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

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, Bela？di, 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