Conformal invariants of QED domains
Let \(\Omega\) be a domain in the extended complex plane \(\overline{{\mathbb C}}\). The aim of this paper is to give inequalities between the boundary quasiextremal distance constant \(M_b(\Omega)\), the quasiextremal distance constant \(M(\Omega)\) and the quasiextremal reflection constant \(R(\Omega)\). These constants were introduced and studied by S. Yang in a series of papers. They are defined as folows. Given two disjoint nondegenerate continua \(A\) and \(B\) in the closure \(\overline{\Omega}\), we let \(\text{mod}(A,B;\Omega)\) be the modulus of the family \(\Gamma(A,B;\Omega)\) of curves that join \(A\) and \(B\) in \(\Omega\). Then, \[ M(\Omega)=\sup{\text{mod}(A,B;{\mathbb C})\over \text{mod}(A,B;\Omega)} \] where the supremum is taken over all pairs \(A\) and \(B\) in \(\overline{\Omega}\). The domain \(\Omega\) is called a QED domain if \(M(\Omega)\) is finite. The notion of QED domain is due to Gehring and Martio. For such a domain, one can define the boundary quasiextremal reflection constant \(R(\Omega)\) by \[ R(\Omega)=\inf K[f], \] the infimum being taken over all quasiconformal reflections \(f\) in \(\partial \Omega\). Finally, one defines \[ M_b(\Omega)=\sup{\text{mod}(A,B;{\mathbb C})\over \text{mod}(A,B;\Omega)} \] where the supremum is taken over all pairs \(A\) and \(B\) in \(\partial \Omega\). The inequality \(M_b(\Omega)\leq M(\Omega)\) is straightforward. \textit{S. Yang} [J. Anal. Math. 75, 173--183 (1998; Zbl 0911.30017)] proved that \(M(\Omega)\leq R(\Omega) +1\) for all smooth domains other than disks and he asked whether \(M(\Omega)\leq R(\Omega) +1\) for all asymptotically conformal extension domains other than disks. Wu and Yang proved that \(M(\Omega)< R(\Omega) +1\) for all asymptotically conformal extension domains other than disks. Here, a Jordan domain \(\Omega\) is said to be an asymptotically conformal extension domain if the Riemann mapping theorem from the unit disk \(D\) to \(\Omega\) has a quasiconformal extension to a neighborhood of \(D\) whose complex dilatation \(\mu\) satisfies \(| \mu(z)|\to 0\) as \(| z|\to 1\). In the paper under review, the author continues the investigation on the relation between the three constants \(M_b(\Omega)\), \(M(\Omega)\) and \(R(\Omega)\) by introducing and studying some closely related constants associated to a quasisymmetric homeomorphism. He obtains a necessary and sufficient condition for the equality \(M_b(\Omega)= R(\Omega) +1\) and he shows that \(M(\Omega)< R(\Omega) +1\) holds for all asymptotically conformal extension domains other than disks. This generalizes the result of Wu and Yang and gives an affirmative answer to the question posed by Yang that we mentioned above.
boundary quasiextremal distance constant
scientific article; zbMATH DE number 2148497
scientific article