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Criteria for univalency and quasiconformal extension for harmonic mappings. (English) Zbl 1475.30048

Summary: 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 L. V. Ahlfors [in: Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, 23–29 (1974; Zbl 0324.30034)], R. Hernández and M. J. Martín [Ann. Acad. Sci. Fenn., Math. 38, No. 2, 617–630 (2013; Zbl 1316.31005)], and X. Chen and Y. Que [J. Aust. Math. Soc. 102, No. 3, 307–315 (2017; Zbl 1376.30017)]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmüller mappings.

MSC:

30C62 Quasiconformal mappings in the complex plane
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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