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On the Distribution of Values of Functions in the Unit Disk

Published online by Cambridge University Press:  22 January 2016

James R. Choike
James R. Choike*
Affiliation:
Oklahoma State University
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Let f(z) be a function analytic and bounded, |f(z)| < 1, in |z| < 1. Then, by Fatou’s theorem the radial limit f*(eiθ ) = limr→1f(reiθ ) exists almost everywhere on |z| = 1. Seidel [8, p. 208] and Calderón, González- Domínguez, and Zygmund [1] (see also [9, pp. 281-282]) proved the following: if f*(eiθ ) is of modulus 1 almost everywhere on an arc a < θ < b of |z| = 1, then either f(z) is analytically continuable across this arc or the values f*(eiθ ), a < d < b, cover the circumference |w| = 1 infinitely many times.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 49 , March 1973 , pp. 77 - 89
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

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