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The Existence of Non-Euclidean Cercles de Remplissage in Certain Subsets of the Unit Disc

Published online by Cambridge University Press:  22 January 2016

L. H. Lange
L. H. Lange*
Affiliation:
San Jose State College
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The concept of a “set of ρ-discs” for complex-valued functions defined in the open unit disc U was introduced in [4; 5, p. 265] and various results concerning the existence of such sets associated with spirals and Stolz angles were derived. These sets are non-Euclidean analogues to the classical cercles de remplissage of Milloux [7; 8]. In the present paper additional theorems about Stolz angle phenomena and the existence of ρ-discs in more general subsets of U are established and new proofs are indicated for some results of Seidel [9, Theorem 2 and Corollary 1].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 19 , October 1961 , pp. 41 - 47
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1961

References

[1] Bagemihl, F. and Seidel, W., Behavior of meromorphic functions on boundary paths, with applications to normal functions. Archiv der Mathematik, 12 (1960), 263–269.CrossRefGoogle Scholar
[2] Carathéodory, C., Conformal representation, Cambridge, 1952.Google Scholar
[3] Collingwood, E. F. and Cartwright, M. L., Boundary theorems for a function meromorphic in the unit circle. Acta Math. 87 (1952), 83–146.CrossRefGoogle Scholar
[4] Lange, L. H., A non-Euclidean analogue to a theorem of H. Milloux and its relationship to a theorem of W. Seidel. Amer. Math. Soc. Notices 5 (1958), 847–848.Google Scholar
[5] Lange, L. H., Sur les cercles de remplissage non-euclidiens. Ann. Sci. Ecol. Norm. Sup. (3), 77 (1960), 257–280.CrossRefGoogle Scholar
[6] Lehto, O. and Virtanen, K. I., Boundary behavior of normal meromorphic functions. Acta Math. 97 (1957), 47–65.Google Scholar
[7] Milloux, H., Le théorème de M. Picard. Suites de fonctions holomorphes. Fonctions méromorphes et fonctions entières. J. Math, pures et appl. (9), 3 (1924), 345–401.Google Scholar
[8] Milloux, H., Sur le théorème de M. Picard. Bull. Soc. Math. Fr. 53 (1925), 181–207.CrossRefGoogle Scholar
[9] Seidel, W., Holomorphic functions with spiral asymptotic paths. Nagoya Math J. 14 (1959), 159–171.CrossRefGoogle Scholar
[10] Valiron, G., Sur les singularités de certaines fonctions holomorphes et de leurs inverses. J. Math, pures et appl. (9), 15 (1936), 423–435.Google Scholar