Abstract

We investigate the uniqueness questions of the difference operator on entire functions and obtain three uniqueness theorems using the idea of weight sharing.

1. Introduction

A function is called meromorphic, if it is analytic in the complex plane except at poles. It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory such as the characteristic function , and proximity function , counting function (see [1, 2]). In addition we use denotes any quantity that satisfies the condition: as possibly outside an exceptional set of finite logarithmic measure.

Let and be two nonconstant meromorphic functions, , we say that and share the value IM (ignoring multiplicities) if and have the same zeros, they share the value CM (counting multiplicities) if and have the same zeros with the same multiplicities. When the zeros of mean the poles of (see [2]).

Let be a positive integer and . We use to denote the counting function of the zeros of (counted with proper multiplicities) whose multiplicities are not bigger than , to denote the counting function of the zeros of whose multiplicities are not less than . and denote their corresponding reduced counting functions (ignoring multiplicities), respectively. We denote by the set of zeros of with multiplicity, the set of zeros of (counted with proper multiplicities) whose multiplicities are not greater than .

In 1997, Yang and Hua (see [3]) studied the uniqueness of the differential monomials and obtained the following result.

Theorem A. Let and be nonconstant entire functions, and let be an integer. If and share 1 CM, then either , , where , , and are constants satisfying or for a constant such that .

Recently, a number of papers (including [1, 3–17]) have focused on complex difference equations and differences analogues of Nevanlinna theory.

In particular, Qi et al. (see [16]) proved Theorem B, which can be considered as a difference counterpart of Theorem A.

Theorem B. Let and be transcendental entire functions with finite order, be a nonzero complex constant and be an integer. If and share 1 CM, then or for some constant and which satisfy and .

In 2011, Zhang et al. (see [17]) investigated the distribution of zeros and shared values of the difference operator on meromorphic functions and uniqueness of difference polynomials with the same 1 points or fixed points. They obtained the following results.

Theorem C. Let and be nonconstant entire functions of finite order, and let be an integer. Suppose that is a nonzero complex constant such that and . If and share 1 CM, and and share 0 CM, then and , where , , and are constants satisfying or for a constant such that .

Theorem D. Let and be nonconstant entire functions of finite order, and let be an integer. Suppose that is a nonzero complex constant such that and . If and share CM, and and share 0 CM, then , where is a constant satisfying .

We investigate the uniqueness theorem of another differences polynomial and prove Theorem 1.

Theorem 1. Let and be nonconstant transcendental entire functions of finite order, and let be an integer. Suppose that is a nonzero real constant such that and . If , then , where is a constant satisfying except that .

In paper [15], Wang et al. improved the Theorem B and proved the following result.

Theorem E. Let and be transcendental entire functions with finite order, be a nonzero complex constant and be an integer. If , then or for some constant and which satisfy and .

The purpose of this paper is to induce the idea of weight sharing to Theorems C and D, the results as follow.

Theorem 2. Let and be nonconstant entire functions of finite order, and let be an integer. Suppose that is a nonzero complex constant such that and . If share 1 CM, and share 0 CM, then and , where , , and are constants satisfying or for a constant such that .

Theorem 3. Let and be nonconstant entire functions of finite order, and let be an integer. Suppose that is a non-zero complex constant such that and . If share  CM, and share 0 CM, then , where is a constant satisfying .

Remark 4. Some ideas of this paper are based on [15, 17].

2. Some Lemmas

In order to prove our theorems, we need the following Lemmas.

Lemma 5 is a difference analogue of the logarithmic derivative lemma, given by Halburd and Korhonen [9] and Chiang and Feng [7] independently.

Lemma 5 (see [9]). Let be a meromorphic function of finite order, and let and . Then for all outside of a possibly exceptional set with finite logarithmic measure.

Lemma 6 (see [2]). Let be a nonconstant meromorphic function, and let , where , are small functions of . Then

Lemma 7 (see [18]). Let and be two nonconstant meromorphic function satisfying for some positive integer . Define as follow
If , then where denotes the counting function of zeros of but not zeros of , and is similarly defined.

Lemma 8 (see [19]). Under the conditions of Lemma 7, we have

Lemma 9 (see [19]). If , then either or provided that where and is a set with infinite linear measure.

Lemma 10. Let be a meromorphic function of finite order, . Then

Proof. Using Lemma 5 and the formula (12) in [12]
Replacing with , we have for every , so we deduce that
From (8) and (10), we obtain that
Thus we completed the proof.

Lemma 11 (see [9]). Let be a nondecreasing continuous function, , , and let be the set of all satisfy
If the logarithmic measure of is infinite, then

3. Proof of Theorems

Proof of Theorem 1. We define
In Lemma 7, we replace and , by and respectively, we claim that . If it is not true, then . From Lemma 8 we have that
From the Nevanlinna second foundational theorem, we can get that
From the definitions of and , the following inequalities are obvious:
Combining (15), (16), and (17), we deduce that
We can apply Lemma 5, Lemma 6, and Lemma 10 to show that which implies
The same augment as above, we have that
From (18), (20), and (21), we can deduce that which is a contraction. Therefore, .
Noting that where .
Because of Lemma 9, we have that or . We will consider the following two cases.
Case 1. Suppose that . Then
Let , we deduce that
If , by the hypothesis , we get that . So which means is a constant, because of .
Then and is a constant satisfying except that .
Case 2. Suppose that . Then
Note that zero is a Picard exceptional value of and , then and , where and are polynomials. In (27), we let , then
It is impossible, because of is a real number.

Proof of Theorem 2. Denoting
In Lemma 7, we replace and , by and respectively. If , by Lemma 8 we deduce that
The same reasons as in the proof of Theorem 1, we have that
Combining (30) and (31), we deduce that
We can apply Lemmas 5, 6, and 10 to show that which implies
We have that , Since and share  CM, then
From (32), (34), and (35), we can deduce that which is a contraction. Therefore, .
Noting that where .
Because of Lemma 9, we have that or .
By using the same methods as in the proof of Theorem  1.10 in [17], we can complete the proof of Theorem 2.

Proof of Theorem 3. The proof is almost literally the same as the proof of Theorem 2, with the methods as in the proof of Theorem  1.9 in [17] replacing the methods as in the proof of Theorem  1.10 in [17].

Acknowledgment

This research was supported by the funds of Taiyuan University of Technology.