A technique of constructing planar harmonic mappings and their properties

Abstract

The analytic part of a planar harmonic mapping plays a vital role in shaping its geometric properties. For a normalized analytic function $f$ defined in the unit disk, define an operator $\Phi[f](z) = f(z) + \overline{f(z)-z}$. In this paper, necessary and sufficient conditions on $f$ are determined for the harmonic function $\Phi[f]$ to be univalent and convex in one direction. Similar results are obtained for $\Phi[f]$ to be starlike and convex in the unit disk. This results in the coefficient estimates, growth results and convolution properties of $\Phi[f]$. In addition, various radii constants associated with $\Phi[f]$ have been computed.