Typically real harmonic functions
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.
scientific article; zbMATH DE number 6035451
scientific article