On Growth of Meromorphic Solutions of Complex Functional Difference Equations

Abstract

The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form , where and are two finite index sets, are distinct complex numbers, and are small functions relative to and is a rational function in with coefficients which are small functions of , of degree . We also give some examples to show that our results are sharp.

1. Introduction and Main Results

Let be a function meromorphic in the complex plane . We assume that the reader is familiar with the standard notations and results in Nevanlinna’s value distribution theory of meromorphic functions such as the characteristic function , proximity function , counting function , and the first and second main theorems (see, e.g., [1–4]). We also use to denote the counting function of the poles of whose every pole is counted only once. The notations and denote the order and the lower order of , respectively. denotes any quantity that satisfies the condition: as possibly outside an exceptional set of of finite linear measure. A meromorphic function is called a small function of or a small function relative to if and only if .

Recently, some papers (see, e.g., [5–7]) focusing on complex difference and functional difference equations emerged. In 2005, Laine et al. [5] firstly considered the growth of meromorphic solutions of the complex functional difference equations by utilizing Nevanlinna theory. They obtained the following result.

Theorem A. Suppose that is a transcendental meromorphic solution of the equation where is a collection of all subsets of , ’s are distinct complex constants, and is a polynomial of degree . Moreover, we assume that the coefficients are small functions relative to and that . Then where .

In 2007, Rieppo [6] gave an estimation of growth of meromorphic solutions of complex functional equations as follows.

Theorem B. Suppose that is a transcendental meromorphic function. Let , be rational functions in with small meromorphic coefficients relative to such that and of degree . If is a solution of the functional equation then , and for any , there exist positive real constants and such that when is large enough.

Rieppo [6] also considered the growth order of meromorphic solutions of functional equation (3) when and got the following.

Theorem C. Suppose that is a transcendental meromorphic solution of (3), where , , and . Then

Two years later, Zheng et al. [7] extended Theorem A to more general type and obtained a similar result of Theorem C. In fact, they got the following two results.

Theorem D. Suppose that is a transcendental meromorphic solution of the equation where is a collection of all nonempty subsets of , are distinct complex constants, of degree , and is a rational function in of . Also suppose that all the coefficients of (6) are small functions relative to . Then , and where .

Theorem E. Suppose that is a transcendental meromorphic solution of (6), where is a collection of all nonempty subsets of , are distinct complex constants, , , and is a rational function in of . Also suppose that all the coefficients of (6) are small functions relative to .(i)If , then we have (ii)If , then we have and (iii)If , then we have .

In this paper, we will consider a more general class of complex functional difference equations. We prove the following results, which generalize the above related results.

Theorem 1. Suppose that is a transcendental meromorphic solution of the functional difference equation where are distinct complex constants, ∈, and   , are two finite index sets, of degree , and is a rational function in of . Also suppose that all the coefficients of (10) are small functions relative to . Denoting Then , and where .

Theorem 2. Suppose that is a transcendental meromorphic solution of the equation where are distinct complex constants, , and , are two finite index sets, , and is a rational function in of . Also suppose that all the coefficients of (10) are small functions relative to . Denoting (i)If , then we have (ii)If , then we have and (iii)If and , then we have .

Next we will give some examples to show that our results are best in some extent.

Example 3. Let , . Then it is easy to check that solves the following equation: Obviously, we have where , and .

Example 3 shows that the estimate in Theorem 2(i) is sharp.

Example 4. It is easy to check that satisfies the equation Clearly, we have where , and .

Example 4 shows that the estimate in Theorem 2(ii) is sharp.

Example 5. satisfies the equation of the form where , , and .

Example 5 shows that the strict inequality in Theorem 2 may occur. Therefore, we do not have the same estimation as in Theorem C for the growth order of meromorphic solutions of (13).

The following Example shows that the restriction in case (iii) in Theorem 2 is necessary.

Example 6. Meromorphic function solves the following equation: where and , but .

Next, we give an example to show that case (iii) in Theorem 2 may hold.

Example 7. Function satisfies the following equation: where and . Obviously, .

2. Main Lemmas

In order to prove our results, we need the following lemmas.

Lemma 1 (see [4, 8]). Let be a meromorphic function. Then for all irreducible rational functions in , such that the meromorphic coefficients , satisfy then one has

From the proof of Theorem  1 in [9], we have the following estimate for the Nevanlinna characteristic.

Lemma 2. Let be distinct meromorphic functions and Then where , and =   are two finite index sets, . and hold for all and satisfy and .

Lemma 3 (see [7]). Let be a complex constant. Given and a meromorphic function , one has for all , where is some positive constant.

Lemma 4 (see [4]). Let , be monotone increasing functions such that outside of an exceptional set of finite linear measure. Then, for any , there exists such that for all .

Lemma 5 (see [10]). Let be a transcendental meromorphic function, and , be a nonconstant polynomial of degree . Given , denote and . Then given and , one has for all large enough.

Lemma 6 (see [11]). Let be positive and bounded in every finite interval, and suppose that holds for all large enough, where , , and are real constants. Then where .

Lemma 7 (see [6]). Let , where , be a monotone increasing function. If for some real constant , there exists a real number such that , then

Lemma 8 (see [12]). Let be a monotone increasing function and let be a nonconstant meromorphic function. If, for some real constant , there exist real constants and such that then

3. Proof of Theorems

Proof of Theorem 1. We assume is a transcendental meromorphic solution of (10). Denoting . According to Lemmas 1, 2, and 3 and the last assertion of Lemma 5, we get that for any , where is large enough and for some . Since holds for large enough for , we may assume to be large enough to satisfy outside a possible exceptional set of finite linear measure. By Lemma 4, we know that whenever , holds for all large enough. Denote ; thus the inequality (37) may be written in the form By Lemma 6, we have where Denoting now and ; thus we obtain the required form.
Finally, we show that . If , then we have . For sufficiently small , we have , which contradicts with the transcendency of . Thus Theorem 1 is proved.

Proof of Theorem 2. Suppose is a transcendental meromorphic solution of (13). Denoting .(i). We may assume that , since the case is trivial by the fact that . By Lemmas 1–3, we have for any and , where is large enough.
By the last assertion of Lemma 5 and (41), we obtain that, for , the following inequality holds, where is large enough outside of a possible set of finite linear measure. By Lemma 4, we get that for any and sufficiently large , Therefore, Since , , and , we have and when is small enough. Using Lemma 7, we see that Letting , , and , we have (ii). By the similar reasoning as is (i), we easily obtain that for all large enough. We may select sufficiently small numbers and , such that and . Thus we have namely, where is large enough possibly outside of a set of finite linear measure. By Lemma 4, we have for any , that is, holds for all sufficiently large . By Lemma 8, we obtain Letting , and , we have (iii) and . The proof of this case is completely similar as in the case in (i). In fact, we set . Similarly, we can get Since , we have .

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The research was supported by Colonel-level topics (JSNU-ZY-01), (Jsie2012zd01), and NSF of China (11271179).