Convolution of a harmonic mapping with $n$-starlike mappings and its partial sums

TL;DR

This paper studies the properties of convolutions involving harmonic mappings and $n$-starlike mappings, focusing on univalency, directional convexity, and partial sums, with new results on these geometric features.

Contribution

It introduces new conditions for univalency and convexity of convolutions with $n$-starlike mappings and analyzes partial sums of specific harmonic convolutions.

Findings

Convolution with $n$-starlike mappings preserves univalency under certain conditions.

Partial sums of convolutions with harmonic mappings exhibit directional convexity.

New criteria for convexity and starlikeness in harmonic convolution contexts.

Abstract

We investigate the univalency and the directional convexity of the convolution $\phi\tilde{*}f=\phi*h+\overline{\phi*g}$ of the harmonic mapping $f=h+\bar{g}$ with a mapping $\phi$ whose convolution with the mapping $z+\sum_{k=2}^{\infty}k^nz^k$ is starlike (and such a mapping $\phi$ is called $n$-starlike). In addition, we investigate the directional convexity of (i) the convolution of an analytic convex mapping with the slanted half-plane mapping, and (ii) the partial sums of the convolution of a $6$-starlike mapping with the harmonic Koebe mapping and the harmonic half-plane mapping.