Typically real harmonic functions
type: Item
instance of: scholarly article
title: Typically real harmonic functions
zbMATH Open document ID: 1250.30009
DOI: 10.1216/RMJ-2012-42-2-567
author: M. Dorff
author: M. Nowak
author: Wojciech Szapiel
published in: Rocky Mountain Journal of Mathematics
publication date: 2012-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0903.1600
full work available at URL: https://projecteuclid.org/euclid.rmjm/1335187166
reviewed by: V. J. Ja'kl
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 6035451
zbMATH Keywords: harmonic mappings
zbMATH Keywords: typically real
zbMATH Keywords: univalence
MaRDI profile type: Publication
arXiv ID: 0903.1600
cites work: Q5593095
cites work: Harmonic typically real mappings
cites work: Harmonic univalent functions
cites work: Q4828548
cites work: The Argument Principle for Harmonic Functions
cites work: The Domain Covered by a Typically-Real Function
cites work: Q4220703
cites work: Extremal Problems for the Typically Real Functions
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cites work: On the non-vanishing of the Jacobian in certain one-to-one mappings
cites work: On Typically-Real Functions in a Cut Plane
cites work: The Radii of Starlikeness and Convexity of Certain Nevanlinna Analytic Functions
cites work: Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen
cites work: Q3231381
cites work: Univalence of Continued Fractions and Stieltjes Transforms
cites work: The radii of starlikeness and convexity of order alpha of typically real functions
cites work: On harmonic typically real mappings
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review text: The authors modify the Clunie-Sheil-Small definition of typically real harmonic mappings on the unit disk \(\mathbb{D}\) in such a way that functions \(f(z)= h(z)+ \overline{g(z)}\) in their class, denoted by \({\mathcal T}_H\), may not be locally univalent and, moreover, all typically real analytic \(f(z)\) are in \({\mathcal T}_H\). Denote by \({\mathcal T}^0_H\) the subclass of \({\mathcal T}_H\) of all those \(f\) for which \(g'(0)= 0\). It is proved that functions in \({\mathcal T}_H\) are strictly increasing on the interval \((-1,1)\) and that \({\mathcal T}^0_H\) is compact, and its extreme points are determined. Next, the authors concentrate on the univalence problem for \({\mathcal T}^0_H\) and show the following:NEWLINENEWLINE (1) for each \(f\) there exists an open set \(V\) with \((-1,1)\subset V\subset \mathbb{D}\) on which \(f\) is univalent;NEWLINENEWLINE (2) each \(f\in{\mathcal T}^0_H\) is univalent on the disk \(D(0,\sqrt{6}-\sqrt{5})\) and on the domain \(\{z\in \mathbb{D}:|2z(1+ z^2)^{-1}|< \sqrt{2}- 1\}\).NEWLINENEWLINE The paper is concluded with conjectures and open problems.
description: scientific article; zbMATH DE number 6035451