Univalence and convexity in one direction of the convolution of harmonic mappings

TL;DR

This paper investigates conditions for harmonic convolutions to be univalent and convex in one direction, focusing on specific subclasses of harmonic functions and providing examples of such mappings.

Contribution

It establishes new criteria for the univalence and convexity in one direction of harmonic convolutions within certain subclasses of harmonic functions.

Findings

Conditions for harmonic convolution univalence

Criteria for convexity in one direction

Examples of harmonic mappings via convolution

Abstract

Let $\mathcal{H}$ denote the class of all complex-valued harmonic functions $f$ in the open unit disk normalized by $f(0)=0=f_{z}(0)-1=f_{\bar{z}}(0)$, and let $\mathcal{A}$ be the subclass of $\mathcal{H}$ consisting of normalized analytic functions. For $\phi \in \mathcal{A}$, let $\mathcal{W}_{H}^{-}(\phi):=\{f=h+\bar{g} \in \mathcal{H}:h-g=\phi\}$ and $\mathcal{W}_{H}^{+}(\phi):=\{f=h+\bar{g} \in \mathcal{H}:h+g=\phi\}$ be subfamilies of $\mathcal{H}$. In this paper, we shall determine the conditions under which the harmonic convolution $f_1*f_2$ is univalent and convex in one direction if $f_1 \in \mathcal{W}_{H}^{-}(z)$ and $f_2 \in \mathcal{W}_{H}^{-}(\phi)$. A similar analysis is carried out if $f_1 \in \mathcal{W}_{H}^{-}(z)$ and $f_2 \in \mathcal{W}_{H}^{+}(\phi)$. Examples of univalent harmonic mappings constructed by way of convolution are also presented.