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

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 $\varphi \in \mathcal{A}$, let ${\mathcal{W}}_H^{-}(\varphi ):=\{f=h+\bar{g}\in \mathcal{H}:h-g=\varphi \}$ and ${\mathcal{W}}_H^{+}(\varphi ):=\{f=h+\bar{g}\in \mathcal{H}:h+g=\varphi \}$ be subfamilies of $\mathcal{H}$. In this paper, we shall determine the conditions under which the harmonic convolution $f_1\ast f_2$ is univalent and convex in one direction if $f_1\in {\mathcal{W}}_H^{-}(z)$ and $f_2\in {\mathcal{W}}_H^{-}(\varphi )$. A similar analysis is carried out if $f_1\in {\mathcal{W}}_H^{-}(z)$ and $f_2\in {\mathcal{W}}_H^{+}(\varphi )$. Examples of univalent harmonic mappings constructed by way of convolution are also presented.

References in corpus (2)

• Construction of subclasses of univalent harmonic mappings
• A subclass of close-to-convex harmonic mappings