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Lipschitz spaces and harmonic mappings. (English) Zbl 1180.30029
Quasiconformal (qc) harmonic mappings were apparently first considered by O. Martio in 1968 [Ann. Acad. Sci. Fenn., Ser. A I 425, 10 p. (1968; Zbl 0162.37902)].
Thirty years later many authors continued this work: D. Partyka and K.-I. Sakan [J. Comput. Appl. Math. 105, No. 1–2, 425–436 (1999; Zbl 0951.30017)], M. Pavlović [Ann. Acad. Sci. Fenn., Math. 27, No. 2, 365–372 (2002; Zbl 1017.30014)], D. Kalaj [Complex Variables, Theory Appl. 48, No. 2, 175–187 (2003; Zbl 1041.30006)], and M. Arsenović, V. Kojić and M. Mateljević [Ann. Acad. Sci. Fenn., Math. 33, No. 1, 315–318 (2008; Zbl 1140.31003)].
Refining his earlier results the author proves that a qc harmonic map \(f: \Omega_1 \to \Omega_2\) between two Jordan domains \( \Omega_j\) with \(C^{j,\alpha}\) boundaries, \(j=1,2\), is bilipschitz.

MSC:
30C55 General theory of univalent and multivalent functions of one complex variable
30C62 Quasiconformal mappings in the complex plane
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