Starlikeness, convexity and Landau type theorem of the real kernel alpha-harmonic mappings

Abstract

In [26], Olofsson introduced a kind of second order homogeneous  partial differential equation. We call the solution of this equation real kernel $\alpha-$harmonic mappings. In this paper,  we  study  some geometric properties of this real kernel $\alpha-$harmonic mappings. We give univalence criteria and sufficient coefficient conditions for real kernel $\alpha-$harmonic mapping that are fully starlike or fully convex of order $\gamma$, $\gamma\in[0,1)$. Furthermore,  we give a Landau type theorem for real kernel $\alpha-$harmonic mappings.