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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.
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Abstract

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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

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