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Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane. (English) Zbl 1071.30016
Let \(\mathbb{U}\) denote the upper half-plane, \(\mathbb{U}= \{z= x+ iy\in\mathbb{C}: y> 0\}\) and \(\overline{\mathbb{U}}\) denote the closure of \(\mathbb{U}\) in \(\overline{\mathbb{C}}= \mathbb{C}\cup\{\infty\}\). By \(QC(\mathbb{U})\) we denote the group of all quasiconformal homeomorphisms of \(\overline{\mathbb{U}}\) onto itself fixing the point \(\infty\).
In this paper among others the following results are proved:
Theorem 1. Let \(f= u+ iv\) be a quasiconformal harmonic mapping of \(\mathbb{U}\) into \(\mathbb{U}\). Then the following assertions are equivalent.
(a) \(f\in QC(\mathbb{U})\).
(b) There are positive constants \(c\) and \(M\) such that \(v(z)= cy\), \(1/M\leq u_x\leq M\) and \(|u_y|\leq M\) for all \(z\in\mathbb{U}\).
(c) \(f\) is a bi-Lipschitz mapping of \(\mathbb{U}\) onto \(\mathbb{U}\).
It is also proved that if \(f\in QC(\mathbb{U})\) is harmonic then \(f\) has a unique representation of the form \[ f(z)= 2\operatorname{Re}\,\int^z_i\varphi(\zeta)\,d\zeta+ b+ ic\operatorname{Im}(z), \] where \(b +ic\) is a point in \(\mathbb{U}\) and \(\varphi\) is a holomorphic function on \(\mathbb{U}\) such that \(\varphi(\mathbb{U})\) is a relatively compact subset of the right half-plane \(H= \{z:\text{Re\,}z> 0\}\).

MSC:
30C55 General theory of univalent and multivalent functions of one complex variable
30C62 Quasiconformal mappings in the complex plane
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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