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

Published online by Cambridge University Press:  22 January 2016

Kikuji Matsumoto
Kikuji Matsumoto*
Affiliation:
Mathematical Institute, Nagoya University
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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]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 24 , June 1964 , pp. 129 - 141
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1964

References

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