Conformal invariants of QED domains

Abstract

Given a Jordan domain $\Omega$ in the extended complex plane $\overline{\kern-1.5pt\Bbb C}$, denote by $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ the boundary quasiextremal distance constant, quasiextremal distance constant and quasiconformal reflection constant of $\Omega$, respectively. It is known that $M_b(\Omega)\le M(\Omega)\le R(\Omega)+1$. In this paper, we will give some further relations among $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ by introducing and studying some other closely related constants. Particularly, we will give a necessary and sufficient condition for $M_b(\Omega)=R(\Omega)+1$ and show that $M(\Omega)<R(\Omega)+1$ for all asymptotically conformal extension domains other than disks. This gives an affirmative answer to a question asked by Yang, showing that the conjecture $M(\Omega)=R(\Omega)+1$ by Garnett and Yang is not true for all asymptotically conformal extension domains other than disks. Our discussion relies heavily on the theory of extremal quasiconformal mappings, which in turn gives some interesting results in the extremal quasiconformal mapping theory as well.