Two-point distortion theorems for univalent functions
type: Item
instance of: scholarly article
title: Two-point distortion theorems for univalent functions
zbMATH Open document ID: 0788.30009
author: D. Minda
author: Seong-A. Kim
published in: Pacific Journal of Mathematics
publication date: 1994-04-26T00:00:00Z
review text: The authors establish a one-parameter family of symmetric, linearly invariant two-point distortion theorems for univalent functions defined on the unit disk. The weakest theorem in the family is a symmetric, linearly invariant form of a classical distortion theorem of Koebe, while another special case is a distortion theorem of \textit{Ch. Blatter} [Comment. Math. Helv. 53, 651-659 (1978; Zbl 0398.30004)]. All of these distortion theorems are necessary and sufficient for univalence. Each of these distortion theorems can be expressed as a two-point comparison theorem between euclidean and hyperbolic geometry on a simply connected region; however, none of these comparison theorems characterize simply connected regions. The authors obtain analogous results for convex univalent functions and convex regions, except that in this context the two-point comparison theorems do characterize convex regions. These results are all proved by an extension of Blatter's method which involves an interplay between differential geometry and univalent function theory.
reviewed by: D. Minda
Mathematics Subject Classification ID: 30C55
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 537377
zbMATH Keywords: convex univalent functions
zbMATH Keywords: distortion theorems
zbMATH Keywords: hyperbolic geometry
MaRDI profile type: Publication
full work available at URL: pjm.1994.163.137
OpenAlex ID: W2078391458
DOI: 10.2140/PJM.1994.163.137
description: scientific article; zbMATH DE number 537377