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The various characteristics of meromorphic functions are main tool in the study of value distribution of meromorphic functions this book will introduce. They are real-valued functions defined on the positive real axis. In this chapter, we discuss certain properties of such real functions for application in later chapters. We begin with the order and the lower order of such functions which include...
We characterize meromorphic functions in terms of points at which they assume some values. The purpose is realized by using their characteristics. In this chapter, we introduce the Nevanlinna’s characteristic in a domain (especially in a disk centered at the origin), the Nevanlinna’s characteristic in an angle and the Tsuji’s characteristic in terms of the generalized Poisson formula, Carleman formula...
A transcendental meromorphic function has a singular property in any neighborhood of its essential singular point, for example, it assumes there infinitely often all but at most two values on the extended complex plane. This property is preserved in any angular domain containing some fixed ray. Such ray is termed as the singular direction of the function considered. In this chapter, we mainly discuss...
We investigate above bound of total number of deficient values of a transcendental meromorphic function and its derivatives of every order if most of its zeros and poles distribute along finitely many rays starting from the origin and prove that the bound is the number of the rays under some assumption, for example, the function considered is of finite lower order. Next we discuss relations between...
A value on the extended complex plane is a radially distributed value of a transcendental meromorphic function if most of points at which the value is assumed distribute closely along a finite number of rays from the origin. In this chapter, we study the growth order of a meromorphic function with two radially distributed values and a distinct deficient value (in other words, this hints a condition...
This chapter is devoted to discussing singular values of a transcendental meromorphic function. The singular value is that in any neighborhood of which the inverse of the function contains a multiple-valued branch. A value is a singular value if and only if it is an asymptotic value or a critical value. We show the construction of the parabolic simply connected Riemann surface associated with a fixed...
This chapter is mainly devoted to introducing the proof of the Nevanlinna’s conjecture which Eremenko provided in terms of the potential theory. This conjecture proposed by F. Nevanlinna in 1929 had been at an important and special position in the Nevanlinna’s value distribution theory. It was proved first by D. Drasin in 1987, but the Drasin’s proof is very complicated. In our attempt to help readers...
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