A convolution property of univalent harmonic right half-plane mappings

Abstract

We consider the convolution of right half-plane harmonic mappings in the unit disk $$\mathbb {D}:=\{z\in \mathbb {C}:\, |z|<1\}$$ with respective dilatations $$ e^{i \alpha }(z + a)/(1 + a z)$$ and $$-z$$ , where $$-1< a < 1$$ and $$\alpha \in \mathbb {R}$$ . We prove that such convolutions are locally univalent and convex in the horizontal direction under certain condition.