Growth of Logarithmic Derivatives and Their Applications in Complex Differential Equations

We continue the study of the behavior of the growth of logarithmic derivatives. In fact, we prove some relations between the value distribution of solutions of linear differential equations and growth of their logarithmic derivatives. We also give an estimate of the growth of the quotient of two differential polynomials generated by solutions of the equation where and are entire functions. 1. Introduction and Main Results 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 [1–4]). In addition, we will denote by (resp., ) and (resp., ) the exponent of convergence of zeros (resp., distinct zeros) and poles (resp., distinct poles) of a meromorphic function , to denote the order of growth of . A meromorphic function is called a small function with respect to if as except possibly a set of of finite linear measure, where is the Nevanlinna characteristic function of . Definition 1 (see [4]). Let be a meromorphic function. Then the hyperorder of is defined as Definition 2 (see [1, 3]). The type of a meromorphic function of order is defined as If is an entire function, then the type of of order is defined as where is the maximum modulus function. Remark 3. For entire function, we can have . For example, if , then we have and . Growth of logarithmic derivative of meromorphic functions has been generously considered during the last decades, among others, by Gol'dberg and Grin？te？n [5], Benbourenane and Korhonen [6], Hinkkanen [7], and Heittokangas et al. [8]. All these considerations have been devoted to getting more detailed estimates to the proximity function than what is the original Nevanlinna estimate that essentially can be written as . The same applies for the case of higher logarithmic derivatives as well. The estimation of logarithmic derivatives plays the key role in theory of differential equations. In his paper Gundersen [9] proved some interesting inequalities on the module of logarithmic derivatives of meromorphic functions. Recently [10], the authors have studied some properties of the behavior of growth of logarithmic derivatives of entire and meromorphic functions and have obtained some relations between the zeros of entire functions and the growth of their logarithmic derivatives. In fact, they have proved the following. Theorem A (see [10]). Let be an entire function with finite number of zeros. Then, for any integer , Theorem B (see [10]). Suppose that is an integer and let be a meromorphic function. Then In [11, pages 457–458], Bank