Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 818052, 11 pages

http://dx.doi.org/10.1155/2012/818052

## Value Distribution and Uniqueness Results of Zero-Order Meromorphic Functions to Their -Shift

^{1}Department of Public Teaching, Wenzhou Vocational College of Science and Technology, Zhejiang, Wenzhou 325000, China^{2}Department of Science and Humanities, Shandong Transport Vocational College, Shandong, Weifang 261206, China^{3}Department of Educational Administration and Supervision, Wenzhou Vocational College of Science and Technology, Zhejiang, Wenzhou 325000, China

Received 21 June 2012; Accepted 18 September 2012

Academic Editor: Risto Korhonen

Copyright © 2012 Haiwa Guan 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.

#### Abstract

We investigate value distribution and uniqueness problems of meromorphic functions with their -shift. We obtain that if is a transcendental meromorphic (or entire) function of zero order, and is a polynomial, then has infinitely many zeros, where , is nonzero constant, and (or ). We also obtain that zero-order meromorphic function share is three distinct values IM with its -difference polynomial , and if , then .

#### 1. Introduction and Main Results

A function is called meromorphic function if it is analytic in the complex plane except at isolated poles. It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory such as the characteristic function , proximity function , and the counting function , see [1–3]. Let us recall the definition of the order and the zeros exponent convergence of function . The order of meromorphic function is said by The zeros of exponent convergence of meromorphic function is said by

In 1959, Hayman proved the following Theorems.

Theorem A (see [4], Theorem 8). *Let be a transcendental entire function, and let be an integer and be a nonzero constant. Then assumes all finite values infinitely often. *

Theorem B (see [4], Theorem 9). *Let be a transcendental meromorphic function, and let be an integer and be a nonzero constant. Then assumes all finite values infinitely often. *

Recently the difference variant of the Nevanlinna theory has been established independently in [5, 6]. Using these theories, value distribution theory uniqueness theory of difference polynomials of finite order transcendental meromorphic functions has been studied as well. We recall the following result by Liu and Laine.

Theorem C (see [7], Theorem 1.1). *Let be a transcendental entire function of finite order not of period , where is a nonzero constant, and let be a nonzero small function of . Then the difference polynomial has infinitely many zeros in the complex plane provided that . *

In 2010, Chen considered the difference counterpart of Hayman's theorem and porved an almost direct difference analogue of Hayman's theorem.

Theorem D (see [8], Theorem 1.1). *Let be a transcendental entire function of finite order not of period , and let be three complex numbers. Then assumes all finite values infinitely often, provided that and for every one has . *

In this paper, we consider the value distribution of zero-order meromorphic functions with their -shirt and prove the following results.

Theorem 1.1. *Let be a transcendental meromorphic function of zero order, and let be a polynomial. If is an integer and , then has infinitely many zeros, where and is nonzero constant. *

Theorem 1.2. *Let be a transcendental entire function of zero order, and let be a polynomial. If is an integer and , then has infinitely many zeros, where , and is nonzero constant. *

It is well known that two meromorphic functions must be equal, if they share five distinct values. Recently, Heittokangas et al. research the uniqueness of meromorphic functions with their shifts in [6]. They got that if and share three distinct values, where is finite order, then . In this paper, we want to get some results on uniqueness of and , where is zero order and . Let us recall the notation of -difference which is written by .

Theorem 1.3. *Let be a meromorphic function of zero order, let , and let be three distinct values. *(a)*If and share CM, then for all .*(b)*If and share CM, and if
then for all .*

Corollary 1.4. *Let be an entire function of zero order, let , and let be two distinct values. *(a)*If and share CM, then for all .*(b)*If and share CM, and if
then for all .*

Corollary 1.5. *Let be a meromorphic function of zero order, and let . If and share CM and a constant CM, and if there exists a constant such that
**
then for all . *

Theorem 1.6. * Let be a meromorphic function of zero order, let , and let
**
where are constants. Let be the number of nonzero coefficients of the -difference polynomial . If and share three distinct finite values IM, and if
**
then . *

Corollary 1.7. *Let be a meromorphic function of zero order, let , and let be three distinct finite values. *(a)*If and share IM, and if
then for all .*(b)*If and share IM and two constants IM, and if there exists a constant such that
then for all . *

Corollary 1.8. *Let be an entire function of zero order, let , and let be three distinct finite values. *(a)*If and share IM, then for all .*(b)*If and share IM, and if there exists a constant such that
then for all . *

#### 2. Auxiliary Results

The following auxiliary results will be instrumental in proving the theorems.

Lemma 2.1 (see [9], Theorem 1.2). *Let be a nonconstant zero-order meromorphic function, and . Then
*

Lemma 2.2 (see [9], Theorem 3.1). *Let be a non-constant meromorphic functions of zero order, let , and let , where , be distinct points. Then
**
where
*

Lemma 2.3 (see [10], Theorem 1.1). *Let be a non-constant zero-order meromorphic function, and . Then
*

Lemma 2.4 (see [10], Theorem 1.3). *Let be a non-constant zero-order meromorphic function, and . Then
*

Lemma 2.5 (see [11], Lemma 4). *If is a piecewise continuous increasing function such that
**
then the set
**
has logarithmic density 0 for all and . *

#### 3. Proof of Theorem 1.1

Let us put

Hence, is not a constant identity. If not, let us suppose that , where is a constant, and then which give us

By using Lemma 2.3, we have , which contradicts the assumption . Hence . By taking logarithmic derivative on two sides of (3.1), we have

If then by integrating two sides of which, we have , where is a nonzero constant, and hence If , then , which is contradiction with , is transcendental function. If , then Lemma 2.3 implies that , which is impossible. Therefore, we can write (3.4) as

Let us put

Now we consider the poles of . The poles of come from the zeros of and the poles of , , and . If is a zero of or a pole of , but not a pole of , then is a simple pole of . If is a common pole of and with multiplicities of and , respectively, then is a pole of with multiplicity no more than . If is a pole of but not a pole of , we obtain that is at most a simple pole of by using (3.7). Hence, Lemma 2.5 implies that

Let us put

Now, let us consider the pole of . The poles of come from the poles of and and the zeros of and . If is a zero of , zero of , or pole of , then is a simple pole of . If is a pole of but not a pole of , by using Laurent series, we obtain that is analytic at . Therefore, we have according to Lemma 2.5. In the coming (3.7) and Lemma 2.1, it implies that

Therefore, we have which shows that has infinite zeros by .

#### 4. Proof of Theorem 1.2

In the same manner as in the proof of Theorem 1.1, we have (3.1)–(3.7), noting that

Now consider the poles of . The poles of come from the zeros of and the poles of . If is a zero of , then is a simple pole of . Hence, Lemma 2.4 implies that

Let us put

Now, let us consider the pole of . The poles of come from the poles of and the zeros of and . If is a zero of , zero of , then is a simple pole of . Therefore, we have according to Lemma 2.5. In the coming (3.7) and Lemma 2.1, it implies that

Therefore, we have which shows that has infinite zeros by .

#### 5. Proof of Theorem 1.3

(a) Suppose first that are three distinct values, and assume conversely to the assertion that . Then Lemma 2.2 yields and so

Since and share , and CM, it follows that

In addition, since is zero-order meromorphic function, from Lemma 2.4, Lemma 2.5 and equations (5.2) and (5.3), we have which is impossible. This contradiction is only avoided when .

Suppose that while and are distinct finite values. Similarly as above, can be obtained. Therefore, for all .

(b) Assume that . Similarly as above, Lemma 2.2 yields and therefore . This together with the condition results in a contradiction. Hence .

#### 6. Proof of Theorem 1.6

Assume on the contrary to the assertion that . In what follows, is small enough and is large enough. From the condition, we have and this together with Lemma 2.4 and Lemma 2.1 gives

Therefore, by the sharing assumption, from above, it follows that which is impossible. This contradiction yields .

#### Acknowledgment

The first author was supported in part by 2012 Zhejiang Educational and Scientific Projects (SCG295).

#### References

- W. K. Hayman,
*Meromorphic Functions*, Clarendon Press, Oxford, UK, 1964. - C.-C. Yang and H.-X. Yi,
*Uniqueness Theory of Meromorphic Functions*, Kluwer Academic, New York, NY, USA, 2003. - I. Laine,
*Nevanlinna Theory and Complex Differential Equations*, Walter de Gruyter, Berlin, Germany, 1993. - W. K. Hayman, “Picard values of meromorphic functions and their derivatives,”
*Annals of Mathematics*, vol. 70, pp. 9–42, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane,”
*Ramanujan Journal*, vol. 16, no. 1, pp. 105–129, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Heittokangas, R. Korhonen, I. Laine, and J. Rieppo, “Uniqueness of meromorphic functions sharing values with their shifts,”
*Complex Variables and Elliptic Equations*, vol. 56, no. 1–4, pp. 81–92, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Liu and I. Laine, “A note on value distribution of difference polynomials,”
*Bulletin of the Australian Mathematical Society*, vol. 81, no. 3, pp. 353–360, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z.-X. Chen, “On value distribution of difference polynomials of meromorphic functions,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 239853, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. C. Barnett, R. G. Halburd, R. J. Korhonen, and W. Morgan, “Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 137, no. 3, pp. 457–474, 2007. View at Publisher · View at Google Scholar - J. L. Zhang and R. Korhonen, “On the Nevanlinna characteristic of $f(qz)$ and its applications,”
*Journal of Mathematical Analysis and Applications*, vol. 369, no. 2, pp. 537–544, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. K. Hayman, “On the characteristic of functions meromorphic in the plane and of their integrals,”
*Proceedings of the London Mathematical Society*, vol. 14, pp. 93–128, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH