Univalence Criteria Starting from the Method of Loewner Chains

摘要

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.