Estimates for Quasiconformal Mappings onto Canonical Domains (II)

Abstract

In this paper we establish estimates for normal K-quasiconformal mappings $z=g(w)$ of any finitely-connected domain in the extended $w$-plane onto the interior or exterior of the unit circle or the extended $z$-plane with $n(\ge 0)$ slits on the circles $|z|=R_j(j=1,...,n)$. The bounds in the estimates for $R_j,|g(w)|$, etc. are explicitly given. They are sharp or asymptotically sharp and deduced mainly from estimates for the inverse mappings of $g$ in our previous paper [10] based on Carleman’s and Grötzsch’s inequalities and partly improved here. A generalization of the Schwarz lemma and improvements of some classical inequalities for conformal mappings are shown.