Mappings with rotation symmetry
type: Item
instance of: scholarly article
title: Mappings with rotation symmetry
zbMATH Open document ID: 0963.30007
author: I. A. Aleksandrov
author: Gulnora Dolimdganovna Sadritdinova
published in: Russian Mathematics
publication date: 2001-06-28T00:00:00Z
review text: Let \(S_p(M)\), \(p\in\mathbb{N}\), \(M> 1\), be the class of functions \(f(z)= z+\cdots\), holomorphic in the unit disc \(E_1\), where \(E_r= \{z:|z|< r\}\), which univalently map \(E_1\) to domains from the disc \(E_M\) and have \(p\)-multiple rotation symmetry with respect to the point \(0\). It is known, that the set \(\widetilde S_p(M)\) of functions of the form \(W= M\zeta(z,\ln M)\), where \(\zeta(z,\tau)\) \((0\leq \tau\leq\ln M)\) is a solution of the Löwner equation [see \textit{I. A. Aleksandrov}, Parametric continuations in the theory of schlicht functions, Nauka, Moscow (1976; Zbl 0462.30008), p. 27] \({d\zeta\over d\tau}= -\zeta{\mu^p(\tau)+ \zeta^p\over \mu^p(\tau)- \zeta^p}\), \(\zeta(z, 0)= z\in E_1\), \(\mu(\tau)\) -- a continuous function with the module equal to \(1\), is a subclass dense in \(S_p(M)\) (in the topology of uniform convergence on compact subsets of \(E_1\)). In this paper the authors give a form of the control function \(\mu\), such that the Löwner equation leads to a formula for \(\zeta(z, \tau)\), which is more general than the hitherto known one.
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1542181
zbMATH Keywords: univalent bounded holomorphic functions
zbMATH Keywords: Löwner equation
zbMATH Keywords: \(p\)-symmetric univalent functions
reviewed by: Zbigniew Jerzy Jakubowski
MaRDI profile type: Publication
Recommended article: A discrete limit theorem for the Matsumoto zeta-function on the complex plane
Recommended article: Q5043176
Recommended article: Q3137672
Recommended article: Construction of a universal Laurent series
Recommended article: Sur l'équation fonctionnelle de Poinsot généralisée.
Recommended article: Sopra una certa serie di Laurent di due variabili.
Recommended article: Q3607537
Recommended article: Q4261816
Recommended article: Q3982760
Recommended article: Radii of starlikeness of certain classes of regular functions
description: scientific article; zbMATH DE number 1542181