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Univalence Criteria Starting from the Method of Loewner Chains

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Abstract

The paper continues the work of Royster (Duke Math J 19:447–457, 1952), Mocanu [Mathematica (Cluj) 22(1):77–83, 1980; Mathematica (Cluj) 29:49–55, 1987], Cristea [Mathematica (Cluj) 36(2):137–144, 1994; Complex Var 42:333–345, 2000; Mathematica (Cluj) 43(1):23–34, 2001; Mathematica (Cluj), 2010, to appear; Teoria Topologica a Functiilor Analitice, Editura Universitatii Bucuresti, Romania, 1999] of extending univalence criteria for complex mappings to C 1 mappings. We improve now the method of Loewner chains which is usually used in complex univalence theory for proving univalence criteria or for proving quasiconformal extensions of holomorphic mappings f : B → C n to C n. The results are surprisingly strong. We show that the usual results from the theory, like Becker’s univalence criteria remain true for C 1 mappings and since we use a stronger form of Loewner’s theory, we obtain results which are stronger even for holomorphic mappings f : B → C n. In our main result (Theorem 4.1) we end the researches dedicated to quasiconformal extensions of K-quasiregular and holomorphic mappings f : B → C n to C n. We show that a C 1 quasiconformal map f : B → C n can be extended to a quasiconformal map F : C n → C n, without any metric condition imposed to the map f.

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  1. Faculty of Mathematics and Computer Sciences, University of Bucharest, Str. Academiei 14, 010014, Bucharest, Romania

    Mihai Cristea

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  1. Mihai Cristea
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Correspondence to Mihai Cristea.

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Communicated by Matti Vuorinen.

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Cristea, M. Univalence Criteria Starting from the Method of Loewner Chains. Complex Anal. Oper. Theory 5, 863–880 (2011). https://doi.org/10.1007/s11785-010-0093-2

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  • DOI: https://doi.org/10.1007/s11785-010-0093-2

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