#### 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).