Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-27T03:31:07.427Z Has data issue: false hasContentIssue false
  • English
  • Français

The Bloch-Nevanlinna conjecture revisited

Published online by Cambridge University Press:  17 April 2009

Douglas M. Campbell  and
Gene H. Wickes
Douglas M. Campbell
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah, USA.
Gene H. Wickes
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah, USA.
Rights & Permissions [Opens in a new window]

Abstract

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.

In 1929 Rolf Nevanlinna posed a problem attributed to Bloch which has since been known as the Bloch-Nevanlinna conjecture. It can be stated as follows: Is the derivative of a function of bounded characteristic of bounded characteristic? A variety of different counterexamples have provided negative answers to this question. The purpose of the paper is to survey these counterexamples and then give a truly elementary proof of the following theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 18 , Issue 3 , June 1978 , pp. 447 - 453
Copyright
Copyright © Australian Mathematical Society 1978

References

[1]Anderson, J.M., “Category theorems for certain Banach spaces of analytic functions”, J. reine angew. Math. 249 (1971), 83–91.Google Scholar
[2]Collingwood, E.F. and Lohwater, A.J., The theory of cluster sets (Cambridge Tracts in Mathematics and Mathematical Physics, 56. Cambridge University Press, Cambridge, 1966).CrossRefGoogle Scholar
[3]Duren, P.L., ”On the Bloch-Nevanlinna conjecture”, Colloq. Math. 20 (1969), 295–297.Google Scholar
[4]Fried, Hans, “On analytic functions with bounded characteristic”, Bull. Amer. Math. Soc. 52 (1946), 694–699.CrossRefGoogle Scholar
[5]Frostman, Otto, “Sur les produits des Blaschke”, Kungl. Fysiografiska Sällskapets i Lund Förhandlingar [Proc. Roy. Physiog. Soc. Lund]. 12 (1942), no. 15, 169–182.Google Scholar
[6]Hahn, Liang-Shin, “On the Bloch-Nevanlinna problem”, Proc. Amer. Math. Soc. 32 (1972), 221–224.Google Scholar
[7]Hayman, W.K., “On the characteristic of functions meromorphic in the unit disk and of their integrals”, Acta Math. 112 (1964), 181–214.CrossRefGoogle Scholar
[8]Lohwater, A.J., Piranian, G. and Rudin, W., “The derivative of a schlicht function”, Math. Scand. 3 (1955), 103–106.CrossRefGoogle Scholar
[9]Nevanlinna, Rolf, Le théorème de Piecrd-Borel et la théorie des fonctions méromorphes (Gauthiers-Villars, Paris, 1929).Google Scholar
[10]Nevanlinna, Rolf, Analytic functions (translated from the second German edition by Emig, Phillip. Die Grundlehren der mathematischen Wissenschaften, 162. Springer-Verlag, Berlin, Heidelberg, New York, 1970).Google Scholar
[11]Rudin, Walter, “On a problem of Bloch and Nevanlinna”, Proc. Amer. Math. Soc. 6 (1955), 202–204.Google Scholar