Directional Convexity of Combinations of Harmonic Half-Plane and Strip Mappings

TL;DR

This paper investigates the univalence and directional convexity of harmonic mappings formed by combining right half-plane and strip mappings, providing conditions under which these combinations are convex.

Contribution

It introduces new conditions ensuring convexity of harmonic combinations of half-plane and strip mappings, extending understanding of their geometric properties.

Findings

Identifies conditions for convexity of harmonic mappings

Shows univalence preservation under certain combinations

Analyzes the effect of complex parameters on convexity

Abstract

For $k=1,2$, let $f_k=h_k+\overline{g_k}$ be normalized harmonic right half-plane or vertical strip mappings. We consider the convex combination $\hat{f}=\eta f_1+(1-\eta)f_2 =\eta h_1+(1-\eta)h_2 +\overline{\overline{\eta} g_1+(1-\overline{\eta})g_2}$ and the combination $\tilde{f}=\eta h_1+(1-\eta)h_2+\overline{\eta g_1+(1-\eta)g_2}$. For real $\eta$, the two mappings $\hat{f}$ and $\tilde{f}$ are the same. We investigate the univalence and directional convexity of $\hat{f}$ and $\tilde{f}$ for $\eta\in\mathbb{C}$. Some sufficient conditions are found for convexity of the combination $\tilde{f}$.