One generalization of the class of helical functions
type: Item
instance of: scholarly article
title: One generalization of the class of helical functions
zbMATH Open document ID: 1246.30031
DOI: 10.3103/S1066369X11100070
author: M. A. Sevodin
published in: Russian Mathematics
publication date: 2011-11-21T00:00:00Z
review text: The paper under review offers a highly technical generalization of a certain class of schlicht functions that were intensely studied during the twentieth century, namely analytic, univalent and normalized functions on the unit disc of the complex plane. For a fairly general introduction to and overview of univalent functions and their properties see [Univalent functions. New York etc.: Springer (1983; Zbl 0514.30001)] by \textit{P. L. Duren} and [Univalent Functions. Vol. I, II. Tampa, FL: Mariner Publishing Co. (1983; Zbl 1041.30500; Zbl 1041.30501)] by \textit{A. W. Goodman}. In this paper, the boundaries of the domains onto which the considered analytic functions map the unit disc are one-parameter families of curves which happen to be generalizations of the logarithmic spirals studied by \textit{F. W. Gehring} [``Spirals and the universal Teichmüller space'', Acta Math. 141, 99--113 (1978; Zbl 0393.30015)]. The proof of the univalence of the studied families relies on arguments similar to those in [\textit{B. N. Rakhmanov}, Dokl. Akad. Nauk SSSR, n. Ser. 97, 973--976 (1954; Zbl 0056.30002)]. One of the enumerated classes of univalent functions thus obtained coincides with the well-known family of starlike functions [\textit{D. J. Hallenbeck} and \textit{T. H. MacGregor}, Linear problems and convexity techniques in geometric function theory. Boston etc.: Pitman Adavanced Publishing Program (1984; Zbl 0581.30001)]. Another class coincides with the family of functions studied in [Rakhmanov, loc. cit.]. Yet another is a subclass of the family of functions that map the unit circle onto a quasiconformal curve [\textit{L. V. Ahlfors}, Lectures on quasiconformal mappings. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1103.30001); \textit{M. Fait, J. G. Krzyz} and \textit{J. Zygmunt}, ``Explicit quasiconformal extensions for some classes of univalent functions'', Comment. Math. Helv. 51, 279--285 (1976; Zbl 0332.30010)]. The reader should be warned that …
reviewed by: K. T. Hallenbeck
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 5975568
zbMATH Keywords: schlicht functions
zbMATH Keywords: univalent functions
zbMATH Keywords: quasiconformal mappings
zbMATH Keywords: logarithmic spirals
zbMATH Keywords: rectifiable quasiconformal boundaries
MaRDI profile type: Publication
cites work: Q4742751
cites work: Spirals and the universal Teichmüller space
cites work: Q5527180
cites work: Explicit quasiconformal extensions for some classes of univalent functions
cites work: Q5512337
cites work: Q3310107
cites work: Q4750947
full work available at URL: s1066369x11100070
OpenAlex ID: W2008633775
description: scientific article; zbMATH DE number 5975568