Abstract

We investigate the zero distribution of -shift difference polynomials of meromorphic functions with zero order and obtain some results that extend previous results of K. Liu et al.

1. Introduction and Main Results

In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna theory (see, e.g., [1, 2]). Let and be two nonconstant meromorphic functions in the complex plane. By , we denote any quantity satisfying as , possibly outside a set of with finite linear measure. Then the meromorphic function is called a small function of , if . If and have same zeros, counting multiplicity (ignoring multiplicity), then we say that and share the small function CM (IM). The logarithmic density of a set is defined as follows:

Currently, many articles have focused on value distribution in difference analogues of meromorphic functions (see, e.g., [3–11]). In particular, there has been an increasing interest in studying the uniqueness problems related to meromorphic functions and their shifts or their difference operators (see, e.g., [8, 12–15]). Our aim in this article is to investigate the uniqueness problems of -difference polynomials.

Recently, Liu et al. [13] considered uniqueness of difference polynomials of meromorphic functions, corresponding to uniqueness theorems of meromorphic functions sharing values (see, e.g., [9, 16]). They got the following.

Theorem A. Let and be two transcendental meromorphic functions with finite order. Suppose that is a nonzero complex constant and is an integer. If and and share , then or , where .

Theorem B. Under the conditions of Theorem A, if and and share , then or , where .

In this paper, we consider the case of -shift difference polynomials and extend Theorem A as follows:

Theorem 1. Let and be two transcendental meromorphic functions with . Suppose that and are two nonzero complex constants and is an integer. If and and share , then or , where .

It is natural to ask whether Theorem 1 holds if and share 1 IM. Corresponding to this question, we get the following result.

Theorem 2. Under the conditions of Theorem 1, if and and share , then or , where .

Next, we consider the uniqueness of -difference products of entire functions and obtain the following results.

Theorem 3. Let and be two transcendental entire functions with , and let and be two nonzero complex constants, and let be a nonzero polynomial, where , are complex constants, and denotes the number of the distinct zero of . If and and share , then one of the following results holds: (1) for a constant such that , where and (2) and satisfy the algebraic equation , where

Remark 4. A similar result can be found in [15], but the method of this paper is more concise, and the condition of this paper is better.

2. Preliminary Lemmas

The following lemma is a -difference analogue of the logarithmic derivative lemma.

Lemma 5 (see [14]). Let be a meromorphic function of zero order, and let and be two nonzero complex numbers. Then one has on a set of logarithmic density 1.

Lemma 6 (see [7]). If is an increasing function such that then the set has logarithmic density 0 for all and .

The following lemma is essential in our proof and is due to Heittokangas et al., see [12, Theorems 6 and 7].

Lemma 7. Let be a meromorphic function of finite order, and let be fixed. Then

Lemma 8. Let be a meromorphic function with , and let and be two nonzero complex numbers. Then

Proof of Lemma 8. We only prove the case . For the case , we can use the same method in the proof. By a simple geometric observation, we obtain Combining with Lemma 6, we obtain on a set of logarithmic density 1. On the other hand, we have Therefore, on a set of logarithmic density 1. From (9) and (12), we have By Lemma 7, we have
Similarly, we have

Lemma 9. Let be a nonconstant meromorphic function of zero order, and let and be two nonzero complex numbers. Then on a set of logarithmic density 1.

Proof of Lemma 9. By Lemmas 5 and 8, we have on a set of logarithmic density 1.

Lemma 10. Let be an entire function with , let and be two fixed nonzero complex constants, and let be a nonzero polynomial, where are complex constants. Then

Proof of Lemma 10. By and Lemma 5, we obtain on a set of logarithmic density 1. Using the similar method as above, we also get on a set of logarithmic density 1.
Hence, we have on a set of logarithmic density 1.

Lemma 11 (see [17]). Let and be two nonconstant meromorphic functions. If and share , then one of the following three cases holds:(i)(ii), (iii),
where denotes the counting function of zero of , such that simple zero are counted once and multiple zeros are counted twice.

In order to prove Theorem 2, we need the following lemma.

Lemma 12 (see [16]). Let and be two nonconstant meromorphic functions, and let and share . Let
If , then

3. Proof of Theorem 1

Let and . Thus, and share 1 CM. Combining the first main theorem with Lemma 9, we obtain

Hence, we obtain Using the similar method as above, we have

From Lemma 9, we have By the second main theorem, Lemma 9, and (28), we obtain Hence, (25) and (29) imply that Similarly, we have Equations (30) and (31) imply that . Together the definition of with Lemma 9, we have

Similarly, Thus, together (21) with (32)-(33), we obtain

Then, by (25), (26), and (34), we obtain which is a contradiction since . By Lemma 11, we have or . If , that is, . Set . Suppose that is not a constant. Then we obtain

Lemma 9 and (36) imply that Hence, must be a nonzero constant, since . Set . By (36), we know . Thus, , where .

If , that is,

Let . Using the similar method as above, we also obtain that must be a nonzero constant. Thus, we have , where .

4. Proof of Theorem 2

Let and , and let be defined in Lemma 12. Using the similar proof as the proof of Theorem 1, we prove that (25)–(33) hold. By Lemma 9, we obtain Similarly, we obtain

Together Lemma 12 with (32), (33), (39), and (40), we have By (25), (26) and (41) yield that which is impossible, since . Hence, we have .

By integrating (22) twice, we have which yields that . From (25)–(28), we obtain

Next, we will prove that or .

Case 1 (). If , by (43), we obtain
Together the Nevanlinna second main theorem with Lemma 9, (28), and (44), we obtain
which yields that , which is impossible, since . Hence, we obtain , so Using the similar method as above, we obtain which is impossible.

Case 2. If and , then follows trivially. Therefore, we may consider the case and . By (43), we have Similarly, we get a contradiction.

Case 3. If , and then   follows trivially. Therefore, we may consider the case and . By (43), we obtain Similarly, we get a contradiction.

5. Proof of Theorem 3

Since and share 1 CM, we obtain where is an entire function. by and , we have as a constant. We can rewrite (52) as follows:

If , by the first main theory, the second main theory, and Lemma 9, we have By Lemma 10 and (54), we have Hence, we have Similarly, we have

Equations (56) and (57) imply that which is impossible, since . Hence, we have . We can rewrite (52) as follows:

Set . We break the rest of the proof into two cases.

Case 1. Suppose that is a constant. Then by substituting into (59), we obtain where are complex constants. By the fact that is transcendental entire function, we have . Hence, we obtain
Equation (61) implies that and when for . Therefore, , where is defined as the assumption of Theorem 3.

Case 2. Suppose that is not a constant, then we know by (59) that and satisfy the algebraic equation , where .

Acknowledgments

The authors would like to thank the referee for his/her helpful suggestions and comments. The work was supported by the NNSF of China (no. 10771121), the NSFC Tianyuan Mathematics Youth Fund (no. 11226094), the NSF of Shandong Province, China (no. ZR2012AQ020 and no. ZR2010AM030), the Fund of Doctoral Program Research of University of Jinan (XBS1211), and Shandong University Graduate Student Independent Innovation Fund (yzc11024).