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

TL;DR

This paper extends previous results on harmonic convolutions by applying Cohn's rule to analyze the univalence and convexity of convolutions with more general dilatations, broadening the class of mappings studied.

Contribution

It generalizes existing theorems on harmonic convolutions by incorporating a wider range of dilatations using Cohn's rule, enhancing understanding of their geometric properties.

Findings

Confirmed univalence and convexity for new dilatation classes

Extended the class of harmonic mappings with proven properties

Provided a new analytical approach using Cohn's rule

Abstract

Dorff et al. [4], proved that the harmonic convolution of right half-plane mapping with dilatation -z and mapping f_\beta = h_\beta + \bar{g}_\beta, where f_\beta is obtained by shearing of analytic vertical strip mapping, with dilatation e^{i\theta}z^n; n = 1,2,\theta \in R, is in S_H^0 and is convex in the direction of the real axis. In this paper, by using Cohn's rule, we generalize this result by considering dilatations (a-z)/(1-az), a\in (-1,1) and e^{i\theta} z^n (n\in N;\theta\in R) of right half-plane mapping and f_\beta, respectively.