Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-10-31T05:21:47.675Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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]).

Information

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

References

[1] Ahlfors, L. V. and Beurling, A.: Conformal invariants and function-theoretic null-sets, Acta Math., 83 (1950), 101–129.Google Scholar
[2] Carathéodory, C.: Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kuve auf einen Kreis, Math. Ann., 73 (1913), 305–320.Google Scholar
[3] Kametani, S.: On Hausdorff’s measures and generalized capacities with some of their applications to the theory of functions, Jap. Journ. Math., 19 (1944-48), 217–257.Google Scholar
[4] Kuroda, T.: On analytic functions on some Riemann surfaces, Nagoya Math. Journ., 10 (1956), 27–50.CrossRefGoogle Scholar
[5] Lavrentieff, M. : Sur quelques problèmes concernant’les fonctions univalentes sur la frontière, Rec. Math., 43 (1936), 816–846 (en russe).Google Scholar
[6] Lohwater, A. J. and Seidel, W.: An example in conformal mapping, Duke Math. Journ., 15 (1948), 137–143.CrossRefGoogle Scholar
[7] Lohwater, A. J. and Piranian, G.: Conformal mapping of a Jordan region whose boundary has positive two-dimensional measure, Michigan Math. Journ., 1 (1952), 1–4.Google Scholar
[8] Lusin, N. and Privaloff, I.: Sur l’unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup., 42 (1925), 143–191.Google Scholar
[9] Moore, R. L. and Kline, J. R.: On the most general plane closed set through which it is possible to pass a simple continuous arc, Ann. of Math., 20 (1918-1919), 218–223.Google Scholar
[10] Nevanlinna, R.: Eindeutige Analytische Funktionen, Berlin, 1936.Google Scholar
[11] Ohtsuka, M.: Théorèmes étoiles de Gross et leurs applications, Ann. Inst. Fourier, 5 (1954), 1–28.Google Scholar
[12] F., and Riesz, M.: Über die Randwerte einer analytischen Funktion, Quatreme Congre des math, scand. à Stockholm (1916), 24–47.Google Scholar
[13] Sario, L.: Questions d’existence au voisinage de la frontière d’une surface de Riemann, C. R. Acad. Sci. Paris 230 (1950), 269–271.Google Scholar