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On a Question of Seidel Concerning Holomorphic Functions Bounded on a Spiral

Published online by Cambridge University Press:  20 November 2018

K. F. Barth  and
W. J. Schneider
K. F. Barth
Affiliation:
Syracuse University, Syracuse, New York
W. J. Schneider
Affiliation:
Syracuse University, Syracuse, New York
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Let S be a spiral contained in D = {|z| < 1} such that S tends to C = {|z| = 1}. For the sake of brevity, by “f is bounded on S” we shall mean that f is holomorphic in D, unbounded, and bounded on S. The existence of such functions was first discussed by Valiron (9; 10); see also (1; 3; 8). Valiron also proved that any function that is “bounded on a spiral” must have the asymptotic value ∞ (10, p. 432). Functions that are bounded on a spiral may also have finite asymptotic values (1, p. 1254). In view of the above, Seidel has raised the question (oral communication): “Does there exist a function bounded on a spiral that has only the asymptotic value ∞?”. The following theorem answers this question affirmatively.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1255 - 1262
Copyright
Copyright © Canadian Mathematical Society 1969

References

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