Injectivity of harmonic mappings with a specified injective holomorphic part

Abstract

Let $F=H+\bar{G}$ be a locally injective and sense-preserving harmonic mapping of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$, where $H$ and $G$ are holomorphic in $\mathbb{D}$ and $G(0)=0$. The aim of this paper is studying interplay between properties of $F_{\epsilon }:=H+\epsilon \bar{G}$, $\epsilon \in \mathbb{C}$, and its holomorphic part $H$. In particular, several results dealing with the injectivity of $F_{\epsilon }$ are obtained.