Directional convexity of harmonic mappings

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 $\varphi (z):={\int }_0^z(1/(1-2\xi {\text{e}}^{\text{i}\mu }\cos \nu +{\xi }^2{\text{e}}^{2\text{i}\mu }))\text{d}\xi$, where $\mu$ and $\nu$ are real numbers. For any real number $\alpha$ and harmonic function $f=h+\overline{g}$, define an analytic function $f_{\alpha }:=h+{\text{e}}^{-2\text{i}\alpha }g$. 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 $f_{{\mu }_1}\ast F_{{\mu }_2}=\varphi$. It is shown that the convolution $f\ast F$ is univalent and convex in the direction of $-\mu$, provided it is locally univalent and sense-preserving. Also, local-univalence of the above convolution $f\ast F$ is shown for some specific analytic dilatations of $f$ and $F$. Furthermore, if $g\equiv 0$ and both the analytic functions $f_{{\mu }_1}$ and $F_{{\mu }_2}$ are convex, then the convolution $f\ast F$ is shown to be convex. These results extends the work done by Dorff \textit{et al.} to a larger class of functions.