Open AccessInjectivity of harmonic mappings with a specified injective holomorphic part
Author(s) -
Dariusz Partyka,
Ken-ichi Sakan
Publication year - 2022
Publication title -
annales fennici mathematici
Language(s) - English
Resource type - Journals
eISSN - 2737-114X
pISSN - 2737-0690
DOI - 10.54330/afm.115432
Subject(s) - holomorphic function , injective function , unit disk , mathematics , harmonic , harmonic function , complex plane , plane (geometry) , unit (ring theory) , pure mathematics , combinatorics , mathematical analysis , physics , geometry , quantum mechanics , mathematics education
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.