Pluriharmonic Mappings with the Convex Holomorphic Part

Abstract

In 2018, Partyka et al. established several equivalent conditions for a sense-preserving locally injective harmonic mapping $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ with convex holomorphic part $h$ to be quasiconformal in terms of the relationships of two-point distortion of $h$ , $g$ , and $f$ . In this study, we first generalize the above result to the case of pluriharmonic mappings $f_A=h+A\overline{g}$ , where $h$ is a convex mapping in the unit ball ${\mathbb{B}}^n$ and $A\in L\left({ℂ}^n,{ℂ}^n\right)$ with $‖A‖=1$ . Then, we establish a relationship of two-point distortion property between $f$ and $f_A$ .