• Bulletin of the Korean Mathematical Society
• /
• v.55 no.3
• /
• pp.819-835
• /
• 2018

Let $f=h+{\bar{g}}$ be a normalized harmonic mapping in the unit disk $\mathbb{D}$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D^{\epsilon}{_f}=zf_z-{\epsilon}{\bar{z}}f_{\bar{z}}({\mid}{\epsilon}{\mid}=1)$ and $F_{\lambda}(z)=(1-{\lambda)f+{\lambda}D^{\epsilon}{_f}(0{\leq}{\lambda}{\leq}1)$ 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 $f=h+{\bar{g}}$. All results are sharp. Some of the results are motivated by the work of Kalaj et al. [8].