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