## Fully starlike and fully convex harmonic mappings of order $\alpha$

### Volume 108 / 2013

#### Abstract

The hereditary properties of convexity and starlikeness for conformal mappings do not generalize to univalent harmonic mappings. This failure leads to the notions of fully starlike and fully convex mappings. In this paper, properties of fully starlike mappings of order $\alpha$ and fully convex mappings of order $\alpha$ $(0\leq \alpha < 1)$ are studied; in particular, the bounds for the radius of full starlikeness of order $\alpha$ as well as the radius of full convexity of order $\alpha$ are determined for certain families of univalent harmonic mappings. Unlike the analytic case, convexity is not preserved under the convolution of univalent harmonic convex mappings. Given two univalent harmonic convex mappings $f$ and $g$, the radius $r_{0}$ such that their harmonic convolution $f*g$ is a univalent harmonic convex mapping in $|z| < r_{0}$ is also investigated.