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On A Property of the Boundary Correspondence under Quasiconformal Mappings

Published online by Cambridge University Press:  22 January 2016

Kazuo Ikoma
Kazuo Ikoma*
Affiliation:
Department of Mathematics, Yamagata University
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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.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 16 , February 1960 , pp. 185 - 188
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1960

References

[1] Ahlfors, L.: On quasiconformal mappings, Journ. d’Anal. Math., 3 (1953/54), 1–58.Google Scholar
[2] Ahlfors, A. Beurling-L.: The boundary correspondence under quasiconformal mapping, Acta Math. 98 (1956), 125–142.Google Scholar
[3] Kuroda, T.: A criterion for a set to be of 1-dimensional measure zero, to appear in Jap. Journ. Math.Google Scholar
[4] Nevanlinna, R.: Eindeutige analytische Funktionen, Springer Verlag, 2 Aufl. (1953).CrossRefGoogle Scholar
[5] Pfluger, A.: Quelques theorems sur une classe de fonctions pseudo-analytiques, C. R. 231 (1950), 1022–1023.Google Scholar
[6] Teichmüller, O.: Untersuchungen über konforme und quasikonforme Abbildung, Deutsche Math. 3 (1938), 621–678.Google Scholar