On A Property of the Boundary Correspondence under Quasiconformal Mappings

Abstract

Let w = f(z) be a quasiconformal mapping, in the sense of Pfluger [5]-Ahlfors [1], with maximal dilatation K, which will be simply referred to a K-QC mapping. As is well known, any K-QC mapping w = f(z) of Im z > 0 onto Im w > 0 can be extended to a homeomorphism from Im z ≧ 0 onto Im w ≧ 0 and hence it transforms any set of logarithmic capacity zero on Im z = 0 into a set with the same property on Im w = 0.

A Criterion for a Set and Its Image under Quasiconformal Mapping to be of α(0 < α ≦ 2)-Dimensional Measure Zero

“…In this paper motivated by the above results, by extending our argument in the previous paper [3] where the following lemma due to Teichmϋller is very useful, we shall give a criterion for both some closed set E in a Jordan domain D and its image set by any if-quasiconformal mapping w = w(z) of D to be of α:-dimensional measure zero, where 0 < a ^ 2.…”