Bulletin of the
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Bull. Korean Math. Soc. 2018; 55(3): 819-835

Online first article March 8, 2018      Printed May 31, 2018

https://doi.org/10.4134/BKMS.b170289

Copyright © The Korean Mathematical Society.

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

ZhiHong Liu, Saminathan Ponnusamy

Honghe University, Indian Institute of Technology Madras

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}.

Keywords: harmonic mappings, harmonic differential operator, coefficient inequality, radius of univalence, fully starlike harmonic mappings, fully convex harmonic mappings

MSC numbers: 30C45, 31C05

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