An application of Cohn's rule to convolutions of univalent harmonic mappings

Abstract

Dorff et al.~\cite {do and no} proved that the harmonic convolutions of the standard right half-plane mapping $F_0=H_0+\overline {G}_0$ (where $H_0+G_0=z/(1-z)$ and $G_0'=-zH_0'$) and mappings $f_\beta =h_\beta +\overline {g}_{\beta }$ (where $f_\beta $ are obtained by shearing of analytic vertical strip mappings with dilatation $e^{i\theta }z^n$, $n=1,2$, $\theta \in \mathbb {R}$) are in $S_H^0$ and are convex in the direction of the real axis. In this paper, by using Cohn's rule, we generalize this result by replacing the standard right half-plane mapping $F_0$ with a family of right half-plane mappings $F_a=H_a+\overline {G}_a$ (with $H_a+G_a=z/(1-z)$ and $G'_a/H'_a= {(a-z)}/{(1-az)}$, $a\in (-1,1)$) and including the cases $n=3$ and $n=4$ (in addition to $n=1$ and $n=2$) for dilatations of $f_\beta $.