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).
Categories: math.CV