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].