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

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.