Difference Theorems for Meromorphic Functions of Several Variables

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 736021, 6 pages http://dx.doi.org/10.1155/2014/736021

Research Article The 𝑞-Difference Theorems for Meromorphic Functions of Several Variables Zhi-Tao Wen Department of Mathematics, Taiyuan University of Technology, Yingze West Street, No. 79, Taiyuan 030024, China Correspondence should be addressed to Zhi-Tao Wen; [email protected] Received 28 March 2014; Revised 9 May 2014; Accepted 22 May 2014; Published 1 June 2014 Academic Editor: Bashir Ahmad Copyright © 2014 Zhi-Tao Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate 𝑞-shift analogue of the lemma on logarithmic derivative of several variables. Let 𝑓 be a meromorphic function in C𝑛 of zero order such that 𝑓(0) ≠ 0, ∞, and let 𝑞 ∈ C𝑛 \{0}. Then we have 𝑚(𝑟, 𝑓(𝑞𝑧)/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.

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 Painlev´e 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 𝑞difference 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 C𝑛 , if 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ) and 𝑧󸀠 = (𝑧1󸀠 , . . . , 𝑧𝑛󸀠 ), then we denote 𝑧1 𝑧1󸀠 = (𝑧1 𝑧1󸀠 , . . . , 𝑧𝑛 𝑧𝑛󸀠 ). 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. Let 𝑓 be a meromorphic function in C𝑛 of zero order such that 𝑓(0) ≠ 0, ∞, and let 𝑞 ∈ C𝑛 \ {0}. Then 𝑚 (𝑟,

𝑓 (𝑞𝑧) ) = 𝑜 (𝑇 (𝑟, 𝑓)) 𝑓 (𝑧)

(1)

on a set of logarithmic density 1. Concerning the sharpness of Theorem 1, 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 Section 2, 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 present 𝑞-difference analogues of the lemma on logarithmic derivative in several complex variables. Let us recall some of the standard notations of Nevanlinna theory in C𝑛 .

2

Abstract and Applied Analysis

Let 𝑀 be a connected complex manifold of dimension 𝑛 and let

Let 𝜏𝑚 = 𝜏0,C𝑛 be the parabolic exhaustion of C𝑛 and define a function on 𝐵𝑛−1 (𝑟) by

2𝑚

𝐴 (𝑀) = ∑ 𝐴𝑛 (𝑀)

(2)

𝑝𝑟 (𝜔) = √𝑟2 − 𝜏𝑛−1 (𝜔) = √𝑟2 − |𝜔|2 .

𝑛=0

be the graded ring of complex valued differential forms on 𝑀. Each set 𝐴𝑛 (𝑀) can be split into a direct sum 𝐴𝑛 (𝑀) = ∑ 𝐴𝑝,𝑞 (𝑀) , 𝑝+𝑞=𝑛

(3)

where 𝐴𝑝,𝑞 (𝑀) is the forms of type (𝑝, 𝑞). The differential operators 𝑑 and 𝑑𝑐 on 𝐴(𝑀) are defined as 1 (𝜕 − 𝜕) , 4𝜋𝑖

𝑑𝑐 :=

𝑑 := 𝜕 + 𝜕,

(4)

For the following discussion, we will need two lemmas from Stoll. Lemma 3 (see [13, Lemma 1.29]). Let 𝑟 > 0, and let ℎ be a function on 𝜕𝐵𝑛 (𝑟) such that ℎ𝜎𝑛 is integrable over 𝜕𝐵𝑛 (𝑟). Then ∫

𝜕𝐵𝑛 (𝑟)

where 𝜕 : 𝐴𝑝,𝑞 (𝑀) 󳨀→ 𝐴𝑝+1,𝑞 (𝑀) , 𝜕 : 𝐴𝑝,𝑞 (𝑀) 󳨀→ 𝐴𝑝,𝑞+1 (𝑀) .

(5)

Let 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ) ∈ C , and let 𝑟 > 0 and fix 𝑎 = (𝑎1 , . . . , 𝑎𝑛 ) ∈ C𝑛 ; an exhaustion function 𝜏𝑎,C𝑛 of C𝑛 is defined by 𝑚

󵄨 󵄨2 𝜏𝑎,C𝑛 (𝑧) = |𝑧 − 𝑎| = ∑ 󵄨󵄨󵄨󵄨𝑧𝑗 − 𝑎𝑗 󵄨󵄨󵄨󵄨 . 2

𝑗=1

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Usually, we let 𝑎 = 0 be the origin and set 𝜏𝑚 (𝑧) = 𝜏0,C𝑛 (𝑧). Let us define a positive measure 𝜎𝑛 (𝑧) with total measure one on the boundary 𝜕𝐵𝑛 (𝑟) := {𝑧 ∈ C𝑛 : |𝑧| = 𝑟} of the ball 𝐵𝑛 (𝑟) := {𝑧 ∈ C𝑛 : |𝑧| < 𝑟} as 𝜔𝑛 (𝑧) := 𝑑𝑑𝑐 log 𝜏𝑛 ,

ℎ (𝑧) 𝜎𝑛 (𝑧)

=

𝑛

𝜎𝑛 (𝑧) := 𝑑𝑐 log 𝜏𝑛 ∧ 𝜔𝑛𝑛−1 (𝑧) . (7)

(10)

1 𝑟2𝑛−2

(11) ∫

∫

𝐵𝑛−1 (𝑟) 𝜕𝐵1 (𝑝𝑟 (𝑤))

ℎ (𝜔, 𝜁) 𝜎1 (𝜁) 𝜌𝑛−1 (𝜔) .

Let 𝑓 be a nonconstant meromorphic function on C𝑛 , take 𝜔 ∈ C𝑛−1 , and define the holomorphic map 𝑗𝑚 : C → C𝑛 by 𝑗𝑤 (𝑧) = (𝑤, 𝑧); thus 𝑓𝜔 (𝑧) = 𝑓(𝜔, 𝑧). Lemma 4 (see [13, Lemma 1.30]). Let 𝑓 be a nonconstant meromorphic function in C𝑛 and take 𝑎 ∈ P1 . If r > 0, then 1 1 1 𝑛 (𝑝𝑟 (𝜔) , ) 𝜎 (𝜔) ≤ 𝑛 (𝑟, ), ∫ 𝑟2𝑛−2 𝜕𝐵𝑛 (𝑟) 𝑓𝜔 − 𝑎 𝑛−1 𝑓−𝑎 1 1 1 𝑚 (𝑝𝑟 (𝜔) , ) 𝜎 (𝜔) = 𝑚 (𝑟, ). ∫ 𝑟2𝑛−2 𝜕𝐵𝑛 (𝑟) 𝑓𝜔 − 𝑎 𝑛−1 𝑓−𝑎 (12)

In addition, we define 𝜐𝑛 (𝑧) := 𝑑𝑑𝑐 𝜏𝑛 (𝑧) ,

𝜌𝑛 (𝑧) : 𝜐𝑛𝑛 (𝑧) .

(8)

It follows that 𝜌𝑛 (𝑧) is the Lebesgue measure on C𝑛 normalized such that the ball 𝐵𝑛 (𝑟) has measure 𝑟2𝑛 . 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. Let 𝑓 be a meromorphic function in C such that 𝑓(0) ≠ 0, ∞, and let 𝑞 ∈ C \ {0}. Then 󵄨󵄨 𝑓 (𝑞𝑧) 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 𝜎 (𝑧) log+ 󵄨󵄨󵄨󵄨 ∫ 󵄨󵄨 𝑓 (𝑧) 󵄨󵄨󵄨 1 𝜕𝐵1 (𝑟) 󵄨𝛿 󵄨 󵄨𝛿 󵄨󵄨 󵄨󵄨𝑞 − 1󵄨󵄨󵄨 (󵄨󵄨󵄨𝑞󵄨󵄨󵄨 + 1) 󵄨󵄨󵄨𝑞 − 1󵄨󵄨󵄨 𝑟 󵄨󵄨󵄨𝑞 − 1󵄨󵄨󵄨 𝑟 󵄨 󵄨 󵄨 󵄨 ≤( 󵄨 󵄨𝛿 + 𝜆 − 󵄨󵄨󵄨𝑞󵄨󵄨󵄨 𝑟 + 𝜆 − 𝑟 ) 𝛿 (1 − 𝛿) 󵄨󵄨󵄨𝑞󵄨󵄨󵄨 󵄨 󵄨 (9) × (𝑛 (𝜆, 𝑓) + 𝑛 (𝜆, +

1 )) 𝑓

󵄨 󵄨 2 󵄨󵄨󵄨𝑞 − 1󵄨󵄨󵄨 𝑟𝜆 1 󵄨󵄨 󵄨󵄨 (𝑚 (𝜆, 𝑓) + 𝑚 (𝜆, 𝑓 )) , (𝜆 − 𝑟) (𝜆 − 󵄨󵄨𝑞󵄨󵄨 𝑟)

where 𝑧 = 𝑟𝑒𝑖𝜙 , 𝜆 > max{𝑟, |𝑞|𝑟}, and 0 < 𝛿 < 1.

We proceed to generalize Lemma 2 to several complex variables by using Lemmas 3 and 4. Lemma 5. Let 𝑓 be a meromorphic function in C𝑛 such that 𝑓(0) ≠ 0, ∞, and let 𝑞̃𝑗 := (1, . . . , 𝑞𝑗 , . . . , 1). Then 󵄨󵄨 󵄨 󵄨󵄨 𝑓 (̃ 𝑞𝑗 𝑧) 󵄨󵄨󵄨 󵄨󵄨 𝜎 (𝑧) log+ 󵄨󵄨󵄨󵄨 󵄨 󵄨󵄨 𝑓 (𝑧) 󵄨󵄨󵄨 𝑛 𝜕𝐵𝑛 (𝑟) 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝛿 󵄨󵄨 󵄨󵄨𝛿 󵄨󵄨𝑞̃𝑗 − 1󵄨󵄨 (󵄨󵄨𝑞̃𝑗 󵄨󵄨 + 1) 󵄨󵄨󵄨𝑞̃ − 1󵄨󵄨󵄨 𝑟 󵄨 󵄨 󵄨 󵄨 󵄨 𝑗 󵄨 ≤( + 󵄨 󵄨󵄨 󵄨󵄨󵄨 + 󵄨󵄨 󵄨󵄨𝛿 𝑅 − 󵄨󵄨󵄨𝑞̃𝑗 󵄨󵄨󵄨 𝑟 𝛿 (1 − 𝛿) 󵄨󵄨󵄨𝑞̃𝑗 󵄨󵄨󵄨

∫

× (𝑛 (𝑅, 𝑓) + 𝑛 (𝑅, +

󵄨󵄨 󵄨 󵄨󵄨𝑞̃𝑗 − 1󵄨󵄨󵄨 𝑟 󵄨 󵄨 ) 𝑅−𝑟

1 )) 𝑓

󵄨 󵄨 2 󵄨󵄨󵄨𝑞 − 1󵄨󵄨󵄨 𝑟𝑅 1 󵄨 󵄨 (𝑚 (𝑅, 𝑓) + 𝑚 (𝑅, 𝑓 )) , (𝑅 − 𝑟) (𝑅 − 󵄨󵄨󵄨𝑞󵄨󵄨󵄨 𝑟)

where 𝑅 > max{𝑟, |̃ 𝑞𝑗 |𝑟} and 0 < 𝛿 < 1.

(13)

Abstract and Applied Analysis

3

Proof. By applying Lemma 3 with ℎ(𝑧) = log+ |𝑓(̃ 𝑞𝑗 𝑧)/𝑓(𝑧)|, we obtain 󵄨 󵄨󵄨 󵄨󵄨 𝑓 (̃ 𝑞𝑗 𝑧) 󵄨󵄨󵄨󵄨 󵄨 󵄨󵄨 𝜎 (𝑧) 󵄨 log 󵄨󵄨 ∫ 󵄨󵄨 𝑓 (𝑧) 󵄨󵄨󵄨 𝑛 𝜕𝐵𝑛 (𝑟) 󵄨 󵄨

where 󵄨󵄨 󵄨𝛿 󵄨 󵄨𝛿 󵄨󵄨𝑞̃𝑗 − 1󵄨󵄨󵄨 (󵄨󵄨󵄨𝑞̃𝑗 󵄨󵄨󵄨 + 1) 󵄨 󵄨 󵄨 󵄨 𝐶 (̃ 𝑞𝑗 , 𝛿) = 󵄨 󵄨𝛿 , 𝛿 (1 − 𝛿) 󵄨󵄨󵄨󵄨𝑞̃𝑗 󵄨󵄨󵄨󵄨

+

=

≤

1 𝑟2𝑛−2 1 𝑟2𝑛−2

󵄨󵄨 󵄨 󵄨󵄨 𝑓𝜔 (̃ 𝑞𝑗 𝑧) 󵄨󵄨󵄨 󵄨󵄨 𝜎 (𝜁) 𝜌 (𝜔) log+ 󵄨󵄨󵄨󵄨 󵄨 𝑛−1 󵄨󵄨 𝑓𝜔 (𝑧) 󵄨󵄨󵄨 1 𝐵𝑛−1 (𝑟) 𝜕𝐵1 (𝑝𝑟 (𝑤)) 󵄨 󵄨 󵄨󵄨 󵄨𝛿 󵄨 󵄨𝛿 󵄨󵄨𝑞̃𝑗 − 1󵄨󵄨󵄨 (󵄨󵄨󵄨𝑞̃𝑗 󵄨󵄨󵄨 + 1) 󵄨 󵄨 󵄨 󵄨 ( ∫ 󵄨 󵄨𝛿 𝐵𝑛−1 (𝑟) 𝛿 (1 − 𝛿) 󵄨󵄨󵄨󵄨𝑞̃𝑗 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨𝑞̃𝑗 − 1󵄨󵄨󵄨 𝑝𝑟 (𝜔) 󵄨 󵄨 + 󵄨 󵄨 𝑝𝑅 (𝜔) − 󵄨󵄨󵄨󵄨𝑞̃𝑗 󵄨󵄨󵄨󵄨 𝑝𝑟 (𝜔) ∫

∫

+

1 ∫ 𝑟2𝑛−2 𝐵𝑛−1 (𝑟)

(17)

1 )) . 𝑓

It implies the assertion. To deal with 𝑚𝑓 (𝑅, 0, ∞), we need the following lemma. Lemma 6 (see [14, Lemma 4]). If 𝑇 : R+ → R+ is a piecewise continuous increasing function such that 𝑟→∞

log 𝑇 (𝑟) = 0, log 𝑟

(18)

then the set

1 )) 𝜌𝑛−1 (𝜔) 𝑓𝜔 (𝑧)

󵄨 󵄨 2 󵄨󵄨󵄨󵄨𝑞̃𝑗 − 1󵄨󵄨󵄨󵄨 𝑝𝑟 (𝜔) 𝑝𝑅 (𝜔) 󵄨 󵄨 (𝑝𝑅 (𝜔) − 𝑝𝑟 (𝜔)) (𝑝𝑅 (𝜔) − 󵄨󵄨󵄨𝑞󵄨󵄨󵄨 𝑝𝑟 (𝜔))

× (𝑚 (𝑝𝑅 (𝜔) , 𝑓𝜔 (𝑧)) + 𝑚 (𝑝𝑅 (𝜔) ,

1 )) , 𝑓

𝑚𝑓 (𝑅, 0, ∞) = (𝑚 (𝑅, 𝑓) + 𝑚 (𝑅,

lim

󵄨󵄨 󵄨 󵄨󵄨𝑞̃𝑗 − 1󵄨󵄨󵄨 𝑝𝑟 (𝜔) 󵄨 + 󵄨 ) 𝑝𝑅 (𝜔) − 𝑝𝑟 (𝜔) × (𝑛 (𝑝𝑅 (𝜔) , 𝑓𝜔 (𝑧)) + 𝑛 (𝑝𝑅 (𝜔) ,

𝑛𝑓 (𝑅, 0, ∞) = (𝑛 (𝑅, 𝑓) + 𝑛 (𝑅,

1 )) 𝜌𝑛−1 (𝜔) 𝑓𝜔 (𝑧)

:= 𝑆𝑛 + 𝑆𝑚 , (14)

𝐸 := {𝑟 : 𝑇 (𝐶1 𝑟) ≥ 𝐶2 𝑇 (𝑟)} has logarithmic density 0 for all 𝐶1 > 1 and 𝐶2 > 1.

To show that 𝑛𝑓 (𝑅, 0, ∞) is small, we need the following two lemmas. Lemma 7 (see [12, Lemma 5.4]). Let 𝑇 : R+ → R+ be an increasing function, and let 𝑈 : R+ → R+ . If there exists a decreasing sequence {𝑐𝑛 }𝑛∈N such that 𝑐𝑛 → 0 as 𝑛 → ∞ and for all 𝑛 ∈ N, the set 𝐹𝑛 := {𝑟 ≥ 1 : 𝑈 (𝑟) < 𝑐𝑛 𝑇 (𝑟)}

for 𝑅 > max{𝑟, |̃ 𝑞𝑗 |𝑟} and 0 < 𝛿 < 1. Now we will estimate terms 𝑆𝑚 and 𝑆𝑛 , respectively. Note that

(19)

(20)

has logarithmic density 1, then 𝑈 (𝑟) = 𝑜 (𝑇 (𝑟))

(21)

on a set of logarithmic density 1. 𝑝𝑟 (𝜔) 𝑟 ≤ 𝑝𝑅 (𝜔) 𝑅

(15)

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

The following lemma is based on the proof of Lemma 5.3 in [12], which is the case of one dimension. Lemma 8. If 𝑓 is a nonconstant meromorphic function in C𝑛 of zero order, then the set 𝐸𝑛 := {𝑟 ≥: 𝑛 (𝑟, 𝑓)
1, it is sufficient to consider only the case |𝑔| ≤ 1. Then 󵄨󵄨 󵄨󵄨󵄨 1 󵄨󵄨󵄨 𝑗] 󵄨󵄨 𝑗0 󵄨 󵄨󵄨 𝑏 ⋅ ⋅ ⋅ 𝑔(𝑞 𝑧) 𝑔(𝑧) ∑ (𝑧) 󵄨󵄨 󵄨󵄨 󵄨󵄨 ] ] 󵄨󵄨 󵄨󵄨𝑔󵄨󵄨 󵄨󵄨𝛾∈𝐽 󵄨󵄨 󵄨 󵄨 󵄨𝑗1 󵄨󵄨 𝑔 (𝑞 𝑧) 󵄨󵄨𝑗] 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 𝑔 (𝑞1 𝑧) 󵄨󵄨󵄨󵄨 󵄨 ] 󵄨 󵄨 ≤ ∑ 󵄨󵄨󵄨𝑏] (𝑧)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ⋅ ⋅ ⋅ 󵄨󵄨󵄨 𝑔 (𝑧) 󵄨󵄨󵄨 , 𝑔 (𝑧) 󵄨 󵄨 󵄨 󵄨 󵄨 𝛾∈𝐽

󵄨󵄨 𝑄 (𝑧, 𝑔) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 𝑔 󵄨󵄨󵄨 = 󵄨 󵄨

(55)

by Theorem 1, (54), and (55), it shows 1 𝑚 (𝑟, ) = 𝑆𝑞 (𝑟, 𝑔) . 𝑔

(56)

Since 𝑔 = 𝑓 − 𝛼, the assertion follows.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

References [1] W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964. [2] C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.

[3] I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1993. [4] A. Vitter, “The lemma of the logarithmic derivative in several complex variables,” Duke Mathematical Journal, vol. 44, no. 1, pp. 89–104, 1977. [5] A. Biancofiore and W. Stoll, “Another proof of the lemma of the logarithmic derivative in several complex variables,” in Recent Developments in Several Complex Variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979), vol. 100 of Annals of Mathematics Studies, pp. 29–45, Princeton University Press, Princeton, NJ, USA, 1981. [6] Z. Ye, “A sharp form of Nevanlinna's second main theorem of several complex variables,” Mathematische Zeitschrift, vol. 222, no. 1, pp. 81–95, 1996. [7] B. Q. Li, “A logarithmic derivative lemma in several complex variables and its applications,” Transactions of the American Mathematical Society, vol. 363, no. 12, pp. 6257–6267, 2011. [8] R. G. Halburd and R. J. Korhonen, “Nevanlinna theory for the difference operator,” Annales Academiae Scientiarum Fennicae: Mathematica, vol. 31, no. 2, pp. 463–478, 2006. [9] R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477–487, 2006. [10] R. G. Halburd, R. Korhonen, and K. Tohge, “Holomorphic curves with shift-invariant hyperplane preimages,” Transactions of the American Mathematical Society, vol. 366, pp. 4267–4298, 2014. [11] R. Korhonen, “A difference Picard theorem for meromorphic functions of several variables,” Computational Methods and Function Theory, vol. 12, no. 1, pp. 343–361, 2012. [12] D. C. Barnett, R. G. Halburd, W. Morgan, and R. J. Korhonen, “Nevanlinna theory for the 𝑞-difference operator and meromorphic solutions of 𝑞-difference equations,” Proceedings of the Royal Society of Edinburgh A: Mathematics, vol. 137, no. 3, pp. 457–474, 2007. [13] P.-C. Hu, P. Li, and C.-C. Yang, Unicity of Meromorphic Mappings, vol. 1 of Advances in Complex Analysis and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. [14] W. K. Hayman, “On the characteristic of functions meromorphic in the plane and of their integrals,” Proceedings of the London Mathematical Society A, vol. 14, pp. 93–128, 1965. [15] J. Clunie, “On integral and meromorphic functions,” Journal of the London Mathematical Society, vol. 37, pp. 17–27, 1962. [16] A. A. Mohon’ko and V. D. Mohon’ko, “Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations,” ˇ vol. 15, pp. 1305–1322, 1431, 1974. Sibirski˘ı Matematiˇceski˘ı Zurnal,

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