## Tohoku Mathematical Journal

### Conformal invariants of QED domains

Yu-Liang Shen

#### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 56, Number 3 (2004), 445-466.

Dates
First available in Project Euclid: 11 April 2005

https://projecteuclid.org/euclid.tmj/1113246678

Digital Object Identifier
doi:10.2748/tmj/1113246678

Mathematical Reviews number (MathSciNet)
MR2075777

Zentralblatt MATH identifier
1063.30017

#### Citation

Shen, Yu-Liang. Conformal invariants of QED domains. Tohoku Math. J. (2) 56 (2004), no. 3, 445--466. doi:10.2748/tmj/1113246678. https://projecteuclid.org/euclid.tmj/1113246678

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