A conformally invariant dilatation of quasisymmetry (Q2769199)

The authors modify the well-known result of Beurling and Ahlfors concerning the boundary values of a quasiconformal self-mapping of the upper half plane by using another condition for the boundary values which is conformally invariant. They call this condition for the boundary values of a quasiconformal self-mapping of a region \(\Omega\) in the extended plane generalized quasisymmetry. The generalized quasisymmetry is based on the second module of a quadrilateral \(Q= \Omega(z_1,z_2,z_3,z_4)\), where the points \(z_1\), \(z_2\), \(z_3\), \(z_4\) lie on the boundary of \(\Omega\). In case of the unit disk (or of the upper half plane) the second module of the quadrilateral \(Q= \Omega(z_1,z_2,z_3,z_4)\) is closely related to the cross-ratio \([z_1,z_2,z_3,z_4]\). The authors also demonstrate how the generalized quasisymmetry leads to some other results in this field.