Abstract and Applied Analysis

Volume 2014 (2014), Article ID 736021, 6 pages

http://dx.doi.org/10.1155/2014/736021

Research Article

The -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

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.

Abstract.

We investigate -shift analogue of the lemma on logarithmic derivative of several variables. Let be a meromorphic function in of zero order such that , , and let . Then we have on a set of logarithmic density 1. The -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 -difference version, Barnett et al. obtain -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 and in , if and , then we denote . Let us state one of the main results as follows, which is -difference analogue of the lemma on logarithmic derivative of several variables.

Theorem 1. Let be a meromorphic function in of zero order such that , and let . Then

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 -difference analogue of the lemma on logarithmic derivative of several variables. In Section 3, we present the -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 -shift of several variables, -shift analogues of the Clunie lemma, and Mohon’ho lemma of several variables.

2. -Difference Logarithmic Derivative Analogue

The purpose of this section is to present -difference analogues of the lemma on logarithmic derivative in several complex variables. Let us recall some of the standard notations of Nevanlinna theory in .

Let be a connected complex manifold of dimension and let

be the graded ring of complex valued differential forms on . Each set can be split into a direct sum

where is the forms of type . The differential operators and on are defined as

where

Let , and let and fix ; an exhaustion function of is defined by

Usually, we let be the origin and set . Let us define a positive measure with total measure one on the boundary of the ball as

In addition, we define

It follows that is the Lebesgue measure on normalized such that the ball has measure .

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 be a meromorphic function in such that , and let . Then

where , , and .

Let be the parabolic exhaustion of and define a function on by

For the following discussion, we will need two lemmas from Stoll.

Lemma 3 (see [13, Lemma 1.29]). Let , and let be a function on such that is integrable over . Then

Let be a nonconstant meromorphic function on , take , and define the holomorphic map by ; thus .

Lemma 4 (see [13, Lemma 1.30]). Let be a nonconstant meromorphic function in and take . If , then

We proceed to generalize Lemma 2 to several complex variables by using Lemmas 3 and 4.

Lemma 5. Let be a meromorphic function in such that , and let . Then

where and .

Proof. By applying Lemma 3 with , we obtain

for and . Now we will estimate terms and , respectively. Note that

for and the fact that is decreasing with for . By using Lemma 4, it follows that

where

It implies the assertion.

To deal with , we need the following lemma.

Lemma 6 (see [14, Lemma 4]). If is a piecewise continuous increasing function such that

then the set

has logarithmic density 0 for all and .

To show that is small, we need the following two lemmas.

Lemma 7 (see [12, Lemma 5.4]). Let be an increasing function, and let . If there exists a decreasing sequence such that as and for all , the set

has logarithmic density 1, then

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 is a nonconstant meromorphic function in of zero order, then the set

has logarithmic density 1 for all .

If we take in Lemma 5, and by using Lemmas 6, 7, and 8, then we have

on a set of logarithmic density 1, since is a meromorphic function in of zero order. According to (23), it follows that

on a set of logarithmic density 1. Therefore, we have -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 and , which is the -shift analogue of the first main theory of several variables.

Theorem 9. Let be a meromorphic function in of zero order such that , and let . Then

on a set of logarithmic density 1.

Proof. By using Lemma 6 and Theorem 1, we have

on a set of logarithmic density 1. We have the assertion.

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

Theorem 10. Let be a meromorphic function in of zero order such that , and let . If are distinct finite constants, then

where

Proof. By using the first main theorem, we obtain

where

Since there exist such that

then

By applying (29), (32), and the fact , we have

Therefore, we have

Since

we have the assertion.

4. Application

Let and be two meromorphic functions in , and let , if and have the same zeros (counting multiplicities), then we say and share CM in .

Theorem 11. Let be a nonconstant meromorphic function of zero order and let . If and share three distinct values CM, then .

Proof. Let us suppose that and share CM, if not, let us make a linearly transformation. Suppose that , Theorem 10 yields

Since and share CM, then

In addition, Lemma 6 implies that , by combining (36) and (37), it follows that

which is impossible.

The -difference polynomials of in are said by the functions which are polynomials in , where , with coefficients such that on a set of logarithmic density 1.

The following theorem is -shift analogue of Clunie lemma [15] of several variables.

Theorem 12. Let be a nonconstant zero order meromorphic solution in of

where and are -difference polynomials in . If the degree of as a polynomial in and its -shift is at most , then

Proof. In calculating the proximity function of , we split the region of integration into two parts. By defining

we have

First, we consider ; we can write as

Thus, we have

Therefore, we obtain

together with Theorem 1 implies that

Now let us consider ; we note that

Hence, we have

by Theorem 1 again, it follows that

The assertion follows by combining (42)–(49).

Let and be meromorphic functions of zero order in such that on a set of logarithmic density 1. Then is called a small function with respect to of zero order in .

The following theorem is -shift analogue of Mohon’ho lemma [16] of several variables.

Theorem 13. Let be a nonconstant zero-order meromorphic solution in of

where is a -difference polynomial in . If for a small function in of , then

Proof. By substituting into (50), we obtain

where

is a -difference polynomial in such that all of its terms are at least degree 1, and , for each . Also , since does not satisfy (50). By using (52), we have

Note that since the integral vanishes on the part of where , it is sufficient to consider only the case . Then

by Theorem 1, (54), and (55), it shows

Since , the assertion follows.

Conflict of Interests

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

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