#### Abstract

In this paper, we shall give some properties of complex differential-difference equation of the form where and are non-negative integers, are distinct, nonzero complex numbers, and are small functions relative to , and is a rational function in with coefficients, which are small functions of . We also consider related complex functional equations in the paper. These results are improvements of some several previous results.

#### 1. Introduction and Main Results

In this note, we assume that the reader is familiar with the standard notations and results about Nevanlinnaâ€™s value distribution theory of meromorphic function , such as the characteristic function , proximity function , and counting function ( e.g., [2, 7, 9, 14]). The notation denotes the counting function of the poles of whose every pole is counted only once and denotes any quantity that satisfies the condition: as possibly outside an exceptional set of of finite logarithmic measure. A meromorphic function is called a small function of if .

As one knows, Malmquist type problem is an important topic in complex differential equations. For the first-order complex differential equations, Malmquist [16] proved the following important result independent of Nevanlinna theory in 1913.

Theorem 1. *Let be birational function. If a differential equation of the formadmits a transcendental meromorphic solution, then the equation can be reduced into a Riccati differential equationwhere are rational functions.*

Later, Yosida [24] gave a proof of Theorem 1 by using Nevanlinna theory. Laine [13], Yang [23], Hille [11], Steinmetz [21], Rieth [19], He and Laine [8], Gackstatter and Laine [4], and Rieth [19] extended Theorem 1 to more general types. In particular, Gackstatter and Laine [4] gave a following generalized result of Theorem 1 in 1980.

Theorem 2. *If the algebraic differential equationwhere is a finite index set, coefficients , , and are meromorphic and small functions relative to , possesses an transcendental meromorphic solution, then reduces to a polynomial in of degree .*

Recently, Zhang and Liao [26] gave a simple proof of Theorem 2, which is different from the methods that of Gackstatter and Laine [4] and Steinmetz [21]. However, Malmquist type problem even for special second-order algebraic differential equations remains open. One may refer to [14, p. 251].

In the past fifteen or more years, people have already had great interest in complex difference equations. In particular, there are a large number of papers on Malmquist type theorem of the complex difference equations, for instance [1, 3, 10, 12, 15, 25]. In 2000, Ablowitz et al. [1] first obtained two results on Malmquist type of complex difference equations by utilizing Nevanlinna theory. One year later, Heittokangas et al. [10] extended these two results to the case of more higher-order complex difference equations. Subsequently, Laine et al. [15] and Huang and Chen [12], respectively, generalized the preceding results. At the same time, Laine et al. [15] also obtained Tumura-Clunie theorem about complex difference equation of Malmquist type and considered the growth of meromorphic solutions of corresponding complex functional equations. In 2010, Zhang and Liao [25] generalized these results to more general complex difference equations and functional equations.

A natural question is whether we can obtain some corresponding results for complex differential-difference equations of Malmquist type? The answer is positive. Now we give our main results as follows.

Theorem 3. *Let be distinct, nonzero complex numbers and suppose that is a transcendental meromorphic solution of the differential-difference equationwhere and are non-negative integers, and are small functions relative to . We denote*

If the order is finite, then .

According to Theorem 3, we can obtain the following two results easily.

Corollary 1. *Let be distinct, nonzero complex numbers and suppose that is a transcendental meromorphic solution of the differential-difference equationwhere are non-negative integers, , and are small functions relative to . We denote*

If the order is finite, then .

Corollary 2. *Let be distinct, nonzero complex numbers and suppose that is a transcendental meromorphic solution of the differential-difference equationwith coefficients and are small functions relative to , where and are two finite index sets, we denote*

If the order is finite, then .

We are now giving the following two examples. Example 1 shows that the estimation in Theorem 3 is sharp and Example 2 shows that the condition is finite in Theorem 3 and cannot be removed.

*Example 1. *Let , . Then, solves the following differential-difference equation:In Example 1, we have and .

*Example 2. *Let , . It is easy to check that satisfies the following differential-difference equation:Clearly, we have and .

The following result is the corresponding Tumura-Clunie theorem on complex differential-difference equation.

Theorem 4. *Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solutions of the following equation:where and are non-negative integers, and the coefficients are nonvanishing small functions relative to and and are relatively prime polynomials in over the field of small functions relative to . We denote*

Moreover, we assume that ,and that, without restricting generality, is a monic polynomial. If there exists such that for all sufficiently large,where , then either the order , orwhere is a small meromorphic function relative to .

Note that by the fundamental property of counting function, we have

Hence, if the left-hand side of (12) in Theorem 4 is replaced by the left-hand side of (6) in Corollary 1, then (16) implies (15). Therefore, we have the following result.

Corollary 3. *Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solutions of the following equation:where are non-negative integers, and the coefficients are nonvanishing small functions relative to and and are relatively prime polynomials in over the field of small functions relative to . We denote*

Moreover, we assume that ,and that, without restricting generality, is a monic polynomial. If there exists such that for all sufficiently large,where , then either the order , orwhere is a small meromorphic function relative to .

Finally, we give a result about the growth of meromorphic solutions of complex functional equations.

Theorem 5. *Let be distinct, nonzero complex numbers and suppose that is a transcendental meromorphic solution of the equation,where are non-negative integers and is a polynomial of degree We denote*

Moreover, we assume that the coefficients and are small functions relative to and that . Then,where .

#### 2. Main Lemmas

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

Lemma 1 (see [17]). *Let be a meromorphic function. Then, for all irreducible rational functions in ,such that the meromorphic coefficients satisfywe have*

The estimation for the Nevanlinna characteristic of some classes of meromorphic functions plays a very important role in the proof of our theorems. According to the Proof of Theorem 3 in [18], we have the following.

Lemma 2 (see [18]). *Let be distinct meromorphic functions andwhere and are non-negative integers, and are small functions of . We set*

Then,for outside of a possible exceptional set of finite linear measure.

Lemma 3 (see [3]). *Let be a meromorphic function with order , and be a fixed nonzero complex number, then for each , we have*

Lemma 4 (see [22]). *Let be a meromorphic function and let be given bywhere are small meromorphic functions relative to . Then, eitheror*

Lemma 5 (see [15, 20]). *Let be a nonconstant meromorphic function and , two polynomials in with meromorphic coefficients small relative to . If and have no common factors of positive degree in over the field of small functions relative to , then*

Lemma 6 (see [5]). *Let be a transcendental meromorphic function, and , be a nonconstant polynomial of degree . Given , we denote and . Then, given and , we havefor all large enough.*

Lemma 7 (see [6]). *Let be positive and bounded in every finite interval, and suppose that holds for all large enough, where and are real constants. Then,where .*

#### 3. Proofs of Theorems

*Proof of Theorem 3. *We assume that is a transcendental meromorphic solution with finite order of equation (4). It follows from Lemma 1 to Lemma 3 that for each ,Therefore, we have .

*Proof of Theorem 4. *Suppose is a transcendental meromorphic solution of equation (12) andThen, by Lemma 4, Lemma 5 and condition (15), we haveThus,Now we assume that the order , then we have andfor . By using Lemma 1, Lemma 2 and condition (16), we conclude that,