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On a Uniqueness Theorem

Published online by Cambridge University Press:  20 November 2018

Mikio Niimura
Mikio Niimura*
Affiliation:
Shibaura Institute of Technology, Tokyo, Japan
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The classical uniqueness theorems of Riesz and Koebe show an important characteristic of boundary behavior of analytic functions in the unit disc. The purpose of this note is to discuss these uniqueness theorems on hyperbolic Riemann surfaces. It will be necessary to give additional hypotheses because Riemann surfaces can be very nasty. So, in this note the Wiener compactification will be used as ideal boundary of Riemann surfaces. The Theorem, Corollaries 1, 2 and 3 are of Riesz type, Riesz-Nevanlinna type, Koebe type and Koebe-Nevanlinna type respectively. Corollaries 4 and 5 are general forms of Corollaries 2 and 3 respectively.

Let f be a mapping from an open Riemann surface R into a Riemann surface R′.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 1072 - 1076
Copyright
Copyright © Canadian Mathematical Society 1979

References

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3. Springer, G., Introduction to Riemann surfaces (Addison-Wesley, Reading, Mass., 1957).Google Scholar