Rotations and convolutions of harmonic convex mappings

TL;DR

This paper explores the properties of harmonic convex mappings, focusing on their rotations, convolutions, and convex combinations, extending existing results to more general settings.

Contribution

It generalizes convolution results for harmonic mappings and proves convexity preservation under convex combinations in a broader class of mappings.

Findings

Convolutions of slanted half-plane and strip mappings are generalized.

Convex combinations of certain harmonic mappings are shown to be convex.

Results extend known properties to more general harmonic mapping classes.

Abstract

In this paper, we consider the convolutions of slanted half-plane mappings and strip mappings and generalize related results in general settings. We also consider a class of harmonic mappings containing slanted half-plane mappings and strip mappings and, as a consequence, we prove that the convex combination of such mappings is convex.