Estimates for quasiconformal mappings onto canonical domains (II)

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 (greater than or equal to 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 Grotzsch's inequalities and partly improved here. A generalization of the Schwarz lemma and improvements of some classical inequalities for conformal mappings are shown.