scholarly journals The zeros of f^{n}f^(k)-a and normal families of meromorphic functions

2020 ◽  
Vol 51 (2) ◽  
pp. 137-144
Author(s):  
Sun Xiong
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Related Result ◽  
Normal Families

In this paper, we first prove that if f be a non-constant meromorphic function, all of whose zeros have multiplicity at least $k$, then f^{n}f^{(k)}-a has at least m+1 distinct zeros, where $k(\geq2),m(\geq1),n(\geq m+1)$ are three integers, and $a\in \mathbb{C}\cup\setminus\{0\}$.Also, in relation to this result, a normality criteria is given, which extends the related result.

Fixed points of composite meromorphic functions and normal families

2004 ◽  
Vol 134 (4) ◽  
pp. 653-660 ◽  
Author(s):  
Walter Bergweiler
Keyword(s):  
Fixed Point ◽  
Meromorphic Function ◽  
Fixed Points ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Normal Families ◽  
The Family

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

Picard values and normal families of meromorphic functions

2009 ◽  
Vol 139 (5) ◽  
pp. 1091-1099 ◽  
Author(s):  
Yan Xu ◽  
Fengqin Wu ◽  
Liangwen Liao
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Complex Number ◽  
Meromorphic Functions ◽  
Normal Families ◽  
Finite Complex ◽  
Transcendental Meromorphic Function ◽  
Positive Integers ◽  
Normal Criterion

Let f be a transcendental meromorphic function on the complex plane ℂ, let a be a non-zero finite complex number and let n and k be two positive integers. In this paper, we prove that if n≥k+1, then $\smash{f+a(f^{(k)})^n}$ assumes each value b∈ℂ infinitely often. Also, the related normal criterion for families of meromorphic functions is given. Our results generalize the related results of Fang and Zalcman.

scholarly journals Nonexceptional functions and normal families of zero-free meromorphic functions

Filomat ◽  
2017 ◽  
Vol 31 (14) ◽  
pp. 4665-4671
Author(s):  
Jun-Fan Chen
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Normal Families

Let k be a positive integer, let F be a family of zero-free meromorphic functions in a domain D, all of whose poles are multiple, and let h be a meromorphic function in D, all of whose poles are simple, h . 0, ?. If for each f ? F, f(k)(z)- h(z) has at most k zeros in D, ignoring multiplicities, then F is normal in D. The examples are provided to show that the result is sharp.

Composite meromorphic functions and normal families

2009 ◽  
Vol 139 (1) ◽  
pp. 57-72 ◽  
Author(s):  
Jianming Chang ◽  
Mingliang Fang ◽  
Lawrence Zalcman
Keyword(s):  
Meromorphic Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Normal Families

We study the normality of families of meromorphic functions defined in terms of certain omitted functions. In particular, we prove the following results. Firstly, if $\mathcal{F}$ is a family of meromorphic functions in a domain D ⊂ ℂ, and a(z), b(z) and c(z) are distinct meromorphic functions in D and if, for all f ∈ $\mathcal{F}$ and all z ∈ D, f(z) ≠ a(z), f(z) ≠ b(z) and f(z) ≠ c(z), then $\mathcal{F}$ is normal in D. Secondly, letting R(w) be a rational function of degree greater than or equal to 3 and $\mathcal{F}$ be a family of functions meromorphic in a domain D ⊂ ℂ, if there exists a non-constant meromorphic function α(z) in D such that, for all f ∈ $\mathcal{F}$ and z ∈ D, R(f(z)) ≠ α(z), then $\mathcal{F}$ is normal in D.

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 .

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