
n
<
q
,
λ
f
p
+
n
,
q
+
n
=
1
for
n
=
1,2
,
…
.An entire function for which(
p
,
q
)th order and (
p
,
q
)th lower order are the same is said to be of regular p
,
q-growth. Functions which are not of regular p
,
q-growth are said to be of irregular p
,
q-growth.Bernal [3] introduced the definition of relative order of f with respect to g, denoted by ρ
g
f as follows:
(8)
ρ
g
f
=
inf
⁡
μ
>
0
:
M
f
r
<
M
g
r
μ
∀
r
>
r
0
μ
>
0
=
lim
⁡
sup
⁡
r
→
∞
log
⁡
M
g
-
1
M
f
r
log
⁡
r
.The definition coincides with the classical one [4] if g
=
exp
⁡.Similarly one can define the relative lower order off with respect to g denoted by λ
g
f as follows:
(9)
λ
g
f
=
lim
⁡
inf
⁡
r
→
∞
log
⁡
M
g
-
1
M
f
r
log
⁡
r
.In the case of relative order, it therefore seems reasonable to define suitably the relativep
,
qth order of entire functions. Lahiri and Banerjee [5] also introduced such definition in the following manner.Definition 3 (see [5]).
Letp and q be any two positive integers with p
>
q. The relative p
,
qth order of f with respect to g is defined by
(10)
ρ
g
p
,
q
f
=
inf
⁡
μ
>
0
:
M
f
r
<
M
g
exp
⁡
p
-
1
μ
log
⁡
q
r
exp
⁡
p
-
1
μ
log
⁡
q
r
∀
r
>
r
0
μ
>
0
=
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
-
1
M
g
-
1
M
f
r
log
⁡
q
r
.
If q
=
1, k
≥
1, and p
=
k
+
1 then ρ
g
p
,
q
f
=
ρ
g
k
f. If g
=
exp
⁡
z then ρ
g
p
,
q
f
=
ρ
f
p
,
q.Sánchez Ruiz et al. [6] gave a more natural definition of relative (
p
,
q
)th order of an entire function in light of index-pair which is as follows.Definition 4.
Letf and g be any two entire functions with index-pairs m
1
,
q and m
2
,
p, respectively, where m
1
=
m
2
=
m and p
,
q, and m are all positive integers such that m
≥
p and m
≥
q. Then the relative p
,
qth order of f with respect to g is defined as
(11)
ρ
g
p
,
q
f
=
inf
⁡
μ
>
0
:
M
f
r
<
M
g
×
exp
⁡
p
log
⁡
m
2
exp
⁡
m
1
μ
log
⁡
q
r
M
f
∀
r
>
r
0
μ
>
0
=
inf
⁡
μ
>
0
:
M
f
r
<
M
g
exp
⁡
p
μ
log
⁡
q
r
exp
⁡
p
μ
log
⁡
q
r
∀
r
>
r
0
μ
>
0
=
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
g
-
1
M
f
r
log
⁡
q
r
.
Similarly one can define the relativep
,
qth lower order of an entire function f with respect to another entire function g denoted by λ
g
p
,
q
f where p and q are any two positive integers in the following way:
(12)
λ
g
p
,
q
f
=
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
g
-
1
M
f
r
log
⁡
q
r
.In fact Definition4 improves Definition 3 ignoring the restriction p
≥
q.In this paper we wish to prove some results related to the growth rates of entire functions on the basis of relativep
,
qth order and relative p
,
qth lower order with respect to another entire function extending some earlier results for any two positive integers p and q.


## 2. Lemmas

In this section we present some lemmas which will be needed in the sequel.Lemma 1 (see [7]).
Iff and g are any two entire functions with g
0
=
0. then
(13)
M
f
∘
g
r
≥
M
g
r
2
f
o
r
a
l
l
s
u
f
f
i
c
i
e
n
t
l
y
l
a
r
g
e
v
a
l
u
e
s
o
f
r
⩾
r
0
.Lemma 2 (see [7]).
Letf be entire and let g be a transcendental entire function of finite lower order. Then, for any δ
>
0,
(14)
M
f
∘
g
r
1
+
δ
≥
M
f
M
g
r
f
o
r
a
l
l
s
u
f
f
i
c
i
e
n
t
l
y
l
a
r
g
e
v
a
l
u
e
s
o
f
r
⩾
r
0
.Lemma 3 (see [8]).
Iff and g are any two entire functions with g
0
=
0. then, for any 0
<
c
<
1,
(15)
M
f
∘
g
r
≥
M
f
c
M
g
r
2
f
o
r
a
l
l
s
u
f
f
i
c
i
e
n
t
l
y
l
a
r
g
e
v
a
l
u
e
s
o
f
r
⩾
r
0
,Lemma 4 (see [9]).
Iff and g are any two entire functions then for all sufficiently large values of r
⩾
r
0
(16)
M
f
∘
g
(
r
)
≤
M
f
M
g
r
.


## 3. Theorems

In this section we present the main results of the paper.Theorem 5.
Letf be an entire function and let g be any polynomial such that f
∘
g has got finite relative p
,
qth order with respect to h where h is a transcendental entire function and p
,
q are any two positive integers. Then ρ
h
p
,
q
f
<
∞.Proof.
Given thatf
∘
g is of finite relative p
,
qth order with respect to h, we have from Definition 4, for a suitable finite number μ
>
0 and for all sufficiently large values of r, that
(17)
M
f
∘
g
r
<
M
h
exp
⁡
p
μ
log
⁡
q
r
.
Now let m be the order of the polynomial g so that
(18)
g
z
=
c
1
z
+
c
2
z
2
+
⋯
+
c
m
z
m
,
c
m
≠
0
.
Then by Cauchy’s inequality we get from (18) that
(19)
c
m
r
m
≤
M
g
r
,
z
=
r
.
Now given 0
<
c
<
1, in view of Lemma 3 and from (17) it follows for all sufficiently large values of r that
(20)
M
f
c
c
m
r
2
m
≤
M
f
∘
g
r
≤
M
h
exp
⁡
p
μ
log
⁡
q
r
.
We rewrite the above to the equivalent for all sufficiently large values of r that
(21)
M
f
r
≤
M
h
exp
⁡
p
μ
log
⁡
q
c
c
m
-
1
2
m
r
1
/
m
.
Therefore from (21) we get for all sufficiently large values of r that
(22)
M
h
-
1
M
f
r
≤
exp
⁡
p
μ
log
⁡
q
c
c
m
-
1
2
m
r
1
/
m
,
i.e.
,
log
⁡
p
M
h
-
1
M
f
r
≤
μ
log
⁡
q
c
c
m
-
1
2
m
r
1
/
m
.
CaseI. Assume q
=
1. Then we have from (22) for all sufficiently large values of r that
(23)
log
⁡
p
M
h
-
1
M
f
r
log
⁡
r
≤
μ
m
log
⁡
r
+
O
1
log
⁡
r
,
where O
1 stands for the constant expression, m
log
⁡
c
c
m
-
1
2
m. Then
(24)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
r
log
⁡
r
≤
μ
m
lim
⁡
sup
⁡
r
→
∞
log
⁡
r
+
O
1
log
⁡
r
,
i.e.
,
ρ
h
p
f
≤
μ
m
<
∞
.
CaseII. Let us now assume q
>
1. Then we obtain from (22) for all sufficiently large values of r that
(25)
log
⁡
p
M
h
-
1
M
f
r
log
⁡
q
r
≤
μ
log
⁡
q
r
+
O
1
log
⁡
q
r
,
where O
1 stands for a bounded quantity. Then
(26)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
r
log
⁡
q
r
≤
μ
lim
⁡
sup
⁡
r
→
∞
log
⁡
q
r
+
O
1
log
⁡
q
r
,
i.e.
,
ρ
h
p
,
q
f
≤
μ
<
∞
.
Thus the theorem follows from (24) and (26).In the forthcoming proofs we will assume the natural numberq to be q
>
1, the reasonings being easily adapted for q
=
1.Theorem 6.
Letf, g, and h be any three transcendental entire functions and let p and q be two positive integers. If, for any α
,
β with 0
<
α
<
1, β
>
0, and α
β
+
1
>
1, it holds that the two limits A
,
B
∈
R
+ of some of either(i)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
(
r
)
/
log
⁡
q
r
α
=
A, lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
β
+
1
=
B,
(ii)
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
(
r
)
/
log
⁡
q
r
α
=
A, lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
β
+
1
=
B, or
(iii)
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
(
r
)
/
log
⁡
q
r
α
=
A, lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
β
+
1
=
B exist, then ρ
h
p
,
q
f
∘
g
=
∞.Proof.
(i) The existence ofA and B implies that given any ɛ
>
0, for sufficiently large values of r,
(27)
log
⁡
p
M
h
-
1
M
g
(
r
)
≥
A
-
ɛ
log
⁡
q
r
α
,
log
⁡
p
M
h
-
1
M
f
(
r
)
≥
B
-
ɛ
log
⁡
p
M
h
-
1
r
β
+
1
.
Since M
g
r is a continuous, increasing, and unbounded function of r, we get from above for all sufficiently large values of r that
(28)
log
⁡
p
M
h
-
1
M
f
(
M
g
r
)
≥
B
-
ɛ
log
⁡
p
M
h
-
1
M
g
r
β
+
1
.
Also M
h
-
1
r is an increasing function of r; it follows from Lemma 2, (27), and (28) that given δ
>
0, for a sequence of values of r tending to infinity, the following holds:
(29)
log
⁡
p
M
h
-
1
M
f
∘
g
r
1
+
δ
≥
log
⁡
p
M
h
-
1
M
f
M
g
r
≥
B
-
ɛ
log
⁡
p
M
h
-
1
M
g
r
β
+
1
≥
B
-
ɛ
A
-
ɛ
log
⁡
q
r
α
β
+
1
,
i.e.
,
log
⁡
p
M
h
-
1
M
f
∘
g
r
1
+
δ
log
⁡
q
r
1
+
δ
≥
B
-
ɛ
A
-
ɛ
β
+
1
log
⁡
q
r
α
β
+
1
log
⁡
q
r
1
+
δ
.
Hence
(30)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
1
+
δ
)
log
⁡
q
r
1
+
δ
≥
lim
⁡
inf
⁡
r
→
∞
B
-
ɛ
A
-
ɛ
β
+
1
log
⁡
q
r
α
β
+
1
log
⁡
q
r
+
O
1
for all sufficiently large values of r. Since ɛ
>
0 is arbitrary and α
β
+
1
>
1 it follows that
(31)
ρ
h
p
,
q
f
∘
g
=
∞
.
Under (ii) or (iii) a similar argument applies.Theorem 7.
Letf, g, and h be any three transcendental entire functions and let p and q be two positive integers. If, for any α
,
β with α
>
1, 0
<
β
<
1, and α
β
>
1, it holds that the two limits A
,
B
∈
R
+ of either(i)
lim
⁡
sup
⁡
r
→
∞
(
log
⁡
p
M
h
-
1
M
g
(
r
)
/
log
⁡
q
+
1
r
α
)
=
A, lim
⁡
inf
⁡
r
→
∞
(
log
⁡
[
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
]
/
log
⁡
p
M
h
-
1
r
β
)
=
B,
(ii)
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
(
r
)
/
log
⁡
q
+
1
r
α
=
A, lim
⁡
sup
⁡
r
→
∞
(
log
⁡
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
/
log
⁡
p
M
h
-
1
r
β
)
=
B, or
(iii)
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
(
r
)
/
log
⁡
q
+
1
r
α
=
A, lim
⁡
inf
⁡
r
→
∞
(
log
⁡
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
/
log
⁡
p
M
h
-
1
r
β
)
=
Bexist, then ρ
h
p
,
q
f
∘
g
=
∞.Proof.
(i) Given anyɛ
>
0, for a sequence of values of r tending to infinity, we get that
(32)
log
⁡
p
M
h
-
1
M
g
(
r
)
≥
A
-
ɛ
log
⁡
q
+
1
r
α
and for all sufficiently large values of r that
(33)
log
⁡
log
⁡
p
M
h
-
1
M
f
(
r
)
log
⁡
p
M
h
-
1
r
≥
B
-
ɛ
log
⁡
p
M
h
-
1
r
β
,
i.e.
,
log
⁡
p
M
h
-
1
M
f
(
r
)
log
⁡
p
M
h
-
1
r
≥
exp
⁡
B
-
ɛ
log
⁡
p
M
h
-
1
r
β
.
Since M
g
r is a continuous, increasing, and unbounded function of r, we get from above for all sufficiently large values of r that
(34)
log
⁡
p
M
h
-
1
M
f
(
M
g
r
)
log
⁡
p
M
h
-
1
M
g
r
≥
exp
⁡
B
-
ɛ
log
⁡
p
M
h
-
1
M
g
r
β
.
Also M
h
-
1
r is an increasing function of r; thus from Lemma 2, (32), and (34) it follows that, given that δ
>
0, for a sequence of values of r tending to infinity,
(35)
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
1
+
δ
)
log
⁡
q
r
1
+
δ
≥
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
q
r
1
+
δ
≥
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
p
M
h
-
1
M
g
r
·
log
⁡
p
M
h
-
1
M
g
r
log
⁡
q
r
+
O
1
.
Therefore
(36)
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
1
+
δ
)
log
⁡
q
r
1
+
δ
≥

exp
⁡
B
-
ɛ
log
⁡
p
M
h
-
1
M
g
r
β
·
A
-
ɛ
log
⁡
q
+
1
r
α
log
⁡
q
r
+
O
1
≥
exp
⁡
B
-
ɛ
A
-
ɛ
β
log
⁡
q
+
1
r
α
β
·
A
-
ɛ
log
⁡
q
+
1
r
α
log
⁡
q
r
+
O
1
=
exp
⁡
B
-
ɛ
A
-
ɛ
β
log
⁡
q
+
1
r
α
β
-
1
log
⁡
q
+
1
r
·
A
-
ɛ
log
⁡
q
+
1
r
α
log
⁡
q
r
+
O
1
≥
log
⁡
q
r
B
-
ɛ
A
-
ɛ
β
log
⁡
q
+
1
r
α
β
-
1
·
A
-
ɛ
log
⁡
q
+
1
r
α
log
⁡
q
r
+
O
1
.
Hence
(37)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
1
+
δ
)
log
⁡
q
r
1
+
δ
≥

lim
⁡
inf
⁡
r
→
∞
log
⁡
q
r
B
-
ɛ
A
-
ɛ
β
log
⁡
q
+
1
r
α
β
-
1
·
A
-
ɛ
log
⁡
q
+
1
r
α
log
⁡
q
r
+
O
1
.
Sinceɛ
>
0 is arbitrary and α
>
1, α
β
>
1, it follows that
(38)
ρ
h
p
,
q
f
∘
g
=
∞
.
Under (ii) or (iii) a similar argument may be used.Theorem 8.
Letf, g, and h be any three transcendental entire functions such that 0
<
λ
h
p
,
q
g
≤
ρ
h
p
,
q
g
<
∞ where p and q are any two positive integers. If the limit A
∈
R exists in either(i)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
=
A or
(ii)
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
(
r
)
/
log
⁡
p
M
h
-
1
r
=
A, then
(39)
λ
h
p
,
q
f
∘
g
≤
A
λ
h
p
,
q
g
≤
ρ
h
p
,
q
f
∘
g
≤
A
ρ
h
p
,
q
g
.Proof.
(i) SinceM
h
-
1
r is an increasing function of r, it follows from Lemmas 2 and 4, given δ
>
0, for all sufficiently large values of r, that
(40)
M
h
-
1
M
f
∘
g
r
1
+
δ
≥
M
h
-
1
M
f
M
g
r
,
(41)
M
h
-
1
M
f
∘
g
r
≤
M
h
-
1
M
f
M
g
r
,
respectively.
Therefore from (40) we get for all sufficiently large values of r that
(42)
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
1
+
δ
)
log
⁡
q
r
1
+
δ
≥
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
q
r
1
+
δ
≥
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
p
M
h
-
1
M
g
r
·
log
⁡
p
M
h
-
1
M
g
r
log
⁡
q
r
+
O
1
.
From here it follows that
(43)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
log
⁡
q
r
≥

lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
p
M
h
-
1
M
g
r
×
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
r
log
⁡
q
r
+
O
1
i.e.
,
ρ
h
p
,
q
f
∘
g
≥
A
λ
h
p
,
q
g
.
Similarly from (41) it follows for all sufficiently large values of r that
(44)
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
≤
log
⁡
p
M
h
-
1
M
f
M
g
r
.
Therefore
(45)
log
⁡
p
M
h
-
1
M
f
∘
g
r
log
⁡
q
r
≤
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
p
M
h
-
1
M
g
r
·
log
⁡
p
M
h
-
1
M
g
r
log
⁡
q
r
.
Hence
(46)
lim
⁡
inf
⁡
r
→
∞
⁡
log
⁡
p
M
h
-
1
M
f
∘
g
r
log
⁡
q
r
≤

lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
p
M
h
-
1
M
g
r
×
lim
⁡
inf
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
r
log
⁡
q
r
,
i.e.
,
λ
h
p
,
q
f
∘
g
≤
A
λ
h
p
,
q
g
.
Also from (45) we obtain for all sufficiently large values of r that
(47)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
log
⁡
q
r
≤
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
M
g
r
log
⁡
p
M
h
-
1
M
g
r
×
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
r
log
⁡
q
r
,
i.e.
,
ρ
h
p
,
q
f
∘
g
≤
A
·
ρ
h
p
,
q
g
.
Then the thesis follows from (43), (46), and (47).
(ii) follows with a similar argument.Theorem 9.
Letf, g, and h be any three transcendental entire functions with g
0
=
0. If p, q, and m are any three positive integers with m
>
q, then ρ
h
p
,
q
f
∘
g
=
∞ under any of the following conditions: (i)
ρ
h
p
,
q
g
=
∞;
(ii)
min
⁡
ρ
h
p
,
q
f
,
λ
g
m
,
q
>
0;
(iii)
min
⁡
(
ρ
g
m
,
q
,
λ
h
p
,
q
f
)
>
0.Proof.
(i) Ifρ
h
p
,
q
g
=
∞, since M
h
-
1
r is an increasing function of r, it follows from Lemma 1, for all sufficiently large values of r, that
(48)
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
≥
log
⁡
p
M
h
-
1
M
g
r
2
.
Therefore
(49)
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
log
⁡
q
r
≥
log
⁡
p
M
h
-
1
M
g
r
/
2
log
⁡
q
r
≥
log
⁡
p
M
h
-
1
M
g
r
/
2
log
⁡
q
r
/
2
+
O
1
.
Then
(50)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
log
⁡
q
≥
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
g
r
/
2
log
⁡
q
r
/
2
+
O
1
,
i.e.
,
ρ
h
p
,
q
f
∘
g
≥
ρ
h
p
,
q
g
=
∞
.
(ii) Supposeρ
h
p
,
q
f
>
0 and λ
g
m
,
q
>
0.
AsM
h
-
1
r is an increasing function of r, we get from Lemma 2 that given δ
>
0 and any ɛ
>
0, for all sufficiently large values of r,
(51)
log
⁡
p
M
h
-
1
M
f
∘
g
r
1
+
δ
≥
log
⁡
p
M
h
-
1
M
f
M
g
r
≥
ρ
h
p
,
q
f
-
ɛ
log
⁡
q
M
g
r
≥
ρ
h
p
,
q
f
-
ɛ
exp
⁡
m
-
q
-
1
log
⁡
q
-
1
r
λ
g
m
,
q
-
ɛ
.
Thus
(52)
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
1
+
δ
)
log
⁡
q
r
1
+
δ
≥
ρ
h
p
,
q
f
-
ɛ
exp
⁡
m
-
q
-
1
log
⁡
q
-
1
r
λ
g
m
,
q
-
ɛ
log
⁡
q
r
1
+
δ
.
Hence
(53)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
1
+
δ
)
log
⁡
q
r
1
+
δ
≥

lim
⁡
inf
⁡
r
→
∞
ρ
h
p
,
q
f
-
ɛ
exp
⁡
m
-
q
-
1
log
⁡
q
-
1
r
λ
g
m
,
q
-
ɛ
log
⁡
q
r
+
O
(
1
)
,
i.e.
,
ρ
h
p
,
q
f
∘
g
=
∞
.
Under (iii) a similar argument to (i) applies.In the line of Theorem9 one can easily prove the following result.Theorem 10.
Letf, g, and h be any three transcendental entire functions with g
0
=
0. If p
,
q, and m are any three positive integers with m
>
q, then λ
h
p
,
q
f
∘
g
=
∞ if any of the following facts happens:(i)
λ
h
p
,
q
g
=
∞;
(ii)
min
⁡
(
λ
h
p
,
q
f
,
λ
g
m
,
q
)
>
0.Theorem 11.
Letf, g, and h be any three transcendental entire functions such that g
0
=
0. If p, q, and m are any three positive integers with m
>
q and any of the following two facts happens(i)
min
⁡
(
ρ
h
p
,
q
f
,
λ
g
m
,
q
)
>
0 or
(ii)
min
⁡
(
λ
h
p
,
q
f
,
λ
g
m
,
q
)
>
0, then
(54)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
log
⁡
p
M
h
-
1
M
f
(
r
)
=
∞
.Proof .
(i) Since(55)
lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
log
⁡
p
M
h
-
1
M
f
(
r
)
≥


lim
⁡
sup
⁡
r
→
∞
log
⁡
p
M
h
-
1
M
f
∘
g
(
r
)
log
⁡
q
r
×
lim
⁡
inf
⁡
r
→
∞
log
⁡
q
r
log
⁡
p
M
h
-
1
M
f
(
r
)
=
ρ
h
p
,
q
f
∘
g
1
ρ
h
p
,
q
f
the result follows from Theorem 9.
(ii) The proof can be carried out in the line of (i) and Theorem10.


## 4. Conclusion

After modifying the notion of relative order of higher dimensions in case of entire functions in [6], where a number of examples of relative order between functions were provided, in this paper we have obtained some growth properties of composite entire functions on the basis of relative (
p
,
q
)th order and relative (
p
,
q
)th lower order. In this process, Theorem 5 and the first part of Theorem 6 and Theorems 7 and 8 can be regarded as extensions of some results of [10].



---

*Source: 164130-2014-11-03.xml*