# Theq-Difference Theorems for Meromorphic Functions of Several Variables

Authors: Zhi-Tao Wen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigateq-shift analogue of the lemma on logarithmic derivative of several variables. Let f be a meromorphic function in ℂ
n of zero order such that f
(
0
)
≠
0,  ∞, and let q
∈
ℂ
n
\
{
0
}. Then we have m
(
r
,
f
(
q
z
)
/
f
(
z
)
)
=
o
(
T
(
r
,
f
)
) on a set of logarithmic density 1. The q-shift analogue of the first and the second main theorems of Nevanlinna theory of several variables and their applications is also shown.


---

## Body

## 1. Introduction

The lemma on logarithmic derivative is a basic tool of Nevanlinna theory; see [1]. It is widely used in value distribution of meromorphic functions [2], differential equations in the complex plane [3], and so forth. The first generalization of the lemma on the logarithmic derivative to several complex variables is given by Vitter [4]. Another proof is given by Biancofiore and Stoll in [5]. Ye obtains a sharp bound for the lemma on logarithmic derivative of several complex variables in [6]. In [7], Li improves the lemma on logarithmic derivative of several variables without exceptional intervals, which makes a significant difference from other estimates.Essentially prompted by the recent interest in discrete Painlevé equations, difference analogues of the lemma on logarithmic derivative appeared to be useful in a number of applications, in particular in complex difference equations; see [8–10]. Later, Korhonen studies the difference analogues of the lemma on logarithmic derivative of several variables in [11]. For q-difference version, Barnett et al. obtain q-difference analogue of the lemma on logarithmic derivative in the complex plane in [12].In this paper, our aim is to generalize the result in [12] to several variables. For any two constants z and z
′ in C
n, if z
=
(
z
1
,
…
,
z
n
) and z
′
=
(
z
1
′
,
…
,
z
n
′
), then we denote z
1
z
1
′
=
(
z
1
z
1
′
,
…
,
z
n
z
n
′
). Let us state one of the main results as follows, which is q-difference analogue of the lemma on logarithmic derivative of several variables.Theorem 1.
Letf be a meromorphic function in C
n of zero order such that f
(
0
)
≠
0
,
∞, and let q
∈
C
n
∖
{
0
}. Then
(1)
m
(
r
,
f
(
q
z
)
f
(
z
)
)
=
o
(
T
(
r
,
f
)
)
on a set of logarithmic density 1.Concerning the sharpness of Theorem1, there is an example in [12, page 459] to show that zero order cannot be replaced by any positive order in the statement of Theorem 1.The remainder of the paper is organized as follows. In Section2, we prove the theorem of q-difference analogue of the lemma on logarithmic derivative of several variables. In Section 3, we present the q-shift analogue of the first and the second main theorems of Nevanlinna theory of several variables. In Section 4, we give three important theorems: uniqueness theorem of meromorphic function with its q-shift of several variables, q-shift analogues of the Clunie lemma, and Mohon’ho lemma of several variables.


## 2.q-Difference Logarithmic Derivative Analogue

The purpose of this section is to presentq-difference analogues of the lemma on logarithmic derivative in several complex variables. Let us recall some of the standard notations of Nevanlinna theory in C
n.LetM be a connected complex manifold of dimension n and let
(2)
A
(
M
)
=
∑
n
=
0
2
m
A
n
(
M
)
be the graded ring of complex valued differential forms on M. Each set A
n
(
M
) can be split into a direct sum
(3)
A
n
(
M
)
=
∑
p
+
q
=
n
A
p
,
q
(
M
)
,
where A
p
,
q
(
M
) is the forms of type (
p
,
q
). The differential operators d and d
c on A
(
M
) are defined as
(4)
d
≔
∂
+
∂
¯
,
d
c
≔
1
4
π
i
(
∂
-
∂
¯
)
,
where
(5)
∂
:
A
p
,
q
(
M
)
⟶
A
p
+
1
,
q
(
M
)
,
∂
¯
:
A
p
,
q
(
M
)
⟶
A
p
,
q
+
1
(
M
)
.
Let z
=
(
z
1
,
…
,
z
n
)
∈
C
n, and let r
>
0 and fix a
=
(
a
1
,
…
,
a
n
)
∈
C
n; an exhaustion function τ
a
,
C
n of C
n is defined by
(6)
τ
a
,
C
n
(
z
)
=
|
z
-
a
|
2
=
∑
j
=
1
m
|
z
j
-
a
j
|
2
.
Usually, we let a
=
0 be the origin and set τ
m
(
z
)
=
τ
0
,
C
n
(
z
). Let us define a positive measure σ
n
(
z
) with total measure one on the boundary ∂
B
n
(
r
)
≔
{
z
∈
C
n
:
|
z
|
=
r
} of the ball B
n
(
r
)
≔
{
z
∈
C
n
:
|
z
|
<
r
} as
(7)
ω
n
(
z
)
≔
d
d
c
log
⁡
τ
n
,
σ
n
(
z
)
≔
d
c
log
⁡
τ
n
∧
ω
n
n
-
1
(
z
)
.
In addition, we define
(8)
υ
n
(
z
)
≔
d
d
c
τ
n
(
z
)
,
ρ
n
(
z
)
:
υ
n
n
(
z
)
.
It follows that ρ
n
(
z
) is the Lebesgue measure on C
n normalized such that the ball B
n
(
r
) has measure r
2
n.Let us start with the one-dimensional case at first, the following result is based on the proof of [12, Lemma 5.1].Lemma 2.
Letf be a meromorphic function in C such that f
(
0
)
≠
0
,
∞, and let q
∈
C
∖
{
0
}. Then
(9)
∫
∂
B
1
(
r
)
log
⁡
+
|
f
(
q
z
)
f
(
z
)
|
σ
1
(
z
)
≤
(
|
q
-
1
|
δ
(
|
q
|
δ
+
1
)
δ
(
1
-
δ
)
|
q
|
δ
+
|
q
-
1
|
r
λ
-
|
q
|
r
+
|
q
-
1
|
r
λ
-
r
)
×
(
n
(
λ
,
f
)
+
n
(
λ
,
1
f
)
)
+
2
|
q
-
1
|
r
λ
(
λ
-
r
)
(
λ
-
|
q
|
r
)
(
m
(
λ
,
f
)
+
m
(
λ
,
1
f
)
)
,
where z
=
r
e
i
ϕ, λ
>
max
⁡
{
r
,
|
q
|
r
}, and 0
<
δ
<
1.Letτ
m
=
τ
0
,
C
n be the parabolic exhaustion of C
n and define a function on B
¯
n
-
1
(
r
) by
(10)
p
r
(
ω
)
=
r
2
-
τ
n
-
1
(
ω
)
=
r
2
-
|
ω
|
2
.
For the following discussion, we will need two lemmas from Stoll.Lemma 3 (see [13, Lemma 1.29]).
Letr
>
0, and let h be a function on ∂
B
n
(
r
) such that h
σ
n is integrable over ∂
B
n
(
r
). Then
(11)
∫
∂
B
n
(
r
)
h
(
z
)
σ
n
(
z
)
=
1
r
2
n
-
2
∫
B
¯
n
-
1
(
r
)
∫
∂
B
1
(
p
r
(
w
)
)
h
(
ω
,
ζ
)
σ
1
(
ζ
)
ρ
n
-
1
(
ω
)
.Letf be a nonconstant meromorphic function on C
n, take ω
∈
C
n
-
1, and define the holomorphic map j
m
:
C
→
C
n by j
w
(
z
)
=
(
w
,
z
); thus f
ω
(
z
)
=
f
(
ω
,
z
).Lemma 4 (see [13, Lemma 1.30]).
Letf be a nonconstant meromorphic function in C
n and take a
∈
P
1. If r
>
0, then
(12)
1
r
2
n
-
2
∫
∂
B
n
(
r
)
n
(
p
r
(
ω
)
,
1
f
ω
-
a
)
σ
n
-
1
(
ω
)
≤
n
(
r
,
1
f
-
a
)
,
1
r
2
n
-
2
∫
∂
B
n
(
r
)
m
(
p
r
(
ω
)
,
1
f
ω
-
a
)
σ
n
-
1
(
ω
)
=
m
(
r
,
1
f
-
a
)
.We proceed to generalize Lemma2 to several complex variables by using Lemmas 3 and 4.Lemma 5.
Letf be a meromorphic function in C
n such that f
(
0
)
≠
0
,
∞, and let q
j
~
≔
(
1
,
…
,
q
j
,
…
,
1
). Then
(13)
∫
∂
B
n
(
r
)
log
⁡
+
|
f
(
q
j
~
z
)
f
(
z
)
|
σ
n
(
z
)
≤
(
|
q
j
~
-
1
|
δ
(
|
q
j
~
|
δ
+
1
)
δ
(
1
-
δ
)
|
q
j
~
|
δ
+
|
q
j
~
-
1
|
r
R
-
|
q
j
~
|
r
+
|
q
j
~
-
1
|
r
R
-
r
)
×
(
n
(
R
,
f
)
+
n
(
R
,
1
f
)
)
+
2
|
q
-
1
|
r
R
(
R
-
r
)
(
R
-
|
q
|
r
)
(
m
(
R
,
f
)
+
m
(
R
,
1
f
)
)
,
where R
>
max
⁡
{
r
,
|
q
j
~
|
r
} and 0
<
δ
<
1.Proof.
By applying Lemma3 with h
(
z
)
=
log
⁡
+
|
f
(
q
j
~
z
)
/
f
(
z
)
|, we obtain
(14)
∫
∂
B
n
(
r
)
‍
log
⁡
+
|
f
(
q
j
~
z
)
f
(
z
)
|
σ
n
(
z
)
=
1
r
2
n
-
2
∫
B
¯
n
-
1
(
r
)
∫
∂
B
1
(
p
r
(
w
)
)
log
⁡
+
|
f
ω
(
q
j
~
z
)
f
ω
(
z
)
|
σ
1
(
ζ
)
ρ
n
-
1
(
ω
)
≤
1
r
2
n
-
2
∫
B
¯
n
-
1
(
r
)
(
|
q
j
~
-
1
|
δ
(
|
q
j
~
|
δ
+
1
)
δ
(
1
-
δ
)
|
q
j
~
|
δ
+
|
q
j
~
-
1
|
p
r
(
ω
)
p
R
(
ω
)
-
|
q
j
~
|
p
r
(
ω
)
+
|
q
j
~
-
1
|
δ
(
|
q
j
~
|
δ
+
1
)
δ
(
1
-
δ
)
|
q
j
~
|
δ
|
q
j
~
-
1
|
p
r
(
ω
)
p
R
(
ω
)
-
p
r
(
ω
)
)
×
(
n
(
p
R
(
ω
)
,
f
ω
(
z
)
)
+
n
(
p
R
(
ω
)
,
1
f
ω
(
z
)
)
)
ρ
n
-
1
(
ω
)
+
1
r
2
n
-
2
∫
B
¯
n
-
1
(
r
)
2
|
q
j
~
-
1
|
p
r
(
ω
)
p
R
(
ω
)
(
p
R
(
ω
)
-
p
r
(
ω
)
)
(
p
R
(
ω
)
-
|
q
|
p
r
(
ω
)
)
×
(
m
(
p
R
(
ω
)
,
f
ω
(
z
)
)
+
m
(
p
R
(
ω
)
,
1
f
ω
(
z
)
)
)
ρ
n
-
1
(
ω
)
≔
S
n
+
S
m
,
for R
>
max
⁡
{
r
,
|
q
j
~
|
r
} and 0
<
δ
<
1. Now we will estimate terms S
m and S
n, respectively. Note that
(15)
p
r
(
ω
)
p
R
(
ω
)
≤
r
R
for R
>
r and the fact that x
/
(
x
-
1
)
(
x
-
t
) is decreasing with x for x
>
max
⁡
{
1
,
t
}. By using Lemma 4, it follows that
(16)
S
n
≤
(
R
r
)
2
n
-
2
(
C
(
q
j
~
,
δ
)
+
|
q
j
~
-
1
|
r
R
-
|
q
j
~
|
r
+
|
q
j
~
-
1
|
r
R
-
r
)
×
n
f
(
R
,
0
,
∞
)
,
S
m
≤
(
R
r
)
2
n
-
2
2
|
q
j
~
-
1
|
r
R
(
R
-
r
)
(
R
-
|
q
|
r
)
m
f
(
R
,
0
,
∞
)
,
where
(17)
C
(
q
j
~
,
δ
)
=
|
q
j
~
-
1
|
δ
(
|
q
j
~
|
δ
+
1
)
δ
(
1
-
δ
)
|
q
j
~
|
δ
,
n
f
(
R
,
0
,
∞
)
=
(
n
(
R
,
f
)
+
n
(
R
,
1
f
)
)
,
m
f
(
R
,
0
,
∞
)
=
(
m
(
R
,
f
)
+
m
(
R
,
1
f
)
)
.
It implies the assertion.To deal withm
f
(
R
,
0
,
∞
), we need the following lemma.Lemma 6 (see [14, Lemma 4]).
IfT
:
R
+
→
R
+ is a piecewise continuous increasing function such that
(18)
lim
⁡
r
→
∞
log
⁡
T
(
r
)
log
⁡
r
=
0
,
then the set
(19)
E
≔
{
r
:
T
(
C
1
r
)
≥
C
2
T
(
r
)
}
has logarithmic density 0 for all C
1
>
1 and C
2
>
1.To show thatn
f
(
R
,
0
,
∞
) is small, we need the following two lemmas.Lemma 7 (see [12, Lemma 5.4]).
LetT
:
R
+
→
R
+ be an increasing function, and let U
:
R
+
→
R
+. If there exists a decreasing sequence {
c
n
}
n
∈
N such that c
n
→
0 as n
→
∞ and for all n
∈
N, the set
(20)
F
n
≔
{
r
≥
1
:
U
(
r
)
<
c
n
T
(
r
)
}
has logarithmic density 1, then
(21)
U
(
r
)
=
o
(
T
(
r
)
)
on a set of logarithmic density 1.The following lemma is based on the proof of Lemma  5.3 in [12], which is the case of one dimension.Lemma 8.
Iff is a nonconstant meromorphic function in C
n of zero order, then the set
(22)
E
n
≔
{
r
≥
:
n
(
r
,
f
)
<
T
(
r
,
f
)
2
n
}
has logarithmic density 1 for all n
∈
N.If we takeR
=
k
r in Lemma 5, and by using Lemmas 6, 7, and 8, then we have
(23)
∫
∂
B
n
(
r
)
log
⁡
+
|
f
(
q
j
~
z
)
f
(
z
)
|
σ
n
(
z
)
=
o
(
T
(
r
,
f
)
)
on a set of logarithmic density 1, since f is a meromorphic function in C
n of zero order. According to (23), it follows that
(24)
∫
∂
B
n
(
r
)
log
⁡
+
|
f
(
q
z
)
f
(
z
)
|
σ
n
(
z
)
=
∫
∂
B
n
(
r
)
log
⁡
+
∏
k
=
1
n
|
f
(
z
∏
j
=
0
k
q
j
~
)
f
(
z
∏
j
=
0
k
-
1
q
j
~
)
|
σ
n
(
z
)
≤
∑
k
=
1
n
‍
∫
∂
B
n
(
r
)
log
⁡
+
|
f
(
z
∏
j
=
0
k
q
j
~
)
f
(
z
∏
j
=
0
k
-
1
q
j
~
)
|
σ
n
(
z
)
=
o
(
T
(
r
,
f
)
)
on a set of logarithmic density 1. Therefore, we have q-difference analogue of the lemma on logarithmic derivative of several variables, which is the proof of Theorem 1.


## 3. First and Second Main Theorems

We will discuss the relation ofT
(
r
,
f
(
q
z
)
) and T
(
r
,
f
(
z
)
), which is the q-shift analogue of the first main theory of several variables.Theorem 9.
Letf be a meromorphic function in C
n of zero order such that f
(
0
)
≠
0
,
∞, and let q
∈
C
n
∖
{
0
}. Then
(25)
T
(
r
,
f
(
q
z
)
)
=
T
(
r
,
f
(
z
)
)
+
o
(
T
(
r
,
f
)
)
on a set of logarithmic density 1.Proof.
By using Lemma6 and Theorem 1, we have
(26)
T
(
r
,
f
(
q
z
)
)
=
m
(
r
,
f
(
q
z
)
)
+
N
(
r
,
f
(
q
z
)
)
=
m
(
r
,
f
(
z
)
)
+
N
(
|
q
|
r
,
f
(
z
)
)
+
m
(
r
,
f
(
q
z
)
f
(
z
)
)
≤
T
(
r
,
f
)
+
o
(
T
(
r
,
f
)
)
,
on a set of logarithmic density 1. We have the assertion.Let us discuss theq-shift analogue of the second main theory of several variables.Theorem 10.
Letf be a meromorphic function in C
n of zero order such that f
(
0
)
≠
0
,
∞, and let q
∈
C
n
∖
{
0
}. If a
i
(
i
=
1
,
…
,
q
)
∈
P
1 are distinct finite constants, then
(27)
m
(
r
,
f
)
+
∑
j
=
1
q
‍
m
(
r
,
1
f
-
a
j
)
≤
2
T
(
r
,
f
)
-
N
p
a
i
r
(
r
,
f
)
+
S
q
(
r
,
f
)
,
where
(28)
N
p
a
i
r
(
r
,
f
)
≔
2
N
(
r
,
f
)
-
N
(
r
,
Δ
q
f
)
+
N
(
r
,
1
Δ
q
f
)
.Proof.
By using the first main theorem, we obtain(29)
∑
j
=
1
q
m
(
r
,
1
f
-
a
j
)
=
∑
j
=
1
q
T
(
r
,
1
f
-
a
j
)
-
∑
j
=
1
q
N
(
r
,
1
f
-
a
j
)
=
q
T
(
r
,
f
)
-
N
(
r
,
1
P
(
f
)
)
+
S
q
(
r
,
f
)
,
where
(30)
P
(
r
,
f
)
=
∏
j
=
1
q
‍
(
f
-
a
j
)
.
Since there exist α
j
∈
P
1 such that
(31)
1
P
(
f
)
=
∑
j
=
1
q
‍
α
j
f
-
a
j
then
(32)
m
(
r
,
1
P
(
f
)
)
≤
m
(
r
,
Δ
q
f
P
(
f
)
)
+
m
(
r
,
1
Δ
q
f
)
≤
m
(
r
,
1
Δ
q
f
)
+
∑
j
=
1
q
‍
m
(
r
,
Δ
q
f
f
-
a
j
)
=
m
(
r
,
1
Δ
q
f
)
+
S
q
(
r
,
f
)
.
By applying (29), (32), and the fact T
(
r
,
P
(
f
)
)
=
q
T
(
r
,
f
)
+
O
(
1
), we have
(33)
∑
j
=
1
q
m
(
r
,
1
f
-
a
j
)
=
T
(
r
,
P
(
f
)
)
-
N
(
r
,
1
P
(
f
)
)
+
S
q
(
r
,
f
)
=
m
(
r
,
1
P
(
f
)
)
+
S
q
(
r
,
f
)
≤
m
(
r
,
1
Δ
q
f
)
+
S
q
(
r
,
f
)
=
T
(
r
,
Δ
q
f
)
-
N
(
r
,
1
Δ
q
f
)
+
S
q
(
r
,
f
)
.
Therefore, we have
(34)
m
(
r
,
f
)
+
∑
j
=
1
q
m
(
r
,
1
f
-
a
j
)
≤
T
(
r
,
f
)
+
N
(
r
,
Δ
q
f
)
+
m
(
r
,
Δ
q
f
)
-
N
(
r
,
1
Δ
q
f
)
-
N
(
r
,
f
)
+
S
q
(
r
,
f
)
.
Since
(35)
m
(
r
,
Δ
q
f
)
≤
m
(
r
,
f
)
+
m
(
r
,
Δ
q
f
f
)
=
m
(
r
,
f
)
+
S
q
(
r
,
f
)
,
we have the assertion.


## 4. Application

Letf and g be two meromorphic functions in C
n, and let a
∈
C
n, if f
-
a and g
-
a have the same zeros (counting multiplicities), then we say f and g share a CM in C
n.Theorem 11.
Letf
:
C
n
→
P
1 be a nonconstant meromorphic function of zero order and let q
∈
C
n
∖
{
0
}. If f
(
z
) and f
(
q
z
) share three distinct values CM, then f
(
z
)
=
f
(
q
z
).Proof.
Let us suppose thatf
(
z
) and f
(
q
z
) share {
0,1
,
∞
} CM, if not, let us make a linearly transformation. Suppose that Δ
q
f
≢
0, Theorem 10 yields
(36)
m
(
r
,
f
)
+
m
(
r
,
1
f
)
+
m
(
r
,
1
f
-
1
)
≤
2
T
(
r
,
f
)
+
N
(
r
,
Δ
q
f
)
-
2
N
(
r
,
f
)
-
N
(
r
,
1
Δ
q
f
)
+
S
q
(
r
,
f
)
.
Since f
(
z
) and f
(
q
z
) share {
0,1
,
∞
} CM, then
(37)
N
(
r
,
f
)
+
N
(
r
,
1
f
)
+
N
(
r
,
1
f
-
1
)
≤
N
(
r
,
1
Δ
q
f
)
.
In addition, Lemma 6 implies that N
(
r
,
Δ
q
f
)
≤
2
N
(
r
,
f
)
+
S
q
(
r
,
f
), by combining (36) and (37), it follows that
(38)
T
(
r
,
f
)
=
S
q
(
r
,
f
)
which is impossible.Theq-difference polynomials of f
(
z
) in C
n are said by the functions which are polynomials in f
(
q
j
z
), where q
j
∈
C
n, with coefficients a
λ
(
z
) such that T
(
r
,
a
λ
(
z
)
)
=
o
(
T
(
r
,
f
)
) on a set of logarithmic density 1.The following theorem isq-shift analogue of Clunie lemma [15] of several variables.Theorem 12.
Letf
:
C
n
→
P
1 be a nonconstant zero order meromorphic solution in C
n of
(39)
f
(
z
)
n
P
(
z
,
f
)
=
Q
(
z
,
f
)
,
where P
(
z
,
f
) and Q
(
z
,
f
) are q-difference polynomials in f
(
z
). If the degree of Q
(
z
,
f
) as a polynomial in f
(
z
) and its q-shift is at most n, then
(40)
m
(
r
,
P
(
z
,
f
)
)
=
S
q
(
r
,
f
)
.Proof.
In calculating the proximity function ofP
(
z
,
f
), we split the region of integration into two parts. By defining
(41)
E
1
≔
{
z
∈
∂
B
n
(
r
)
∣
|
f
(
z
)
|
<
1
}
,
E
2
≔
∂
B
n
(
r
)
∖
E
1
,
we have
(42)
m
(
r
,
P
(
z
,
f
)
)
=
∫
E
1
log
⁡
+
|
P
(
z
,
f
)
|
σ
n
(
z
)
+
∫
E
2
log
⁡
+
|
P
(
z
,
f
)
|
σ
n
(
z
)
.
First, we consider E
1; we can write P
(
z
,
f
) as
(43)
P
(
z
,
f
)
=
∑
c
i
(
z
)
f
(
z
)
i
0
f
(
q
z
)
i
1
⋯
f
(
q
ν
z
)
l
ν
.
Thus, we have
(44)
|
P
(
z
,
f
)
|
≤
∑
|
c
i
(
z
)
|
|
f
(
q
z
)
i
1
f
(
z
)
|
l
1
⋯
|
f
(
q
ν
z
)
l
ν
f
(
z
)
|
.
Therefore, we obtain
(45)
∫
E
1
log
⁡
+
|
P
(
z
,
f
)
|
σ
n
(
z
)
≤
∑
m
(
r
,
c
i
)
+
O
(
∑
i
=
1
q
m
(
r
,
f
(
q
z
)
f
(
z
)
)
)
,together with Theorem 1 implies that(46)
∫
E
1
log
⁡
+
|
P
(
z
,
f
)
|
σ
n
(
z
)
=
S
q
(
r
,
f
)
.
Now let us consider E
2; we note that
(47)
Q
(
z
,
f
)
=
∑
b
γ
(
z
)
f
(
q
0
z
)
γ
0
⋯
f
(
q
μ
z
)
γ
μ
.
Hence, we have
(48)
|
P
(
z
,
f
)
|
=
|
1
f
(
z
)
n
∑
b
γ
(
z
)
f
(
q
0
z
)
γ
0
⋯
f
(
q
μ
z
)
γ
μ
|
≤
∑
|
b
γ
(
z
)
|
|
f
(
q
1
z
)
f
(
z
)
|
γ
1
⋯
|
f
(
q
μ
z
)
f
(
z
)
|
γ
μ
,
by Theorem 1 again, it follows that
(49)
∫
E
2
log
⁡
+
|
P
(
z
,
f
)
|
σ
n
(
z
)
=
S
q
(
r
,
f
)
.
The assertion follows by combining (42)–(49).Letα and f be meromorphic functions of zero order in C
n such that T
(
r
,
α
)
=
o
(
T
(
r
,
f
)
) on a set of logarithmic density 1. Then α is called a small function with respect to f of zero order in C
n.The following theorem isq-shift analogue of Mohon’ho lemma [16] of several variables.Theorem 13.
Letf
(
z
) be a nonconstant zero-order meromorphic solution in C
n of
(50)
P
(
z
,
f
)
=
0
,
where P
(
z
,
f
) is a q-difference polynomial in f
(
z
). If P
(
z
,
α
)
≢
0 for a small function α in C
n of f
(
z
), then
(51)
m
(
r
,
1
f
-
α
)
=
S
q
(
r
,
f
)
.Proof.
By substitutingf
=
g
+
α into (50), we obtain
(52)
Q
(
z
,
g
)
+
D
(
z
)
=
0
,
where
(53)
Q
(
z
,
g
)
=
∑
γ
=
(
j
0
,
…
,
j
ν
)
∈
J
b
ν
(
z
)
g
(
z
)
j
0
⋯
g
(
q
ν
z
)
j
ν
is a q-difference polynomial in g such that all of its terms are at least degree 1, and T
(
r
,
D
)
=
S
q
(
r
,
g
), T
(
r
,
b
ν
(
z
)
)
=
S
q
(
r
,
g
) for each ν
∈
J. Also D
≢
0, since α does not satisfy (50). By using (52), we have
(54)
m
(
r
,
1
g
)
≤
m
(
r
,
D
g
)
+
m
(
r
,
1
D
)
=
m
(
r
,
Q
(
z
,
g
)
g
)
+
m
(
r
,
1
D
)
.
Note that since the integral m
(
r
,
1
/
g
) vanishes on the part of ∂
B
n
(
r
) where |
g
|
>
1, it is sufficient to consider only the case |
g
|
≤
1. Then
(55)
|
Q
(
z
,
g
)
g
|
=
1
|
g
|
|
∑
γ
∈
J
b
ν
(
z
)
g
(
z
)
j
0
⋯
g
(
q
ν
z
)
j
ν
|
≤
∑
γ
∈
J
‍
|
b
ν
(
z
)
|
|
g
(
q
1
z
)
g
(
z
)
|
j
1
⋯
|
g
(
q
ν
z
)
g
(
z
)
|
j
ν
,
by Theorem 1, (54), and (55), it shows
(56)
m
(
r
,
1
g
)
=
S
q
(
r
,
g
)
.
Since g
=
f
-
α, the assertion follows.



---

*Source: 736021-2014-06-01.xml*