scholarly journals On Exceptional Values of Meromorphic Functions with the Set of Singularities of Capacity Zero

1961 ◽  
Vol 18 ◽  
pp. 171-191 ◽  
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Essential Singularity ◽  
Compact Set ◽  
Capacity Zero ◽  
A Value

LetEbe a compact set in thez-plane and letΩbe its complement with respect to the extendedz-plane. Suppose thatEis of capacity zero. ThenΩis a domain and we shall consider a single-valued meromorphic functionw=f(z) onΩwhich has an essential singularity at each point ofE. We shall say that a valuewis exceptional forf(z)at a point ζ ∈Eif there exists a neighborhood of C where the functionf(z)does not take this valuew.

scholarly journals Some Note on Exceptional Values of Meromorphic Functions

1963 ◽  
Vol 22 ◽  
pp. 189-201 ◽  
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Essential Singularity ◽  
Compact Set ◽  
A Value ◽  
Cluster Set ◽  
Totally Disconnected ◽  
Null Set

LetEbe a totally-disconnected compact set in thez-plane and letΩbe its complement with respect to the extendedz-plane. ThenΩis a domain and we can consider a single-valued meromorphic functionw = f(z)onΩwhich has a transcendental singularity at each point ofE. Suppose thatEis a null-set of the classWin the sense of Kametani [4] (the classNBin the sense of Ahlfors and Beurling [1]). Then the cluster set off(z)at each transcendental singularity is the wholew-plane, and hencef(z)has an essential singularity at each point ofE. We shall say that a valuewis exceptional forf(z)at an essential singularity ζ ∈Eif there exists a neighborhood of ζ where the functionf(z)does not take this valuew.

scholarly journals Exceptionally ramified meromorphic functions with a non-enumerable set of essential singularities

1982 ◽  
Vol 88 ◽  
pp. 133-154 ◽  
Author(s):  
Toshiko Kurokawa
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Function Theory ◽  
Complex Function ◽  
Essential Singularity ◽  
Complex Function Theory

In the complex function theory, Picard’s Great Theorem plays an essential and important role. It is well-known as generalizations of this theorem that in a neighborhood of an isolated essential singularity, a meromorphic function cannot be exceptionally ramified (see W. Gross [2]) and that even it cannot be normal (see O. Lehto and K. I. Virtanen [7]). We are therefore interested in the behaviour of meromorphic functions with non-isolated essential singularities as well as in generalizations of the Gross’ result. Several approaches in this direction have been made by G. af Hällström [3], S. Kametani [4], K. Noshiro [13], K. Matsumoto [8], [9], [10], [11], [12], S. Toppila [15], etc..

scholarly journals A note on power of meromorphic function and its shift operator of certain hyper-order sharing one small function and a value

2021 ◽  
Vol 55 (1) ◽  
pp. 57-63
Author(s):  
A. Banerjee ◽  
A. Roy
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Shift Operator ◽  
Small Function ◽  
A Value ◽  
Hyper Order

In this article, we obtain two results on $n$ the power of a meromorphic function and its shift operator sharing a small function together with a value which improve and complement some earlier results. In particular, more or less we have improved and extended two results of Qi-Yang [Meromorphic functions that share values with their shifts or their $n$-th order differences, Analysis Math., 46(4)2020, 843-865] by dispelling the superfluous conclusions in them.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

scholarly journals On Properties of Differences Polynomials about Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Jianming Qi ◽  
Jie Ding ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Special Class ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Class Difference ◽  
Difference Polynomial

We study the value distribution of a special class difference polynomial about finite order meromorphic function. Our methods of the proof are also different from ones in the previous results by Chen (2011), Liu and Laine (2010), and Liu and Yang (2009).

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals A Criterion for Normalcy

1968 ◽  
Vol 32 ◽  
pp. 277-282 ◽  
Author(s):  
Paul Gauthier
Keyword(s):  
Meromorphic Function ◽  
Holomorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions

Gavrilov [2] has shown that a holomorphic function f(z) in the unit disc |z|<1 is normal, in the sense of Lehto and Virtanen [5, p. 86], if and only if f(z) does not possess a sequence of ρ-points in the sense of Lange [4]. Gavrilov has also obtained an analagous result for meromorphic functions by introducing the property that a meromorphic function in the unit disc have a sequence of P-points. He has shown that a meromorphic function in the unit disc is normal if and only if it does not possess a sequence of P-points.

scholarly journals Growth of Meromorphic Function Sharing Functions and Some Uniqueness Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Jianming Qi ◽  
Fanning Meng ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Meromorphic Functions ◽  
Meromorphic Solutions ◽  
Complex Differential Equations ◽  
Function Sharing

Estimating the growth of meromorphic solutions has been an important topic of research in complex differential equations. In this paper, we devoted to considering uniqueness problems by estimating the growth of meromorphic functions. Further, some examples are given to show that the conclusions are meaningful.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

scholarly journals Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane

2020 ◽  
Vol 54 (2) ◽  
pp. 172-187
Author(s):  
I.E. Chyzhykov ◽  
A.Z. Mokhon'ko
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Sharp Estimates ◽  
Exceptional Sets ◽  
Upper Half Plane ◽  
Logarithmic Derivatives

We established new sharp estimates outside exceptional sets for of the logarithmic derivatives $\frac{d^ {k} \log f(z)}{dz^k}$ and its generalization $\frac{f^{(k)}(z)}{f^{(j)}(z)}$, where $f$ is a meromorphic function $f$ in the upper half-plane, $k>j\ge0$ are integers. These estimates improve known estimates due to the second author in the class of meromorphic functions of finite order.Examples show that size of exceptional sets are best possible in some sense.

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