Conformal invariants of smooth domains and extremal quasiconformal mappings of ellipses
For a domain \(\Omega\) in \(\mathbb{C}\) let \(M(\Omega\) and \(R(\Omega)\) denote the quasiextremal distance (QED) constant and the quasiconformal reflection constant of \(\Omega\), respectively. Here \(M(\Omega)= \sup \bmod (A,B; \mathbb{C})/ \bmod (A,B; \Omega)\) and the supremum is taken over all nondegenerate continua \(A,B \subset \overline \Omega\), see [\textit{F. Gehring}, \textit{O. Martio}: J. Anal. Math. 45, 181-206 (1985; Zbl 0596.30031)] and \(R(\Omega)\) is the infimum of \(K(f)\) over all homeomorphic reflections in \(\Omega\). The author computes \(R(\Omega)\) for ellipses \(\Omega\) and shows that \(M(\Omega)\neq R (\Omega) +1\) for ellipses. Let \(\Omega\) be a domain with analytic boundary. The author shows that then \(M(\Omega)= M_b (\Omega)\) or \(M(\Omega) <R(\Omega) +1\); here \(M_b(\Omega)\) refers to the boundary QED constant of \(\Omega\) whose definition involves continua \(A,B \subset \partial \Omega\) only.
scientific article; zbMATH DE number 1043158