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.

scholarly journals On Some Properties of Normal Meromorphic Functions in the Unit Disc

1968 ◽  
Vol 33 ◽  
pp. 153-164 ◽  
Author(s):  
Toshiko Zinno
Keyword(s):  
Meromorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Conformal Mappings ◽  
The Family ◽  
One To One

We denote by D the unit disc {z; |z| < 1} and by the totality of one to one conformal mappings z′ = s(z) of D onto itself. A meromorphic function f(z) in D is normal if and only if the family is a normal family in D in the sense of Montel. We denote by the totality of the normal meromorphic functions in D.

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 Fixed Points of Difference Operator of Meromorphic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Zhaojun Wu ◽  
Hongyan Xu
Keyword(s):  
Meromorphic Function ◽  
Fixed Points ◽  
Meromorphic Functions ◽  
Difference Operator ◽  
Transcendental Meromorphic Function ◽  
Exact Difference ◽  
Borel Exceptional Values ◽  
Nevanlinna Deficiency

Letfbe a transcendental meromorphic function of order less than one. The authors prove that the exact differenceΔf=fz+1-fzhas infinitely many fixed points, ifa∈ℂand∞are Borel exceptional values (or Nevanlinna deficiency values) off. These results extend the related results obtained by Chen and Shon.

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.

scholarly journals Stratified, Weak Stratified, and Three-Valued Semantics1

1990 ◽  
Vol 13 (1) ◽  
pp. 19-33
Author(s):  
Melvin Fitting ◽  
Marion Ben-Jacob
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Horn Clause ◽  
Logic Programs ◽  
The Family ◽  
Fixed Point Semantics ◽  
Truth Value ◽  
The Relationship ◽  
Classical Truth

We investigate the relationship between three-valued Kripke/Kleene semantics and stratified semantics for stratifiable logic programs. We first show these are compatible, in the sense that if the three-valued semantics assigns a classical truth value, the stratified approach will assign the same value. Next, the familiar fixed point semantics for pure Horn clause programs gives both smallest and biggest fixed points fundamental roles. We show how to extend this idea to the family of stratifiable logic programs, producing a semantics we call weak stratified. Finally, we show weak stratified semantics coincides exactly with the three-valued approach on stratifiable programs, though the three-valued version is generally applicable, and does not require stratification assumptions.

Exceptional functions and normal families of holomorphic functions with multiple zeros

2011 ◽  
Vol 18 (1) ◽  
pp. 31-38
Author(s):  
Jun-Fan Chen
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Holomorphic Functions ◽  
Normal Families ◽  
Multiple Zeros ◽  
Zeros Of Functions

Abstract Let k be a positive integer, and let ℱ be a family of functions holomorphic on a domain D in C, all of whose zeros are of multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each ƒ ∈ ℱ, ƒ(k)(z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D. The condition that the zeros of functions in ℱ are of multiplicity at least k + 1 cannot be weakened, and the corresponding result for families of meromorphic functions is no longer true.

scholarly journals Fixed points of holomorphic mappings for domains in Banach spaces

2003 ◽  
Vol 2003 (5) ◽  
pp. 261-274 ◽  
Author(s):  
Lawrence A. Harris
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Holomorphic Function ◽  
Fixed Points ◽  
Banach Spaces ◽  
Point Theorem ◽  
Numerical Range ◽  
Holomorphic Functions ◽  
Holomorphic Mappings

We discuss the Earle-Hamilton fixed-point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer-Phillips theorem estimating resolvents to dissipative holomorphic functions.

A Quantitative Estimate on Fixed-Points of Composite Meromorphic Functions

1995 ◽  
Vol 38 (4) ◽  
pp. 490-495 ◽  
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Fixed Points ◽  
Lower Order ◽  
Quantitative Estimate ◽  
Meromorphic Functions ◽  
Transcendental Entire Function ◽  
Logarithmic Density ◽  
Image Position ◽  
Finite Lower Order

AbstractLet ƒ(z) be a transcendental meromorphic function of finite order, g(z) a transcendental entire function of finite lower order and let α(z) be a non-constant meromorphic function with T(r, α) = S(r,g). As an extension of the main result of [7], we prove thatwhere J has a positive lower logarithmic density.

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