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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 152069, 6 pages
Value Sharing Results for -Shifts Difference Polynomials
1Department of Mathematics, Shaoxing College of Arts and Sciences, Shaoxing, Zhejiang 312000, China
2Department of Physics and Mathematics, Joensuu Campus, University of Eastern Finland, P.O. Box 111, Joensuu 80101, Finland
3School of Mathematics and Information Sciences, Henan Polytechnic University, Jiaozuo, Henan 454000, China
4Department of Mathematics, Jinan University, Jinan, Shandong 250022, China
5School of Mathematics, Shandong University, Jinan, Shandong 250100, China
Received 15 January 2013; Accepted 26 March 2013
Academic Editor: Risto Korhonen
Copyright © 2013 Yong Liu et al. 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.
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.  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 , 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 ). 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 ). 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
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.
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 ). Let and be two nonconstant meromorphic functions. If and share , then one of the following three cases holds:(i)(ii),
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 ). 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
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
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:
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 .
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).
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