Planar harmonic mappings in a family of functions convex in one direction

Abstract

Let $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ be the open unit disk, and $h$ and $g$ be two analytic functions in $\mathbb{D}$. Suppose that $f=h+\overline{g}$ is a harmonic mapping in $\mathbb{D}$ with the usual normalization $h(0)=0=g(0)$ and $h'(0)=1$. In this paper, we consider harmonic mappings $f$ by restricting its analytic part to a family of functions convex in one direction and, in particular, starlike. Some sharp and optimal estimates for coefficient bounds, growth, covering and area bounds are investigated for the class of functions under consideration. Also, we obtain optimal radii of fully convexity, fully starlikeness, uniformly convexity, and uniformly starlikeness of functions belonging to those family.