Some Properties of Solutions of Second-Order Linear Differential Equations

Journal of Complex Analysis, Jan 2013

We study the growth and oscillation of , where and are entire functions of finite order not all vanishing identically and and are two linearly independent solutions of the linear differential equation .

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Some Properties of Solutions of Second-Order Linear Differential Equations

Some Properties of Solutions of Second-Order Linear Differential Equations Zinelaâbidine Latreuch and Benharrat Belaïdi Laboratory of Pure and Applied Mathematics, Department of Mathematics, University of Mostaganem (UMAB), BP 227, 27000 Mostaganem, Algeria Received 13 August 2012; Accepted 25 September 2012 Academic Editor: Rabha W. Ibrahim Copyright © 2013 Zinelaâbidine Latreuch and Benharrat Belaïdi. 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 study the growth and oscillation of , where and are entire functions of finite order not all vanishing identically and and are two linearly independent solutions of the linear differential equation . 1. Introduction and Main Results Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory (see [1–4]). In addition, we will use and to denote, respectively, the exponents of convergence of the zero sequence and distinct zeros of a meromorphic function , to denote the order of growth of . Definition 1 (see [4, 5]). Let be a meromorphic function. Then the hyperorder of is defined by Definition 2 (see [4, 5]). Let be a meromorphic function. Then the hyper-exponent of convergence of zeros sequence of is defined by where is the counting function of zeros of in . Similarly, the hyperexponent of convergence of the sequence of distinct zeros of is defined by where is the counting function of distinct zeros of in . Suppose that and are two linearly independent solutions of the complex linear differential equation and the polynomial of solutions where and are entire functions of finite order in the complex plane. It is clear that if are complex numbers or where is a complex number, then is a solution of (4) or has the same properties of the solutions. It is natural to ask what can be said about the properties of in the case when where is a complex number and under what conditions keeps the same properties of the solutions of (4). In [6], Chen studied the fixed points and hyper-order of solutions of second-order linear differential equations with entire coefficients and obtained the following results. Theorem A (see [6]). For all nontrivial solutions of (4) the following hold.(i)If is a polynomial with , then one has (ii)If is transcendental and , then one has Before we state our results we define and by where is entire function of finite order and The subject of this paper is to study the controllability of solutions of the differential equation (4). In fact, we study the growth and oscillation of where  and are two linearly independent solutions of (4) and   and are entire functions of finite order not all vanishing identically and satisfying where is a complex number, and we obtain the following results. Theorem 3. Let be a transcendental entire function of finite order. Let be finite-order entire functions that are not all vanishing identically such that . If and   are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies Theorem 4. Under the hypotheses of Theorem 3, let be an entire function with finite order such that . If and are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies Theorem 5. Let be a polynomial of . Let    be finite-order entire functions that are not all vanishing identically such that and . If are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies Theorem 6. Under the hypotheses of Theorem 5, let be an entire function with such that . If and are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies 2. Auxiliary Lemmas Lemma 7 (see [7, 8]). Let be finite-order meromorphic functions. If is a meromorphic solution of the equation with and , then satisfies Here, we give a special case of the result due to Cao et al. in [9]. Lemma 8. Let be finite-order meromorphic functions. If is a meromorphic solution of (14) with then 3. Proofs of the Theorems Proof of Theorem 3. Suppose that and are two linearly independent solutions of (4). Then by Theorem A, we have Suppose that , where is a complex number. Then, by (5) we obtain Since is a solution of (4) and , then we have Suppose now that where is a complex number. Differentiating both sides of (5), we obtain Differentiating both sides of (21), we obtain Substituting into (22), we obtain Differentiating both sides of (23) and by substituting , we obtain By (5), (21), (23), and (24) we have To solve this system of equations, we need first to prove that . By simple calculations we obtain To show that , we suppose that Dividing both sides of (27) by , we obtain equivalent to whi (...truncated)


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Zinelaâbidine Latreuch, Benharrat Belaïdi. Some Properties of Solutions of Second-Order Linear Differential Equations, Journal of Complex Analysis, 2013, 2013, DOI: 10.1155/2013/253168