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A note on a result of A. J. Lohwater and George Piranian

Published online by Cambridge University Press:  22 January 2016

G. L. Csordas
G. L. Csordas*
Affiliation:
University of Hawaii
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Let I denote the set of all inner functions in H∞, where H∞ is the Banach algebra of all bounded analytic functions on the open unit disk D. Let I* denote the set of all functions f(z) in H∞ for which the cluster set C(f,α) at any point α on the circumference C = {α| |α| = 1} is either the closed unit disk |w| ≤ 1 or else a single point of modulus one. Clearly, I is a subset of I*. In [3] the author has proved that I is properly contained in I*. Recently, Lohwater and Piranian [7] have shown that there is an outer function in I*. The purpose of this note is to point out some applications of this result. In particular we shall show in Theorem 2.3 that there exist outer functions whose boundary behavior is similar to that of inner functions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 56 , January 1975 , pp. 171 - 174
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

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