Abstract:
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

Abstract:
We give an alternative and simpler method for getting pointwise estimate of meromorphic solutions of homogeneous linear differential equations with coefficients meromorphic in a finite disk or in the open plane originally obtained by Hayman and the author. In particular, our estimates generally give better upper bounds for higher order derivatives of the meromorphic solutions under consideration, are valid, however, outside an exceptional set of finite logarithmic density. The estimates again show that the growth of meromorphic solutions with a positive deficiency at infinity can be estimated in terms of initial conditions of the solution at or near the origin and the characteristic functions of the coefficients.

Abstract:
This paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.

Abstract:
The purpose of this investigation is first to reveal some relations between certain complex (differential) equations and nonnormalized meromorphic functions and then to point some of their useful consequences out. 1. Introduction, Notations, Definitions, and Motivation As is known, a differential equation is an equation that involves the derivatives of a function as well as the function itself. If partial derivatives are involved, the equation is called a partial differential equation; if only ordinary derivatives are present, the equation is called an ordinary differential equation (ODE). Differential equations play an extremely important and useful role in applied math, engineering, and physics, and more mathematical and numerical machineries have been developed for the solution to differential equations. As we also know, ODE is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real- or complex-valued function, but more generally, it may be vector-valued or matrix-valued; this corresponds to considering a system of ordinary differential equations for a single function. Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. There are many methods to compute numerical solutions to differential equations and studies on the properties of differential equations involving approximation theory of the solution to a differential equation by the solution to a corresponding differential equation. In this work, as a novel investigation, we want to focus on certain types of (linear or nonlinear) first-order complex differential equation. Especially, for functions and being analytic in certain domains of complex domain, we want to take cognizance of a few types of certain complex equations including several first-order complex differential equations like and to reveal some of their useful implications between certain complex (differential) equations and nonnormalized multivalent functions which are analytic in the punctured unit disk and

Abstract:
Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational, periodic, elliptic) are found for a wide class of autonomous nonlinear ordinary differential equations.

Abstract:
Thirty research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.

Abstract:
In this paper, we study the problem of meromorphic functions that share one small function of differential polynomial with their derivatives and prove one theorem. The theorem improves the results of Jin-Dong Li and Guang-Xin Huang [1].

Abstract:
In this paper, we study the complex oscillation of solutions and their derivatives of the differential equation F’’+ A(z)f’+ B (z)f = F (z), where A(z),B (z) and F (z) are meromorphic functions of finite iterated p-order in the unit disc = {z : |z| < 1}.

Abstract:
In this paper, we consider the behaviour, when $q$ goes to $1$, of the set of a convenient basis of meromorphic solutions of a family of linear $q$-difference equations. In particular, we show that, under convenient assumptions, such basis of meromorphic solutions converges, when $q$ goes to $1$, to a basis of meromorphic solutions of a linear differential equation. We also explain that given a linear differential equation of order at least two, which has a Newton polygon that has only slopes of multiplicities one, and a basis of meromorphic solutions, we may build a family of linear $q$-difference equations that discretizes the linear differential equation, such that a convenient family of basis of meromorphic solutions is a $q$-deformation of the given basis of meromorphic solutions of the linear differential equation.

Abstract:
We investigate the growth of solutions of higher-order nonhomogeneous linear differential equations with meromorphic coefficients. We also discuss the relationship between small functions and solutions of such equations.