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Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk. (English) Zbl 1017.30014
Let \(f\) be a harmonic homeomorphism of the unit disk onto itself. The author proves that the following conditions are equivalent: (a) \(f\) is quasiconformal; (b) \(f\) is bi-Lipschitz in the Euclidean metric; and (c) the boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in \(L^{\infty }\). This extends an earlier result of O. Martio [Ann. Acad. Sci. Fenn., Ser. A I 425, 1-10 (1968; Zbl 0162.37902)].

MSC:
30C55 General theory of univalent and multivalent functions of one complex variable
30C62 Quasiconformal mappings in the complex plane
Citations:
Zbl 0162.37902
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