Quasiconformality of harmonic extensions
Let \(\Hom^+({\mathbb T})\) denote the class of all sense preserving homeomorphic self-mappings of the unit circle \({\mathbb T}\). The authors show that a homeomorphism \(\gamma\in\Hom^+({\mathbb T})\) with bounded derivative has a quasi-conformal extension, if and only if, its derivative is bounded away from zero and the Cauchy-Stieltjes transform of its differential is bounded. It is also shown that the foregoing result extends an earlier result due to \textit{O. Martio} [Ann. Acad. Sci. Fenn., Ser. A. I. Math. 425, 1-10 (1968; Zbl 0162.37902)].
scientific article; zbMATH DE number 1334884