Abstract
In this paper, we study the univalency and quasiconformal extension of sense-preserving harmonic mappings $f$ in the unit disk. For $f$, we introduce a quantity similar to Ahlfors's criteria and obtain a criterion of univalency and quasiconformal extension of $f$, which can be regarded as generalizations of the results obtained by Ahlfors [Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974), 23-29], Hernández and Martín [Quasiconformal extensions of harmonic mappings in the plane, Ann. Acad. Sci. Fenn. Math. 38 (2013), 617-630], and Chen and Que [Quasiconformal extension of harmonic mappings with a complex parameter, J. Aust. Math. Soc. 102 (2017), 307-315]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmüller mappings.
Acknowledgment
The authors would like to thank the referee for his/her helpful comments and suggestions. This work was supported by National Natural Science Foundation of China (Nos. 11871215, 11571172).
Citation
Zhenyong Hu. Jinhua Fan. "Criteria for univalency and quasiconformal extension for harmonic mappings." Kodai Math. J. 44 (2) 273 - 289, June 2021. https://doi.org/10.2996/kmj44203
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