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A Localization Principle for a Class of Analytic Functions

Published online by Cambridge University Press:  22 January 2016

D. A. Storvick
D. A. Storvick*
Affiliation:
Mathematics Research Center U.S. Army, University of Wisconsin
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It has been shown by Kiyoshi Noshiro [8; p. 35] that a bounded analytic function w = f(z) in |z| < 1 having radial limit values of modulus one almost everywhere satisfies a localization principle of the following type. Let (c) be any circular disk: | w − α | < ρ lying inside |w| < 1 whose periphery may be tangent to the circumference |w| = 1. Denote by Δ any component of the inverse image of (c) under w = f(z) and by z = z(ξ) a function which maps |ξ| < 1 onto the simply connected domain Δ in a one-to-one conformal manner. Then, the function

is also a bounded analytic function in | ξ | < 1 with radial limits of modulus one almost everywhere.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 23 , December 1963 , pp. 207 - 212
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

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