Unicity of Entire Functions concerning Shifts and Difference Operators

Abstract

We prove a unicity theorem of entire functions that share two distinct small functions with their shifts. The corollary of the theorem confirms the conjecture posed by Li and Gao (2011).

1. Introduction

Let be a nonconstant meromorphic function in the complex plane . We will use the standard notations in Nevanlinna theory of meromorphic functions such as , , and (see [1, 2]). The notation is defined to be any quantity satisfying as possibly outside a set of finite linear measures. A meromorphic function is called a small function related to provided that .

Let and be two nonconstant meromorphic functions, and let be a small function related to both and . We say that and share CM if and have the same zeros with the same multiplicities. and are said to share IM if and have the same zeros ignoring multiplicities.

Let be the counting functions of all common zeros with the same multiplicities of and . If then we say that and share CM almost.

For a nonzero complex constant , we define difference operators as and , , .

In 1977, Rubel and Yang [3] proved the following result.

Theorem A. Let be a nonconstant entire function. If and share two distinct finite values CM, then .

In fact, the conclusion still holds if the two CM values are replaced by two IM values (see Gundersen [4, 5], Mues and Steinmetz [6]).

Recently, a number of articles focused on value distribution in shifts or difference operators of meromorphic functions (see [7–11]). In particular, some papers studied the unicity of meromorphic functions sharing values with their shifts or difference operators (see [12–14]). In 2009, Heittokangas et al. [12] proved the following result concerning shifts.

Theorem B. Let be a nonconstant entire function of finite order, . If and share two distinct finite values CM, then .

In 2011, Li and Gao [14] proved the following result concerning difference operators.

Theorem C. Let be a nonconstant entire function of finite order, , and let be a positive integer. Suppose that and share two distinct finite values , CM and one of the following cases is satisfied:(i);(ii) and .
Then .

In [14], Li and Gao conjectured that the restriction for the case can be removed. In this paper, we confirm their conjecture. In fact, we prove the following more general results.

Theorem 1. Let be a nonconstant entire function of finite order, let be a positive integer, let , be two distinct small functions related to , let be nonzero complex numbers and distinct finite values, and let If and share , CM, then .

Corollary 2. Let be a nonconstant entire function of finite order, let be a nonzero finite complex number, let be a positive integer, and let , be two distinct finite values. If and share , CM, then .

Remark 3. Corollary 2 confirms the conjecture of Li and Gao in [14].

Corollary 4. Let be a nonconstant entire function of finite order, let be a nonzero finite complex number, and let , be two distinct small functions related to . If and share , CM, then .

2. Some Lemmas

For the proof of Theorem 1, we require the following results.

Lemma 5 (see [15]). Let and be two nonconstant meromorphic functions satisfying If and share CM almost, then either or .

Lemma 6 (see [15]). Let and be two nonconstant meromorphic functions satisfying If and share and CM almost, and where is a set of infinitely linear measure, then where , , , and are constants satisfying .

Lemma 7 (see [10]). Let be a nonconstant meromorphic function of finite order, . Then for all outside a possible exceptional set with finite logarithmic measure .

In the following, denotes any function satisfying as , possibly outside a set with finite logarithmic measure.

3. Proof of Theorem 1

We prove Theorem 1 by contradiction. Suppose that . Then it follows from and being two distinct entire functions that and share , , and CM. By the Nevanlinna second fundamental theorem for three small functions, we have Similarly, we have . Therefore, .

Set Thus , share , , and CM almost.

Obviously, we have By Nevanlinna’s second fundamental theorem, we have

Since   , thus By (11), we have It follows that On the other hand, by Nevanlinna first fundamental theorem, we have So we get

Set

If , we can deduce by (16) that

If , set

Then we have

It followed from (16) that

By (18) and (21), we can deduce that

It follows from (14) and (22) that

By Lemma 6, we have where , , , and are complex numbers satisfying .

Now, we consider three cases.

Case 1. Consider  . Thus

Similarly, we have

By Lemma 5, we get that either or .

If , we can easily deduce that , which is a contradiction with our assumption.

If , that is then we have

From (28), we have

It follows that , a contradiction.

Case 2. Consider  . Using the same argument as used in Case 1, we deduce that , a contradiction.

Case 3. Consider  , . Since and share , CM almost, we deduce from (24) that

If , then ; that is, , a contradiction.

Hence . Thus we have

Obviously, , . Thus by Nevanlinna second fundamental theorem and (14), we get

It follows that , a contradiction. Thus we prove that . This completes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for careful reading of the paper, pointing out a gap in the previous version of this paper, and giving many valuable suggestions. Research is supported by the NNSF of China (Grant no. 11371149) and NSF of Guangdong Province, China (Grant no. S2012010010121).