Geodesic
Injectivity of harmonic mappings with a specified injective holomorphic part
Dariusz Partyka1 ; Ken-ichi Sakan2
1The John Paul II Catholic University of Lublin, Department of Mathematical Analysis, and The State School of Higher Education in Chełm, Institute of Mathematics and Information Technology
2Osaka Metropolitan University, Graduate School of Science
Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 573-586

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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.
DOI : 10.54330/afm.115432
Keywords: Harmonic mappings, quasiconformal mappings, Lipschitz condition, bi-Lipschitz condition, co-Lipschitz condition, injectivity of harmonic mappings

Dariusz Partyka  1   ; Ken-ichi Sakan  2

1 The John Paul II Catholic University of Lublin, Department of Mathematical Analysis, and The State School of Higher Education in Chełm, Institute of Mathematics and Information Technology
2 Osaka Metropolitan University, Graduate School of Science 
Dariusz Partyka; Ken-ichi Sakan. Injectivity of harmonic mappings with a specified injective holomorphic part. Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 573-586. doi: 10.54330/afm.115432
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