Value Distribution of Meromorphic Functions

Abstract

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, SpainPrinted on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This book is devoted to the study of value distribution of functions which are meromorphic on the complex plane or in an angular domain with vertex at the origin. We characterize such meromorphic functions in terms of distribution of some of their value points. The study, together with certain related topics, is known as theory of value distribution of meromorphic functions. The theory is too vast to be justified within a single work. Therefore we selected and organized the content based on their significant importance to our understanding and interests in this book. I gladly acknowledge my indebtedness in particular to the books of M. Tsuji, A. A. Goldberg and I. V. Ostrovskii, Yang L. and the papers of A. Eremenko.An outline of the book is provided below. The introduction of the Nevanlinna characteristic to the study of meromorphic functions is a new starting symbol of the theory of value distribution. The Nevanlinna characteristic is powerful, and its thought has been used to produce various characteristics such as the Nevanlinna characteristic and Tsuji characteristic for an angular domain. And from geometric point of view, namely the Ahlfors theory of covering surfaces, the Ahlfors-Shimizu characteristic have also been introduced. These characteristics are real-valued functions defined on the positive real axis. Therefore, in the first chapter, we collect the basic results about positive real functions that are often used in the study of meromorphic function theory. Some of these results are distributed in other books, some in published papers, and some have been newly established in order to serve our specific objectives in the book.In the present book, we discuss value distribution not only in the complex plane, but also in an angular domain. Therefore, we introduce, in the second chapter, various characteristics of a meromorphic function: The Nevanlinna characteristic for a disk, the Nevanlinna characteristic for an angle, the Tsuji characteristic and the Ahlfors-Shimizu characteristic for an angle. Although they were distributed in another books, we collected all of them, and more importantly, we carefully compared them with one another to reveal their relations that enabled us to produce new results and applications. We establish the first and second fundamental theorems for the various characteristics and the corresponding integrated counting functions, and provide an estimate of the error term related to the Nevanlinna characteristic for an ii Preface angle in terms of the Nevanlinna characteristic in a larger angle. We discuss in an angle the growth order of a meromorphic func...

Value distributions of solutions to complex linear differential equations in angular domains

“…In this article, we assume the reader is familiar with standard notations and basic results of Nevanlinna's value distribution theory in the unit disk D fz W jzj < 1g, in an angular region, and in the complex plane C respectively; see [1][2][3][4][5]. The order .f / and lower order .f / of f which is meromorphic in C or are defined as follows: …”

Nevanlinna’s Five Values Theorem on Annuli

“…[17]). Let f and g be two nonconstant meromorphic functions in an angular domain Ω(α, β) = {z : α < arg z < β}(0 < β − α < 2π), and:…”

Radial Oscillation of Linear Differential Equation

“…In this paper, the meromorphic function always means a function being meromorphic in the whole complex plane C. Assume that the basic definitions, theorems and standard notations of the Nevanlinna theory for meromorphic function (see [11], [22] or [24]) are known. There have appeared many papers on the global theory of complex differential equations which were studied from the point of view of Nevanlinna theory, since 1982 when the article by Bank and Laine [1] appeared in Trans.…”

“…When k ≥ 2 and p = 1, the equivalence of (ii) and (iii) has been obtained by Huang and Chen [8] in an weakly form. Most recently, Zheng [24] give a system research about the equivalence of (i) and (ii) by using the proximate order of Qinglai Hiong (see [5]). Here, the innovative point is the equivalence of (ii) and (iii).…”