The 𝑞-Difference Theorems for Meromorphic Functions of Several Variables

Abstract

We investigate 𝑞-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(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 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=(𝑧₁,…,𝑧_𝑛) and 𝑧^′=(𝑧^′₁,…,𝑧^′_𝑛), then we denote 𝑧₁𝑧^′₁=(𝑧₁𝑧^′₁,…,𝑧_𝑛𝑧^′_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. Let 𝑓 be a meromorphic function in ℂ^𝑛 of zero order such that 𝑓(0)≠0,∞, and let 𝑞∈ℂ^𝑛⧵{0}. Then 𝑚𝑓𝑟,𝑞𝑧𝑓(𝑧)=𝑜𝑇𝑟,𝑓 (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 ℂ^𝑛.

Let 𝑀 be a connected complex manifold of dimension 𝑛 and let 𝐴(𝑀)=^2𝑚_𝑛=0𝐴^𝑛(𝑀) (2) be the graded ring of complex valued differential forms on 𝑀. Each set 𝐴^𝑛(𝑀) can be split into a direct sum 𝐴^𝑛(𝑀)=_{𝑝+𝑞=𝑛}𝐴^𝑝,𝑞(𝑀), (3) 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 𝑑∶=𝜕+𝜕,𝑑^𝑐1∶=4𝜋𝑖𝜕−𝜕, (4) where 𝜕∶𝐴^𝑝,𝑞(𝑀)⟶𝐴^𝑝+1,𝑞(𝑀),𝜕∶𝐴^𝑝,𝑞(𝑀)⟶𝐴^𝑝,𝑞+1(𝑀). (5) Let 𝑧=(𝑧₁,…,𝑧_𝑛)∈ℂ^𝑛, and let 𝑟>0 and fix 𝑎=(𝑎₁,…,𝑎_𝑛)∈ℂ^𝑛; an exhaustion function 𝜏_𝑎,ℂ^𝑛 of ℂ^𝑛 is defined by 𝜏_𝑎,ℂ^𝑛(𝑧)=|𝑧−𝑎|²=^𝑚_𝑗=1𝑧_𝑗−𝑎_𝑗². (6) 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 𝜔_𝑛(𝑧)∶=𝑑𝑑^𝑐log𝜏_𝑛,𝜎_𝑛(𝑧)∶=𝑑^𝑐log𝜏_𝑛∧𝜔_𝑛^𝑛−1(𝑧). (7) In addition, we define 𝜐_𝑛(𝑧)∶=𝑑𝑑^𝑐𝜏_𝑛(𝑧),𝜌_𝑛(𝑧)∶𝜐^𝑛_𝑛(𝑧). (8) It follows that 𝜌_𝑛(𝑧) is the Lebesgue measure on ℂ^𝑛 normalized such that the ball 𝐵_𝑛(𝑟) has measure 𝑟^2𝑛.

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 _{𝜕𝐵1(𝑟)}log⁺𝑓𝑞𝑧𝑓(𝑧)𝜎₁(𝑧)≤𝑞−1^𝛿𝑞^𝛿+1𝛿(1−𝛿)𝑞^𝛿+𝑞−1𝑟𝑞𝑟+𝑟𝜆−𝑞−1×𝑛1𝜆−𝑟𝜆,𝑓+𝑛𝜆,𝑓+2𝑞−1𝑟𝜆()𝑞𝑟𝑚1𝜆−𝑟𝜆−𝜆,𝑓+𝑚𝜆,𝑓, (9) where 𝑧=𝑟𝑒^𝑖𝜙, 𝜆>max{𝑟,|𝑞|𝑟}, and 0<𝛿<1.

Let 𝜏_𝑚=𝜏_0,ℂ^𝑛 be the parabolic exhaustion of ℂ^𝑛 and define a function on 𝐵_𝑛−1(𝑟) by 𝑝_𝑟(𝜔)=𝑟²−𝜏_𝑛−1(𝜔)=𝑟²−|𝜔|². (10) 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 𝜕𝐵_𝑛(𝑟) such that ℎ𝜎_𝑛 is integrable over 𝜕𝐵_𝑛(𝑟). Then _{𝜕𝐵𝑛(𝑟)}ℎ(𝑧)𝜎_𝑛(𝑧)=1𝑟^2𝑛−2_{𝐵𝑛−1(𝑟)𝜕𝐵1(𝑝𝑟(𝑤))}ℎ(𝜔,𝜁)𝜎₁(𝜁)𝜌_𝑛−1(𝜔). (11)

Let 𝑓 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]). Let 𝑓 be a nonconstant meromorphic function in ℂ^𝑛 and take 𝑎∈ℙ¹. If 𝑟>0, then 1𝑟^2𝑛−2_{𝜕𝐵𝑛(𝑟)}𝑛𝑝_𝑟(𝜔),1𝑓_𝜔𝜎−𝑎^𝑛−1(𝜔)1≤𝑛𝑟,,1𝑓−𝑎𝑟^2𝑛−2_{𝜕𝐵𝑛(𝑟)}𝑚𝑝_𝑟(𝜔),1𝑓_𝜔𝜎−𝑎^𝑛−1(𝜔)1=𝑚𝑟,.𝑓−𝑎 (12)

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 ℂ^𝑛 such that 𝑓(0)≠0,∞, and let 𝑞_𝑗∶=(1,…,𝑞_𝑗,…,1). Then _{𝜕𝐵𝑛(𝑟)}log⁺𝑓𝑞_𝑗𝑧𝑓(𝑧)𝜎_𝑛(𝑧)≤𝑞_𝑗−1^𝛿𝑞_𝑗^𝛿+1𝛿(1−𝛿)𝑞_𝑗^𝛿+𝑞_𝑗−1𝑟𝑅−𝑞_𝑗𝑟+𝑞_𝑗−1𝑟×𝑛1𝑅−𝑟𝑅,𝑓+𝑛𝑅,𝑓+2𝑞−1𝑟𝑅()𝑞𝑟𝑚1𝑅−𝑟𝑅−𝑅,𝑓+𝑚𝑅,𝑓, (13) where 𝑅>max{𝑟,|𝑞_𝑗|𝑟} and 0<𝛿<1.

Proof. By applying Lemma 3 with ℎ(𝑧)=log⁺|𝑓(𝑞_𝑗𝑧)/𝑓(𝑧)|, we obtain _{𝜕𝐵𝑛(𝑟)}log⁺𝑓𝑞_𝑗𝑧𝑓(𝑧)𝜎_𝑛(𝑧)=1𝑟^2𝑛−2_{𝐵𝑛−1(𝑟)𝜕𝐵1(𝑝𝑟(𝑤))}log⁺𝑓_𝜔𝑞_𝑗𝑧𝑓_𝜔(𝑧)𝜎₁(𝜁)𝜌_𝑛−1(𝜔)≤1𝑟^2𝑛−2_{𝐵𝑛−1(𝑟)}𝑞_𝑗−1^𝛿𝑞_𝑗^𝛿+1𝛿(1−𝛿)𝑞_𝑗^𝛿+𝑞_𝑗−1𝑝_𝑟(𝜔)𝑝_𝑅(𝜔)−𝑞_𝑗𝑝_𝑟(𝜔)+𝑞_𝑗−1𝑝_𝑟(𝜔)𝑝_𝑅(𝜔)−𝑝_𝑟(𝜔)×𝑛𝑝_𝑅(𝜔),𝑓_𝜔(𝑧)+𝑛𝑝_𝑅(𝜔),1𝑓_𝜔(𝑧)𝜌_𝑛−1(𝜔)+1𝑟^2𝑛−2_{𝐵𝑛−1(𝑟)}2𝑞_𝑗−1𝑝_𝑟(𝜔)𝑝_𝑅(𝜔)𝑝_𝑅(𝜔)−𝑝_𝑟(𝜔)𝑝_𝑅(𝜔)−𝑞𝑝_𝑟(𝜔)×𝑚𝑝_𝑅(𝜔),𝑓_𝜔(𝑧)+𝑚𝑝_𝑅(𝜔),1𝑓_𝜔(𝑧)𝜌_𝑛−1(𝜔)∶=𝑆_𝑛+𝑆_𝑚, (14) for 𝑅>max{𝑟,|𝑞_𝑗|𝑟} and 0<𝛿<1. Now we will estimate terms 𝑆_𝑚 and 𝑆_𝑛, respectively. Note that 𝑝_𝑟(𝜔)𝑝_𝑅(𝜔)≤𝑟𝑅 (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 𝑆_𝑛≤𝑅𝑟^2𝑛−2𝐶𝑞_𝑗+𝑞,𝛿^𝑗−1𝑟𝑅−𝑞_𝑗𝑟+𝑞_𝑗−1𝑟𝑅−𝑟×𝑛_𝑓(),𝑆𝑅,0,∞_𝑚≤𝑅𝑟^2𝑛−22𝑞_𝑗−1𝑟𝑅()𝑞𝑟𝑚𝑅−𝑟𝑅−^𝑓(𝑅,0,∞), (16) where 𝐶𝑞_𝑗=𝑞,𝛿^𝑗−1^𝛿𝑞_𝑗^𝛿+1𝛿(1−𝛿)𝑞_𝑗^𝛿,𝑛_𝑓()=𝑛1𝑅,0,∞𝑅,𝑓+𝑛𝑅,𝑓,𝑚_𝑓()=𝑚1𝑅,0,∞𝑅,𝑓+𝑚𝑅,𝑓. (17) It implies the assertion.

To deal with 𝑚_𝑓(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 lim_𝑟→∞log𝑇(𝑟)log𝑟=0, (18) then the set 𝐸∶=𝑟∶𝑇𝐶₁𝑟≥𝐶₂𝑇(𝑟) (19) has logarithmic density 0 for all 𝐶₁>1 and 𝐶₂>1.

To show that 𝑛_𝑓(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 {𝑐_𝑛}_𝑛∈ℕ such that 𝑐_𝑛→0 as 𝑛→∞ and for all 𝑛∈ℕ, the set 𝐹_𝑛∶=𝑟≥1∶𝑈(𝑟)<𝑐_𝑛𝑇(𝑟) (20) has logarithmic density 1, then 𝑈(𝑟)=𝑜(𝑇(𝑟)) (21) 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. If 𝑓 is a nonconstant meromorphic function in ℂ^𝑛 of zero order, then the set 𝐸_𝑛<𝑇∶=𝑟≥∶𝑛𝑟,𝑓𝑟,𝑓2^𝑛 (22) has logarithmic density 1 for all 𝑛∈ℕ.

If we take 𝑅=𝑘𝑟 in Lemma 5, and by using Lemmas 6, 7, and 8, then we have _{𝜕𝐵𝑛(𝑟)}log⁺𝑓𝑞_𝑗𝑧𝑓(𝑧)𝜎_𝑛(𝑧)=𝑜𝑇𝑟,𝑓 (23) 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 _{𝜕𝐵𝑛(𝑟)}log⁺𝑓𝑞𝑧𝑓(𝑧)𝜎_𝑛(𝑧)=_{𝜕𝐵𝑛(𝑟)}log^+𝑛_𝑘=1𝑓𝑧∏^𝑘_𝑗=0𝑞_𝑗𝑓𝑧∏^𝑘−1_𝑗=0𝑞_𝑗𝜎_𝑛(𝑧)≤^𝑛_{𝑘=1𝜕𝐵𝑛(𝑟)}log⁺𝑓𝑧∏^𝑘_𝑗=0𝑞_𝑗𝑓𝑧∏^𝑘−1_𝑗=0𝑞_𝑗𝜎_𝑛(𝑧)=𝑜𝑇𝑟,𝑓 (24) 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 of 𝑇(𝑟,𝑓(𝑞𝑧)) and 𝑇(𝑟,𝑓(𝑧)), which is the 𝑞-shift analogue of the first main theory of several variables.

Theorem 9. Let 𝑓 be a meromorphic function in ℂ^𝑛 of zero order such that 𝑓(0)≠0,∞, and let 𝑞∈ℂ^𝑛⧵{0}. Then 𝑇𝑟,𝑓𝑞𝑧=𝑇𝑟,𝑓(𝑧)+𝑜𝑇𝑟,𝑓 (25) on a set of logarithmic density 1.

Proof. By using Lemma 6 and Theorem 1, we have 𝑇(𝑧)𝑞(𝑧)𝑓𝑟,𝑓𝑞𝑧=𝑚𝑟,𝑓𝑞𝑧+𝑁𝑟,𝑓𝑞𝑧=𝑚𝑟,𝑓+𝑁𝑟,𝑓+𝑚𝑟,𝑞𝑧𝑓(𝑧)≤𝑇𝑟,𝑓+𝑜𝑇𝑟,𝑓, (26) on a set of logarithmic density 1. We have the assertion.

Let us discuss the 𝑞-shift analogue of the second main theory of several variables.

Theorem 10. Let 𝑓 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)∈ℙ¹ are distinct finite constants, then 𝑚𝑟,𝑓+^𝑞_𝑗=1𝑚1𝑟,𝑓−𝑎_𝑗≤2𝑇𝑟,𝑓−𝑁_{𝑝𝑎𝑖𝑟}𝑟,𝑓+𝑆_𝑞𝑟,𝑓, (27) where 𝑁_{𝑝𝑎𝑖𝑟}𝑟,𝑓∶=2𝑁𝑟,𝑓−𝑁𝑟,Δ_𝑞𝑓1+𝑁𝑟,Δ_𝑞𝑓. (28)

Proof. By using the first main theorem, we obtain ^𝑞_𝑗=1𝑚1𝑟,𝑓−𝑎_𝑗=^𝑞_𝑗=1𝑇1𝑟,𝑓−𝑎_𝑗−^𝑞_𝑗=1𝑁1𝑟,𝑓−𝑎_𝑗1=𝑞𝑇𝑟,𝑓−𝑁𝑟,𝑃𝑓+𝑆_𝑞𝑟,𝑓, (29) where 𝑃𝑟,𝑓=^𝑞_𝑗=1𝑓−𝑎_𝑗. (30) Since there exist 𝛼_𝑗∈ℙ¹ such that 1𝑃𝑓=^𝑞_𝑗=1𝛼_𝑗𝑓−𝑎_𝑗 (31) then 𝑚1𝑟,𝑃𝑓Δ≤𝑚𝑟,^𝑞𝑓𝑃𝑓1+𝑚𝑟,Δ_𝑞𝑓1≤𝑚𝑟,Δ_𝑞𝑓+^𝑞_𝑗=1𝑚Δ𝑟,^𝑞𝑓𝑓−𝑎_𝑗1=𝑚𝑟,Δ_𝑞𝑓+𝑆_𝑞𝑟,𝑓. (32) By applying (29), (32), and the fact 𝑇(𝑟,𝑃(𝑓))=𝑞𝑇(𝑟,𝑓)+𝑂(1), we have ^𝑞_𝑗=1𝑚1𝑟,𝑓−𝑎_𝑗𝑓1=𝑇𝑟,𝑃−𝑁𝑟,𝑃𝑓+𝑆_𝑞1𝑟,𝑓=𝑚𝑟,𝑃𝑓+𝑆_𝑞1𝑟,𝑓≤𝑚𝑟,Δ_𝑞𝑓+𝑆_𝑞𝑟,𝑓=𝑇𝑟,Δ_𝑞𝑓1−𝑁𝑟,Δ_𝑞𝑓+𝑆_𝑞𝑟,𝑓. (33) Therefore, we have 𝑚𝑟,𝑓+^𝑞_𝑗=1𝑚1𝑟,𝑓−𝑎_𝑗≤𝑇𝑟,𝑓+𝑁𝑟,Δ_𝑞𝑓+𝑚𝑟,Δ_𝑞𝑓1−𝑁𝑟,Δ_𝑞𝑓−𝑁𝑟,𝑓+𝑆_𝑞𝑟,𝑓. (34) Since 𝑚𝑟,Δ_𝑞𝑓Δ≤𝑚𝑟,𝑓+𝑚𝑟,^𝑞𝑓𝑓=𝑚𝑟,𝑓+𝑆_𝑞𝑟,𝑓, (35) we have the assertion.

4. Application

Let 𝑓 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. Let 𝑓∶ℂ^𝑛→ℙ¹ be a nonconstant meromorphic function of zero order and let 𝑞∈ℂ^𝑛⧵{0}. If 𝑓(𝑧) and 𝑓(𝑞𝑧) share three distinct values CM, then 𝑓(𝑧)=𝑓(𝑞𝑧).

Proof. Let us suppose that 𝑓(𝑧) and 𝑓(𝑞𝑧) share {0,1,∞} CM, if not, let us make a linearly transformation. Suppose that Δ_𝑞𝑓≢0, Theorem 10 yields 𝑚1𝑟,𝑓+𝑚𝑟,𝑓1+𝑚𝑟,𝑓−1≤2𝑇𝑟,𝑓+𝑁𝑟,Δ_𝑞𝑓1−2𝑁𝑟,𝑓−𝑁𝑟,Δ_𝑞𝑓+𝑆_𝑞𝑟,𝑓. (36) Since 𝑓(𝑧) and 𝑓(𝑞𝑧) share {0,1,∞} CM, then 𝑁1𝑟,𝑓+𝑁𝑟,𝑓1+𝑁𝑟,1𝑓−1≤𝑁𝑟,Δ_𝑞𝑓. (37) In addition, Lemma 6 implies that 𝑁(𝑟,Δ_𝑞𝑓)≤2𝑁(𝑟,𝑓)+𝑆_𝑞(𝑟,𝑓), by combining (36) and (37), it follows that 𝑇𝑟,𝑓=𝑆_𝑞𝑟,𝑓 (38) which is impossible.

The 𝑞-difference polynomials of 𝑓(𝑧) in ℂ^𝑛 are said by the functions which are polynomials in 𝑓(𝑞_𝑗𝑧), 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 is 𝑞-shift analogue of Clunie lemma [15] of several variables.

Theorem 12. Let 𝑓∶ℂ^𝑛→ℙ¹ be a nonconstant zero order meromorphic solution in ℂ^𝑛 of 𝑓(𝑧)^𝑛𝑃𝑧,𝑓=𝑄𝑧,𝑓, (39) 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 𝑚𝑟,𝑃𝑧,𝑓=𝑆_𝑞𝑟,𝑓. (40)

Proof. In calculating the proximity function of 𝑃(𝑧,𝑓), we split the region of integration into two parts. By defining 𝐸₁∶=𝑧∈𝜕𝐵_𝑛(𝑟)∣𝑓(𝑧)<1,𝐸₂∶=𝜕𝐵_𝑛(𝑟)⧵𝐸₁, (41) we have 𝑚𝑟,𝑃𝑧,𝑓=_𝐸1log⁺𝑃𝑧,𝑓𝜎_𝑛(𝑧)+_𝐸2log⁺𝑃𝑧,𝑓𝜎_𝑛(𝑧). (42) First, we consider 𝐸₁; we can write 𝑃(𝑧,𝑓) as 𝑃𝑧,𝑓=𝑐_𝑖(𝑧)𝑓(𝑧)^𝑖0𝑓𝑞𝑧^𝑖1⋯𝑓𝑞_𝜈𝑧^𝑙𝜈. (43) Thus, we have 𝑃𝑧,𝑓≤𝑐_𝑖(𝑧𝑓)𝑞𝑧^𝑖1𝑓(𝑧)^𝑙1⋯𝑓𝑞_𝜈𝑧^𝑙𝜈𝑓(𝑧). (44) Therefore, we obtain _𝐸1log⁺𝑃𝑧,𝑓𝜎_𝑛(𝑧)≤𝑚𝑟,𝑐_𝑖+𝑂^𝑞_𝑖=1𝑚𝑓𝑟,𝑞𝑧𝑓(𝑧), (45)together with Theorem 1 implies that_𝐸1log⁺𝑃𝑧,𝑓𝜎_𝑛(𝑧)=𝑆_𝑞𝑟,𝑓. (46) Now let us consider 𝐸₂; we note that 𝑄𝑧,𝑓=𝑏_𝛾(𝑧)𝑓𝑞₀𝑧^𝛾0⋯𝑓𝑞_𝜇𝑧^𝛾𝜇. (47) Hence, we have 𝑃=1𝑧,𝑓𝑓(𝑧)^𝑛𝑏_𝛾(𝑧)𝑓𝑞₀𝑧^𝛾0⋯𝑓𝑞_𝜇𝑧^𝛾𝜇≤𝑏_𝛾(𝑧𝑓𝑞)¹𝑧𝑓(𝑧)^𝛾1⋯𝑓𝑞_𝜇𝑧𝑓(𝑧)^𝛾𝜇, (48) by Theorem 1 again, it follows that _𝐸2log⁺𝑃𝑧,𝑓𝜎_𝑛(𝑧)=𝑆_𝑞𝑟,𝑓. (49) The assertion follows by combining (42)–(49).

Let 𝛼 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 is 𝑞-shift analogue of Mohon’ho lemma [16] of several variables.

Theorem 13. Let 𝑓(𝑧) be a nonconstant zero-order meromorphic solution in ℂ^𝑛 of 𝑃𝑧,𝑓=0, (50) 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 𝑚1𝑟,𝑓−𝛼=𝑆_𝑞𝑟,𝑓. (51)

Proof. By substituting 𝑓=𝑔+𝛼 into (50), we obtain 𝑄𝑧,𝑔+𝐷(𝑧)=0, (52) where 𝑄𝑧,𝑔=_{𝛾=(𝑗0,…,𝑗𝜈)∈𝐽}𝑏_𝜈(𝑧)𝑔(𝑧)^𝑗0⋯𝑔𝑞_𝜈𝑧^𝑗𝜈 (53) 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 𝑚1𝑟,𝑔𝐷≤𝑚𝑟,𝑔1+𝑚𝑟,𝐷𝑄=𝑚𝑟,𝑧,𝑔𝑔1+𝑚𝑟,𝐷. (54) 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 𝑄𝑧,𝑔𝑔=1𝑔_𝛾∈𝐽𝑏_𝜈(𝑧)𝑔(𝑧)^𝑗0⋯𝑔𝑞_𝜈𝑧^𝑗𝜈≤_𝛾∈𝐽𝑏_𝜈(𝑧𝑔𝑞)¹𝑧𝑔(𝑧)^𝑗1⋯𝑔𝑞_𝜈𝑧𝑔(𝑧)^𝑗𝜈, (55) by Theorem 1, (54), and (55), it shows 𝑚1𝑟,𝑔=𝑆_𝑞𝑟,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.