On Convolution of Harmonic Mappings

Abstract

For $$j=1,2,$$ let the sense-preserving locally univalent harmonic mappings $${\mathcal {F}}_j={\mathcal {H}}_j+\overline{{\mathcal {G}}_j}$$ on $${\mathcal {D}}:=\left\{ z\in {\mathbb {C}}: |z|<1\right\} $$ be such that $$\overline{{\mathcal {F}}_j({\overline{z}})}={\mathcal {F}}_j(z)$$ and the mappings $$z({\mathcal {H}}_j+{\mathcal {G}}_j)'$$ are either odd starlike or starlike of order 1/2. It is shown that the convolution $${\mathcal {F}}_1*{\mathcal {F}}_2$$ of $${\mathcal {F}}_1$$ and $${\mathcal {F}}_2$$ is directional convex univalent mapping if it is locally univalent sense-preserving. Also, some examples are given where $${\mathcal {F}}_1*{\mathcal {F}}_2$$ is locally univalent sense-preserving.