Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-02T19:33:42.011Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuity of Cima and Rung's extension and non normal meromorphic functions

Published online by Cambridge University Press:  17 April 2009

Douglas M. Campbell
Douglas M. Campbell
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 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.

A function meromorphic in |z| < 1 is constructed such that on every curve in |z| < 1 which goes to |z| = 1 the set of limit points of the function is the entire complex plane. This example is used to prove the existence of non-normal meromorphic functions in |z| < 1 which have continuous set valued extensions. Cima and Rung had introduced a set valued extension for meromorphic functions and proved that all normal meromorphic functions have a continuous extension while all functions with a continuous extension have the Lindelöf property. For a long time it was thought that this might characterize normal meromorphic functions. This paper proves that it is not possible to determine the normality of a meromorphic function by the continuity of Cima and Rung's set valued extension. The paper closes with the open problem: do there exist non-normal analytic functions for which Cima and Rung's set valued extension is continuous?

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 1 , January 1979 , pp. 139 - 143
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Barth, K.F., “Asymptotic values of meromorphic functions”, Michigan Math. J. 13 (1966), 321–340.CrossRefGoogle Scholar
[2]Bonar, D.D., On annular functions (Mathematische Forschungsberichte, 24. VEB Deutscher Verlag der Wissenschaften, Berlin, 1971).Google Scholar
[3]Brown, Leon and Gauthier, Paul, “Behavior of normal meromorphic functions on the maximal ideal space of H∞“, Michigan Math. J. 18 (1971), 365–371.CrossRefGoogle Scholar
[4]Cima, J.A. and Rung, D.C., “Normal functions and a class of associated boundary functions”, Israel J. Math. 4 (1966), 119–126.CrossRefGoogle Scholar
[5]Lehto, Olli and Virtanen, K.I., “Boundary behaviour and normal meromorphic functions”, Acta Math. 97 (1957), 47–65.CrossRefGoogle Scholar
[6]Rudin, Walter, Real and complex analysis, 2nd edition (MacGraw-Hill, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1974).Google Scholar