# zbMATH — the first resource for mathematics

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
Zbl 0162.37902
Full Text: