Some Remarks on Angular Derivatives and Julia's Lemma

1966 ◽  
Vol 9 (2) ◽  
pp. 233-241 ◽  
Author(s):  
H. L. Jackson
Keyword(s):  
Unit Disk ◽  
Angular Derivative ◽  
Angular Limit ◽  
Angular Derivatives

Let w = f(z) be holomorphic on the unit disk D = { z: | z | < 1}, with the additional restrictions that | f ( z ) | < l and , where denotes the (outer) angular limit of f (z) at z = 1. Let us now define and then focus our attention on the behaviour of g(z) in an arbitrary angular neighbourhood of z = 1. Whenever exists, this limit is commonly referred to as the angular derivative of f(z) at z = 1.

scholarly journals Analytic flows on the unit disk: angular derivatives and boundary fixed points

2005 ◽  
Vol 222 (2) ◽  
pp. 253-286 ◽  
Author(s):  
Manuel D. Contreras ◽  
Santiago Díaz-Madrigal
Keyword(s):  
Fixed Points ◽  
Unit Disk ◽  
Angular Derivatives ◽  
Boundary Fixed Points

Angular Derivatives and Compact Composition Operators on the Hardy and Bergman Spaces

1986 ◽  
Vol 38 (4) ◽  
pp. 878-906 ◽  
Author(s):  
Barbara D. MacCluer ◽  
Joel H. Shapiro
Keyword(s):  
Holomorphic Function ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Open Unit ◽  
Angular Derivative ◽  
Restricted Class ◽  
Open Unit Disc ◽  
Angular Derivatives ◽  
Hardy And Bergman Spaces

Let U denote the open unit disc of the complex plane, and φ a holomorphic function taking U into itself. In this paper we study the linear composition operator Cφ defined by Cφf = f º φ for f holomorphic on U. Our goal is to determine, in terms of geometric properties of φ, when Cφ is a compact operator on the Hardy and Bergman spaces of φ. For Bergman spaces we solve the problem completely in terms of the angular derivative of φ, and for a slightly restricted class of φ (which includes the univalent ones) we obtain the same solution for the Hardy spaces Hp (0 < p < ∞). We are able to use these results to provide interesting new examples and to give unified explanations of some previously discovered phenomena.

scholarly journals On boundary critical points for semigroups of analytic functions

2006 ◽  
Vol 98 (1) ◽  
pp. 125 ◽  
Author(s):  
M. D. Contreras ◽  
S. Díaz-Madrigal ◽  
Ch. Pommerenke
Keyword(s):  
Fixed Points ◽  
Unit Disk ◽  
Critical Points ◽  
Analytic Functions ◽  
Infinitesimal Generators ◽  
Contact Points ◽  
Angular Derivatives ◽  
Boundary Fixed Points ◽  
The Relationship ◽  
New Inequalities

We analyze the relationship between boundary fixed points of semigroups of analytic functions and boundary critical points of their infinitesimal generators. As a consequence, we show two new inequalities for analytic self-maps of the unit disk. The first one is about angular derivatives at fixed points of functions belonging to semigroups of analytic functions. The second one deals with angular derivatives at contact points of arbitrary analytic functions from the unit disk into itself.

scholarly journals Digons and angular derivatives of analytic self-maps of the unit disk

2007 ◽  
Vol 52 (8) ◽  
pp. 685-691 ◽  
Author(s):  
Manuel D. Contreras ◽  
Santiago Díaz-Madrigal ◽  
Alexander Vasil'ev
Keyword(s):  
Unit Disk ◽  
Angular Derivatives ◽  
Derivatives Of

scholarly journals Second angular derivatives and parabolic iteration in the unit disk

2009 ◽  
Vol 362 (01) ◽  
pp. 357-388 ◽  
Author(s):  
Manuel D. Contreras ◽  
Santiago Díaz-Madrigal ◽  
Christian Pommerenke
Keyword(s):  
Unit Disk ◽  
Angular Derivatives

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

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