Fully Starlike and Convex Harmonic Mappings of order \alpha

TL;DR

This paper introduces the concepts of fully starlike and convex harmonic mappings of order lpha, establishing bounds for their radii and exploring properties like convolution preservation, which differ from conformal mappings.

Contribution

It defines fully starlike and convex harmonic mappings of order lpha, determines their radii bounds, and investigates convolution properties in the harmonic setting.

Findings

Bounds for the radius of fully starlikeness and convexity are established.

Convexity is not preserved under convolution of harmonic mappings.

Conditions for convolution to preserve univalent harmonic convexity are analyzed.

Abstract

The hereditary property of convexity and starlikeness for conformal mappings does not generalize to univalent harmonic mappings. This failure leads us to the notion of fully starlike and convex mappings of order \alpha, (0\leq \alpha<1). A bound for the radius of fully starlikeness and fully convexity of order \alpha is determined for certain families of univalent harmonic mappings. Convexity is not preserved under the convolution of univalent harmonic convex mappings, unlike in the analytic case. Given two univalent harmonic convex mappings f and g, the problem of finding the radius r_{0} such that f*g is a univalent harmonic convex mapping in |z|<r_{0}, is being considered.