Estimates for Quasiconformal Mappings onto Canonical Domains

Abstract

In this paper we establish estimates for $K$-quasiconformal mappings $z = g(w)$ of a domain bounded by two circles $|w| = 1, |w| = q$ and $n$ continua situated in $q < |w| < 1$ onto a circular ring $Q(g) < |z| < 1$ that has been slit along $n$ arcs on the circles $|z| = R_j(g) (j = 1,... ,n)$ such that $|z| = 1$ and $|z| = Q$ correspond to $|w| = 1$ and $|w| = q$, respectively. The bounds in the estimates for $Q, R_j$ and $|g(w)|$ are explicitly given, most of them are optimal. They are deduced mainly from [17].