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

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## Abstract

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

### Keywords

Difference Equation Meromorphic Function Monic Polynomial Meromorphic Solution Small Function## 1. Introduction and Main Results

Let Open image in new window 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 Open image in new window , proximity function Open image in new window , counting function Open image in new window , the first and second main theorems (see, e.g., [1, 2, 3, 4]). We also use Open image in new window to denote the counting function of the poles of Open image in new window whose every pole is counted only once. The notation Open image in new window denotes any quantity that satisfies the condition: Open image in new window as Open image in new window possibly outside an exceptional set of Open image in new window of finite linear measure. A meromorphic function Open image in new window is called a small function of Open image in new window if and only if Open image in new window

Recently, a number of papers (see, e.g., [5, 6, 7, 8, 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 Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ), admits a transcendental meromorphic solution of finite order, then Open image in new window

Theorem B.

with polynomial coefficients Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ), admits a transcendental meromorphic solution of finite order, then Open image in new window

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 Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ) admits a transcendental meromorphic solution of finite order, then Open image in new window

Theorem D.

with the coefficients of rational functions Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ) admits a transcendental meromorphic solution of finite order, then Open image in new window

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

Theorem E.

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

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

Theorem F.

where Open image in new window is a small meromorphic function relative to Open image in new window .

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.

where Open image in new window

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.

If the order Open image in new window is finite, then Open image in new window

Corollary 1.3.

If the order Open image in new window is finite, then Open image in new window

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 Open image in new window and Open image in new window Therefore, the estimations in Theorem 1.2 and Corollary 1.3 are sharp.

Theorem 1.7.

where Open image in new window is a small meromorphic function relative to Open image in new window .

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

Corollary 1.8.

where Open image in new window is a small meromorphic function relative to Open image in new window .

Finally, we give a result corresponding to Theorem G.

Theorem 1.9.

where Open image in new window

## 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 Open image in new window and Open image in new window are two finite index sets, and Open image in new window ( Open image in new window ).

Remark 2.3.

Lemma 2.4 (see [6]).

Lemma 2.5 (see [12]).

Lemma 2.7 (see [14]).

for all Open image in new window large enough.

Lemma 2.8 (see [15]).

where Open image in new window .

## 3. Proof of Theorems

Proof of Theorem 1.2.

This yields the asserted result.

Proof of Theorem 1.7.

Finally, let Open image in new window and we conclude that the order Open image in new window Therefore, we get a contradiction and the assertion follows.

Proof of Theorem 1.9.

Denoting now Open image in new window , thus we obtain the required form. Theorem 1.9 is proved.

## Notes

### 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).

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