Some Properties of Solutions of Second-Order Linear Differential Equations

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