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On Completeness With Respect to the Carathéodory Metric*

Published online by Cambridge University Press:  20 November 2018

M. A. Selby
M. A. Selby*
Affiliation:
University of Windsor, Windsor, Ontario
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Let X be a bounded region in the plane. Define

where f is an analytic function on X bounded in modulus by 1. We call d the Carathéodory metric.

In this note we give necessary and sufficient conditions to ensure that this metric be complete.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 261 - 263
Copyright
Copyright © Canadian Mathematical Society 1974

Footnotes

*

This is partly taken from the author’s Ph.D dissertation at Cornell University. Preparation of this paper was supported in part by the National Research Council. Grant No. A8088.

References

1. Gamelin, T. and Garnett, J., Distinguished homomorphisms and fibre algebras. Amer. J. Math. 92 (1970), 455-474.Google Scholar
2. Gleason, A., Function algebras, Seminar on Analytic Function., Institute for Advanced Study, 1957, Vol. II.Google Scholar
3. Hille, E., Analytic function theory, Ginn.Google Scholar
4. Zalcman, L., Analytic capacity and rational approximation. Lecture Notes in Mathematics, No. 50, Springer-Verlag.Google Scholar