# Directional convexity of harmonic mappings

## TL;DR

This paper investigates the convolution properties of harmonic functions in the unit disk, establishing conditions under which the convolution is univalent and convex in a specified direction, extending previous results to broader function classes.

## Contribution

It introduces new convolution results for harmonic functions involving harmonic shearing and extends prior work to larger classes of functions.

## Key findings

- Convolution $f*F$ is univalent and convex in the direction of $-mu$ under certain conditions.
- Local-univalence of the convolution is established for specific analytic dilatations.
- If $g\equiv0$ and both functions are convex, then their convolution is convex.

## Abstract

The convolution properties are discussed for the complex-valued harmonic functions in the unit disk $\mathbb{D}$ constructed from the harmonic shearing of the analytic function $\phi(z):=\int_0^z (1/(1-2\xi\textit{e}^{\textit{i}\mu}\cos\nu+\xi^2\textit{e}^{2\textit{i}\mu}))\textit{d}\xi$, where $\mu$ and $\nu$ are real numbers. For any real number $\alpha$ and harmonic function $f=h+\overline{g}$, define an analytic function $f_{\alpha}:=h+\textit{e}^{-2\textit{i}\alpha}g$. Let $\mu_1$ and $\mu_2$ $(\mu_1+\mu_2=\mu)$ be real numbers, and $f=h+\overline{g}$ and $F=H+\overline{G}$ be locally-univalent and sense-preserving harmonic functions such that $f_{\mu_1}*F_{\mu_2}=\phi$. It is shown that the convolution $f*F$ is univalent and convex in the direction of $-\mu$, provided it is locally univalent and sense-preserving. Also, local-univalence of the above convolution $f*F$ is shown for some specific analytic dilatations of $f$ and $F$. Furthermore, if $g\equiv0$ and both the analytic functions $f_{\mu_1}$ and $F_{\mu_2}$ are convex, then the convolution $f*F$ is shown to be convex. These results extends the work done by Dorff \textit{et al.} to a larger class of functions.