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On a Class of Conformal Metrics

Published online by Cambridge University Press:  22 January 2016

Maurice Heins
Maurice Heins*
Affiliation:
University of Illinois
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Last year when I was preparing for course lectures the work of Ahlfors [1] which establishes that the Bloch constant is at least as large as it appeared to me that the resources of the theory of metrics of negative curvature offered rich possibilities from a function-theoretic point of view. The parallelism between certain properties of subharmonic functions and those of the metrics introduced by Ahlfors [1] is so striking that we are led to ask whether one can introduce a class of metrics including the metrics of Ahlfors for which not only does a Schwarz-Pick-Ahlfors lemma hold, but also requirements of differentiability disappear, as in the modern theory of subharmonic functions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 21 , December 1962 , pp. 1 - 60
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1962

References

[1] Ahlfors, L. V. An extension of Schwarz’s Lemma. Trans. A.M.S. 43 (1938), 359–364.Google Scholar
[2] Bieberbach, L. Göttinger Nachrichten (1912), 599–602.Google Scholar
[3] Carathéodory, C. Conformal Representation. 1st ed. Cambridge, 1932.Google Scholar
[4] Courant, R. and Hilbert, D. Methoden der Mathematischen Physik. Vol. 2, Berlin, 1937.CrossRefGoogle Scholar
[5] Deny, J. Sur les infinis d’un potentiel. C. R. Acad. Sci. Paris 224 (1947), 524–525.Google Scholar
[6] Frostman, O. Sur les produits de Blaschke. K. Fys. Sall, i Lund For. 12 (1943). Nr. 15, 1–14.Google Scholar
[7] Heins, M. Lindelöfian maps. Ann. Math. 62 (1955), 418–446.CrossRefGoogle Scholar
[8] Hurewicz, W. and Wallman, H. Dimension Theory. Princeton, 1948.Google Scholar
[9] Nehari, Z. A generalization of Schwarz’ Lemma. Duke M.J. 14 (1947), 1035–1049.Google Scholar
[10] Nevanlinna, R. Eindeutige analytische Funktionen. 2d ed. Berlin, 1953.Google Scholar
[11] Nevanlinna, R. Le Theoreme de Picard-Borel et la Theorie des Fonctions Meromorphes. Paris, 1929.Google Scholar
[12] Parreau, M. Fonction caracteristique dune application conforme. Ann. Fac. Sci. Toulouse 19 (1956), 175–189.CrossRefGoogle Scholar
[13] Picard, E. Traité d Analyse, vol. 3. Paris, 1928.Google Scholar
[14] Robinson, R. M. Bloch functions. Duke M. J. 2 (1936), 453–458.Google Scholar
[15] Robinson, R. M. A generalization of Picard’s and related theorems. Duke M. J. 5 (1939), 118–131.Google Scholar
[16] Saks, S. and Zygmund, A. Analytic Functions Warsaw, Wroclaw, 1952.Google Scholar
[17] Tsuji, M. Potential Theory in Modern Function Theory. Tokyo, 1959.Google Scholar
[18] Virtanen, K I. Eine Bemerkung uber die Anwendung hyperbolischer Massbestimmungen in der Wertverteilungslehre der meromorphen Funktionen. Math. Scand. 1 (1953).Google Scholar