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

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 $D_f^{\epsilon }=z{f}_z-\epsilon \overline{z}{f}_{\overline{z}}(|\epsilon |=1)$ and $F_{\lambda }(z)=(1-\lambda )f+\lambda D_f^{\epsilon }(0\le \lambda \le 1)$ 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).

14 pages; To appear in the Bulletin of the Korean Mathematical Society

Cited by in corpus (1)

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