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 Nevanlinna’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,