# Further Extending Results of Some Classes of Complex Difference and Functional Equations

- Jian-jun Zhang
^{1}and - Liang-wen Liao
^{1}Email author

**2010**:404582

**DOI: **10.1155/2010/404582

© Jian-jun Zhang and Liang-wen Liao. 2010

**Received: **29 March 2010

**Accepted: **21 September 2010

**Published: **27 September 2010

## Abstract

The main purpose of this paper is to present some properties of the meromorphic solutions of complex difference equation of the form , where and are two finite index sets, are distinct, nonzero complex numbers, and are small functions relative to is a rational function in with coefficients which are small functions of . We also consider related complex functional equations in the paper.

## 1. Introduction and Main Results

Let be a meromorphic function 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 , 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 notation 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 if and only if

Recently, a number of papers (see, e.g., [5–9]) focusing on Malmquist type theorem of the complex difference equations emerged. In 2000, Ablowitz et al. [5] proved some results on the classical Malmquist theorem of the complex difference equations in the complex differential equation by utilizing Nevanlinna theory. They obtained the following two results.

Theorem A.

with polynomial coefficients ( ) and ( ), admits a transcendental meromorphic solution of finite order, then

Theorem B.

with polynomial coefficients ( ) and ( ), admits a transcendental meromorphic solution of finite order, then

One year later, Heittokangas et al. [7] extended the above two results to the case of higher-order difference equations of more general type. They got the following.

Theorem C.

with the coefficients of rational functions ( ) and ( ) admits a transcendental meromorphic solution of finite order, then

Theorem D.

with the coefficients of rational functions ( ) and ( ) admits a transcendental meromorphic solution of finite order, then

Laine et al. [9] and Huang and Chen [8], respectively, generalized the above results. They obtained the following theorem.

Theorem E.

with coefficients ( ) and ( ), which are small functions relative to where is a collection of all subsets of . If the order is finite, then .

In the same paper, Laine et al. also obtained Tumura-Clunie theorem about difference equation.

Theorem F.

where is a small meromorphic function relative to .

Remark 1.1.

Huang and Chen [8] proved that the Theorem F remains true when the left hand side of (1.6) is replaced by the left hand side of (1.5), meanwhile, the condition (1.8) would be replaced by a corresponding form.

Moreover, Laine et al. [9] also gave the following result.

Theorem G.

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

Theorem 1.2.

Corollary 1.3.

Remark 1.4.

then Corollary 1.3 becomes Theorem E. Therefore, Theorem 1.2 is a generalization of Theorem E.

Example 1.5.

Example 1.6.

In above two examples, we both have and Therefore, the estimations in Theorem 1.2 and Corollary 1.3 are sharp.

Theorem 1.7.

where is a small meromorphic function relative to .

by the fundamental property of counting function. Therefore, we get the following result easily.

Corollary 1.8.

where is a small meromorphic function relative to .

Finally, we give a result corresponding to Theorem G.

Theorem 1.9.

## 2. Main Lemmas

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

Lemma 2.1 (see [10]).

Lemma 2.2 (see [11]).

where and are two finite index sets, and ( ).

Remark 2.3.

Lemma 2.4 (see [6]).

Lemma 2.5 (see [12]).

Lemma 2.7 (see [14]).

Lemma 2.8 (see [15]).

## 3. Proof of Theorems

Proof of Theorem 1.2.

This yields the asserted result.

Proof of Theorem 1.7.

Finally, let and we conclude that the order Therefore, we get a contradiction and the assertion follows.

Proof of Theorem 1.9.

Denoting now , thus we obtain the required form. Theorem 1.9 is proved.

## Declarations

### Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The research was supported by NSF of China (Grant no. 10871089).

## Authors’ Affiliations

## References

- Cherry W, Ye Z:
*Nevanlinna's Theory of Value Distribution, Springer Monographs in Mathematics*. Springer, Berlin, Germany; 2001:xii+201.View ArticleGoogle Scholar - Hayman WK:
*Meromorphic Functions, Oxford Mathematical Monographs*. Clarendon Press, Oxford, UK; 1964:xiv+191.Google Scholar - He YZ, Xiao XZ:
*Algebroid Functions and Ordinary Differential Equations*. Science Press, Beijing, China; 1988.Google Scholar - Laine I:
*Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics*.*Volume 15*. Walter de Gruyter, Berlin, Germany; 1993:viii+341.View ArticleGoogle Scholar - Ablowitz MJ, Halburd R, Herbst B:
**On the extension of the Painlevé property to difference equations.***Nonlinearity*2000,**13**(3):889-905. 10.1088/0951-7715/13/3/321MathSciNetView ArticleMATHGoogle Scholar - Chiang Y-M, Feng S-J:
**On the Nevanlinna characteristic of****and difference equations in the complex plane.***Ramanujan Journal*2008,**16**(1):105-129. 10.1007/s11139-007-9101-1MathSciNetView ArticleMATHGoogle Scholar - Heittokangas J, Korhonen R, Laine I, Rieppo J, Tohge K:
**Complex difference equations of Malmquist type.***Computational Methods and Function Theory*2001,**1**(1):27-39.MathSciNetView ArticleMATHGoogle Scholar - Huang Z-B, Chen Z-X:
**Meromorphic solutions of some complex difference equations.***Advances in Difference Equations*2009,**2009:**-10.Google Scholar - Laine I, Rieppo J, Silvennoinen H:
**Remarks on complex difference equations.***Computational Methods and Function Theory*2005,**5**(1):77-88.MathSciNetView ArticleMATHGoogle Scholar - Mohon'ko AZ:
**The Nevanlinna characteristics of certain meromorphic functions.***Teorija Funkciĭ, Funkcional'nyĭ Analiz i ih Priloženija*1971, (14):83-87. - Mohon'ko AZ, Mohon'ko VD:
**Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations.***Sibirskiĭ Matematičeskiĭ Žurnal*1974,**15:**1305-1322.MathSciNetGoogle Scholar - Weissenborn G:
**On the theorem of Tumura and Clunie.***The Bulletin of the London Mathematical Society*1986,**18**(4):371-373. 10.1112/blms/18.4.371MathSciNetView ArticleMATHGoogle Scholar - Shidlovskii AB:
*Transcendental Numbers, de Gruyter Studies in Mathematics*.*Volume 12*. Walter de Gruyter, Berlin, Germany; 1989:xx+466.Google Scholar - Goldstein R:
**Some results on factorisation of meromorphic functions.***Journal of the London Mathematical Society*1971,**4**(2):357-364. 10.1112/jlms/s2-4.2.357View ArticleMATHGoogle Scholar - Goldstein R:
**On meromorphic solutions of certain functional equations.***Aequationes Mathematicae*1978,**18**(1-2):112-157. 10.1007/BF01844071MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.