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 \(H= \{z:\text{Re\,}z> 0\}\).
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 \(H= \{z:\text{Re\,}z> 0\}\).
Reviewer: Jan Stankiewicz (Rzeszów)
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.) |