Radius of fully starlikeness and fully convexity of harmonic linear differential operator

TL;DR

This paper determines the exact radii for univalence, fully starlikeness, and fully convexity of certain harmonic linear differential operators in the unit disk, under specific coefficient conditions, extending previous work and providing sharp results.

Contribution

It introduces sharp radii bounds for harmonic linear differential operators' geometric properties under harmonic Bieberbach coefficient conditions, expanding understanding of harmonic mappings.

Findings

Sharp radius of univalence established

Sharp radius of fully starlikeness determined

Sharp radius of fully convexity obtained

Abstract

Let $f=h+\overline{g}$ be a normalized harmonic mapping in the unit disk $\ID$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D_f^{\epsilon}=zf_{z}-\epsilon\overline{z}f_{\overline{z}}~(|\epsilon|=1)$ and $F_{\lambda}(z)=(1-\lambda)f+\lambda D_f^{\epsilon}~(0\leq\lambda\leq 1)$ when the coefficients of $h$ and $g$ satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of $h$ and $g$ satisfy the corresponding necessary conditions of the harmonic convex function $f=h+\overline{g}$. All results are sharp. Some of the results are motivated by the work of Kalaj et al. \cite{Kalaj2014} (Complex Var. Elliptic Equ. 59(4) (2014), 539--552).