ISRN Mathematical AnalysisVolume 2013 (2013), Article ID 497921, 5 pageshttp://dx.doi.org/10.1155/2013/497921

Research Article

Value Distribution of Meromorphic Solutions and Their Derivatives of Complex Differential Equations

Abdallah El Farissi

Faculty of Sciences and Technology, Bechar University, Bechar, Algeria

Received 11 May 2013; Accepted 16 August 2013

Academic Editors: G. Ólafsson and C. Zhu

Copyright © 2013 Abdallah El Farissi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We deal with the relationship between the small functions and the derivatives of solutions of higher-order linear differential equations ,   , where    are meromorphic functions. The theorems of this paper improve the previous results given by El Farissi, Belaïdi, Wang, Lu, Liu, and Zhang.

1. Introduction and Statement of Result

Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna's value distribution theory (see [1, 2]). In addition, we will use and to denote, respectively, the exponents of convergence of the zero sequence and the pole sequence of a meromorphic function , to denote the order of growth of , to denote the type of the entire function with , and and to denote, respectively, the exponents of convergence of the sequence of distinct zeros and distinct poles of . A meromorphic function is called a small function of a meromorphic function if as , where is the Nevanlinna characteristic function of . In order to express the rate of growth of meromorphic solutions of infinite order, we recall the following definitions.

Definition 1 (see [2–4]). Let be a meromorphic function, and let , such that , be the sequence of the fixed points of , with each point being repeated only once. The exponent of convergence of the sequence of distinct fixed points of is defined by the following: Clearly, where is the counting function of distinct fixed points of in .

Definition 2 (see [4–6]). Let be a meromorphic function. Then the hyperorder of is defined by the following:

Definition 3 (see [4, 5]). Let be a meromorphic function. Then the hyperexponent of convergence of the sequence of distinct zeros of is defined by the following: where is the counting function of distinct zeros of in .

For , we consider the following linear differential equation: where is a transcendental meromorphic function of finite order . Many important results have been obtained on the fixed points of general transcendental meromorphic functions for almost four decades (see [7]). However, there are a few studies on the fixed points of solutions of differential equations. In [8], Wang and Lü have investigated the fixed points and hyperorder of solutions of second-order linear differential equations with meromorphic coefficients and their derivatives, and they have obtained the following result.

Theorem A (see [8]). Suppose that is a transcendental meromorphic function satisfying , . Then, every meromorphic solution of the equation satisfies that and , all have infinitely many fixed points and

Theorem A has been generalized to higher-order differential equations by Liu Ming-Sheng and Zhang Xiao-Mei as follows.

Theorem B (see [4]). Suppose that and are transcendental meromorphic functions satisfying , . Then every meromorphic solution of (5) satisfies that and all have infinitely many fixed points and

In [9], Belaïdi and El Farissi extended the result of Theorem B, and they gave the following theorem.

Theorem C (see [9]). Suppose that and are transcendental meromorphic functions satisfying and   . If is a meromorphic function with finite order , then every meromorphic solution of (5) satisfies

Recently, Xu et al. [10] investigated the relationship between small functions and the derivatives of solutions of the following equation: and obtained the following theorems which improve the results given by Chen, Wang, Lu, Liu, Zhang, Belaïdi, and El Farissi.

Theorem D (see [10]). Let and be entire functions with finite order and satisfy one of the following conditions: (i), (ii) and     ;Then, for every solution of (10) and for any entire function satisfying , one has the following:

Theorem E (see [10]). Let and be meromorphic functions satisfying and . Then for every meromorphic solution of (10) and for any meromorphic function satisfying , one has .

In this paper, we will deal with the above equation, investigate the relationship between small functions and derivatives of solutions of (10), obtain some results, which improve the previous results given by Xu, Tu, and Zheng, and prove the following theorems.

Theorem 4. Let and be meromorphic functions of finite order such that all solutions of (10) satisfy and . Then if is a meromorphic function with , then every solution of (10) satisfies

Theorem 5. Let and be meromorphic functions of finite order such that all solutions of (10) satisfy . Then if is a meromorphic function with , then every solution of (10) satisfies (12). Furthermore, if , and , then

Remark 6. The following example shows that Theorem 4 is not valid when (10) has a finite order solution.

Example 7. For the following equation we can easily get that has solutions , , and . Solution satisfies and . Take ; then . Thus, we can get that and .

Remark 8. The proof of Theorem 4 is quite different from that of Theorem D (see [10]). The main ingredient in the proof is Lemma 17.

Corollary 9. Under the assumptions of Theorem 4, if , then for every solution of (10), one has the following:

Remark 10. Theorem 4 is the improvement of Theorems A, B, C, and D, and Theorem 5 is the improvement of Theorem E.

Corollary 11. Let and be entire functions that have finite order such that . Let be polynomials, where , ; are complex numbers, and suppose that or , where , . If is an entire function with , then every solution of the equation satisfies (12) with .

2. Auxiliary Lemmas

The following lemmas will be used in the proofs of Theorem 4 and Corollary 11.

Lemma 12 (see [11]). Let and be entire functions that have finite order such that . Let  +  +  +  + be polynomials, where , are complex numbers, and suppose that or , where , . Then every solution of (15) of infinite order and .

Lemma 13 (see [3]). Let ,   be finite order meromorphic functions. If is a meromorphic solution with of the following equation: then .

Lemma 14 (see [12]). Let ,   be finite order meromorphic functions. If is a meromorphic solution of (16), then we have the following statements: (i)if , then ;(ii)if , then .

Let be meromorphic functions. We define the sequences of functions by the following: where .

Remark 15. In the case where one of the functions of    is equal to zero, then .

Lemma 16. Suppose that is a solution of (10). Then is a solution of the following equation: where are given by (17).

Proof. Assume that is a solution of (10), and let . We prove that is an entire solution of (18). Our proof is by induction. For , differentiating both sides of (10), we obtain and replacing by we get That is, Suppose that the assertion is true for the values which are strictly smaller than a certain . We suppose is a solution of the following equation: Differentiating (23), we can write the following: In (24), replacing by yields the following: That is, Lemma 16 is thus proved.

Lemma 17. Let be meromorphic functions of finite order such that all solutions of (10) have infinite order and , and let be defined as in (17). Then the nontrivial meromorphic solution of the equation satisfies and .

Proof. Let be a fundamental system of solutions of (10). We show that is a fundamental system of solutions of (28). By Lemma 16, it follows that are solutions of (28). Let be constants such that Then, we have where is a polynomial of a degree less than . Since is a solution of (10), then is a solution of (10), and by the conditions of the lemma, we conclude that is an infinite order solution of (10); this leads to a contradiction. Therefore, is a trivial solution. We deduce that . Using the fact that is a fundamental solution of (10), we get . Now, let be a nontrivial solution of (28). Then, using the fact that is a fundamental solution of (28), we claim that there exist constants not all equal to zero, such that . Let , be a solution of (10), and . Hence, by conditions of the lemma, we get and .

Proof of Theorem 4. Assume that is a solution of (10). By the conditions of the theorem, we get and . Taking , then and . Now, let , where is a meromorphic function with .Then and . In order to prove and , we need to prove only that and . Using the fact that and by Lemma 16 we get the following: By , , and Lemma 17, we get and . By Lemmas 13 and 14 we get and . The proof of Theorem 4 is complete.

Proof of Theorem 5. By the same reasoning as before we can prove Theorem 5.

Proof of Corollary 11. By Lemma 12, all solutions of (15) satisfy and . Applying Theorem 4 we get , and , .

Acknowledgment

The author would like to thank the referee for his/her helpful remarks and suggestions to improve the paper. This paper is supported by ANDRU (Agence Nationale pour le Développement de la Recherche Universitaire) and University of Mostaganem (UMAB) (PNR Project Code 8/u27/3144).

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