Convolutions of harmonic convex mappings

TL;DR

This paper investigates the conditions under which harmonic convolutions of certain convex mappings are univalent, providing counterexamples and specific cases where univalence and convexity are preserved.

Contribution

It demonstrates that local univalence cannot be assumed in general for harmonic convolutions, and identifies cases where univalence holds, including examples mapping onto the plane with slits.

Findings

Local univalence is not guaranteed without assumptions.

Some harmonic convolutions are locally univalent under specific conditions.

Constructed examples include mappings onto the plane with two parallel slits.

Abstract

The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right half-plane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show that in some cases these harmonic convolutions are locally univalent. Using this we obtain interesting examples of univalent harmonic maps one of which is a map onto the plane with two parallel slits.