On the Product of Planar Harmonic Mappings

Abstract

Given two harmonic functions $$f_{1}=h_{1}+{\bar{g}}_{1}$$ and $$f_{2}=h_{2}+{\bar{g}}_{2}$$ defined on the open unit disk of the complex plane, the geometric properties of the product $$f_{1}\otimes f_{2}$$ defined by $$\begin{aligned} f_{1}\otimes f_{2}=h_{1}*h_{2}-\overline{g_{1}*g_{2}}+ i{\text {Im}}(h_{1}*g_{2}+h_{2}*g_{1}) \end{aligned}$$ are discussed. Here $$*$$ denotes the analytic convolution. Sufficient conditions are obtained for the product to be univalent and convex in the direction of the real axis. In addition, a convolution theorem, coefficient inequalities and closure properties for the product $$\otimes $$ are proved.