Injectivity of harmonic mappings with a specified injective holomorphic part

Abstract

Let \(F=H+\overline{G}\) 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 \(F_\varepsilon:=H+\varepsilon\overline G\), \(\varepsilon\in\mathbb{C}\), and its holomorphic part \(H\). In particular, several results dealing with the injectivity of \(F_\varepsilon\) are obtained.