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 Φ[f](z) = f(z) + $\overline{f(z)-z}$. In this paper, necessary and sufficient conditions on f are determined for the harmonic function Φ[f] to be univalent and convex in one direction. Similar results are obtained for Φ[f] to be starlike and convex in the unit disk. This results in the coefficient estimates, growth results and convolution properties of Φ[f]. In addition, various radii constants associated with Φ[f] have been computed.