On Some Boundary Problems in the Theory of Conformal Mappings of Jordan Domains

Abstract

It is a well-known result in the theory of conformal mappings of Jordan domains that if a domain D in the z-plane bounded by a closed Jordan curve C is mapped conformally on the disc |w;|<1 by a function w = f(z), analytic and univalent in D, then f(z) will be continuous on the closure of D and will map C on |w| =1 in a one to one manner (Carathéodory [2]), and that if C is rectifiable, then f(z) will map sets E of points of linear measure zero on C onto sets of linear measure zero on the circumference |w| = 1 and sets E of positive linear measure onto sets of positive linear measure on |w| =1 (F. and M. Riesz [12] and Lusin and Privaloff [8]).