Convolution properties of some harmonic mappings in the right-half plane

TL;DR

This paper generalizes previous results on the convolution of harmonic right-half plane mappings, establishing convexity in the direction of the real axis under broader conditions.

Contribution

It extends earlier findings by removing certain restrictions, broadening the class of harmonic mappings for which convolution convexity holds.

Findings

Convolution of harmonic right-half plane mappings is convex in the real axis direction.

The conditions for local univalence and sense-preservation can be relaxed.

The main theorem generalizes previous specific cases.

Abstract

Dorff, proved in [2] that the convolution of two harmonic right-half plane mappings is convex in the direction of real axis provided that the convolution is locally univalent and sense preserving. Later, it was shown in [3] that the condition of locally univalent and sense preserving can be dropped in some special cases. In this paper, we generalize the main result from [3].