Existence and growth of meromorphic solutions of some nonlinear q-difference equations
© Zheng and Tu; licensee Springer. 2013
Received: 3 August 2012
Accepted: 13 January 2013
Published: 11 February 2013
In this paper, we investigate the existence of transcendental meromorphic solutions of order zero of some nonlinear q-difference equations and some more general equations. We also give some estimates on the growth of transcendental meromorphic solutions of these equations. Some examples are given to illustrate the sharpness of some of our results.
1 Introduction and main results
In this paper, a meromorphic function means being meromorphic in the whole complex plane. We also assume that the readers are familiar with the usual notations of Nevanlinna theory (see, e.g., [1–3]). Especially, we use notations and for the order and the lower order of a meromorphic function f. We also denote by any quantity satisfying for all r outside of a possible exceptional set of finite logarithmic measure. Moreover, the standard definitions of logarithmic measure and logarithmic density can be found in .
In last two decades, there has been a renewed interest in the complex analytic properties of complex differences and meromorphic solutions of complex difference equations owing to the introduction of Nevanlinna theory in this field. And the study of complex q-differences and q-difference equations is an important component of the study of complex differences and difference equations.
Recently, a number of papers (including [8–23]) focus on complex differences and complex q-differences. These papers also investigate the existence and the growth of meromorphic solutions of complex difference equations and complex q-difference equations.
In particular, Yang-Laine  pointed out some similarities between results on the existence and uniqueness of finite order entire solutions of the nonlinear differential equations and differential-difference equations. They obtained the following results.
has no transcendental entire solutions of finite order.
possesses at most one admissible transcendental entire solution of finite order such that all coefficients of are small functions of f. If such a solution f exists, then f is of the same order as h.
Some generalizations of Theorems A and B can be found in Peng-Chen , and we omit those results here.
In this paper, we consider a similar problem on the existence and the growth of transcendental meromorphic solutions of complex q-difference equations (resp. differential-q-difference equations) instead of complex difference equation (1.2) (resp. differential-difference equation (1.3)). And the involved equations are more general than (1.1). Moreover, the fact that all meromorphic solutions of the Riccati q-difference equation and linear q-difference equation, both with rational coefficients, are of order zero, shows that it is of great importance to investigate meromorphic solutions of order zero of q-difference equations.
- (i)If , then the nonlinear q-difference equation(1.4)
If , then nonlinear q-difference equation (1.4) has no transcendental entire solutions of order zero.
where , and . The above two examples show that the estimates on the growth of meromorphic solutions of equations (1.4) and (1.5) are sharp.
In particular, if each monomial of is of the same degree, then we call a homogeneous differential-q-difference polynomial in f.
has no transcendental entire solutions of order zero.
Corollary 1.2 Let be an integer, (≢0) be a linear differential-q-difference polynomial in f, with all meromorphic coefficients of growth , and be a rational function. Then differential-q-difference equation (1.6) has no transcendental entire solutions of order zero.
has no transcendental entire solutions of order zero. Furthermore, if , then nonlinear q-difference equation (1.7) has no transcendental meromorphic solutions of order zero, and any transcendental meromorphic solution f of positive order of (1.7) satisfies .
Remark 1.4 The results concerning the existence of meromorphic solutions of order zero in Theorem 1.1(i) and Corollary 1.1 are not only special cases of Theorem 1.2 and Corollary 1.2 respectively, but also more precise.
Remark 1.5 Clearly, equations (1.4)-(1.7) can have rational solutions.
2 Preliminary lemmas
Lemma 2.1 (See )
on a set of lower logarithmic density 1.
The following two lemmas can be seen as special cases of [, Theorem 3]. In fact, the present versions we give here are more precise than the original one. To facilitate the readers, we give the corresponding proofs here.
where K (>0) is a constant. Thus, the lower order of f satisfies .
where (>0) is a constant. Thus, we get . □
where K (>0) is a constant. Thus, the lower order of f satisfies .
Proof Note that (2.3) holds for both (2.1) and (2.8), then the proof of Lemma 2.3 is similar to that of Lemma 2.2. □
Lemma 2.4 (See )
on a set of logarithmic density 1.
Lemma 2.5 (See )
3 Proofs of Theorem 1.1 and Corollary 1.1
3.1 Proof of Theorem 1.1
- (i)Suppose that is a transcendental meromorphic solution of order zero of (1.4). It follows by (1.4) and Lemma 2.1 that(3.1)
on a set of lower logarithmic density 1. It is clear that (3.1) is a contradiction since . Thus, (1.4) has no transcendental meromorphic solutions of order zero if .
- (ii)Suppose that is a transcendental entire solution of order zero of (1.4). By (3.2) and Lemmas 2.1, 2.4, we have(3.3)
on a set of logarithmic density 1. It is clear that (3.3) is a contradiction since . Thus, (1.4) has no transcendental meromorphic solutions of order zero if .
3.2 Proof of Corollary 1.1
4 Proofs of Theorem 1.2 and Corollaries 1.2, 1.3
4.1 Proof of Theorem 1.2
on a set of logarithmic density 1. It is clear that (4.2) is a contradiction since . Thus, (1.6) has no transcendental entire solution of order zero if .
4.2 Proof of Corollary 1.2
Since is a linear differential-q-difference polynomial in f, the degree of is . Thus, by , we immediately know by Theorem 1.2 that (1.6) has no transcendental entire solutions of order zero.
4.3 Proof of Corollary 1.3
The first part result of Corollary 1.3 is a special case of Corollary 1.2. The second part result of Corollary 1.3 can be proved similarly to Theorem 1.1(i) by replacing Lemma 2.2 with Lemma 2.3.
The authors thank the referees and the editors for their valuable comments to improve the readability of our paper. And this work was supported by the National Natural Science Foundation of China (11126145 and 11171119) and the Natural Science Foundation of Jiangxi Province in China (20114BAB211003 and 20122BAB211005).
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