Abstract:

For harmonic mappings $f_{i}(z)=h_{i}(z)+\overline{g_{i}(z)}$($i=1, 2$) defined in the unit disk satisfying the given coefficient conditions, we consider the radii of full convexity and full starlikeness of order $\alpha$ for the convex combination $(1-t)L^{\epsilon}_{f_{1}}+tL^{\epsilon}_{f_{2}}$, where $L^{\epsilon}_{f_{i}}=z\frac{\partial f_{i}}{\partial z}-\epsilon\overline{z}\frac{\partial f_{i}}{\partial\overline{z}}(|\epsilon|=1)$ denotes the differential operator of $f_{i}$. In addition, we obtain the radii of fully convex and full starlikeness of order $\alpha$ for convolution of harmonic mappings under the differential operator. All results are sharp.

Key words: Harmonic mappings, Convex combination, Fully convex of order α, Fully starlike of order α