Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T23:36:49.029Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1] Collingwood, E. F. and Cartwright, M. L., Boundary theorems for a function meromorphic in the unit circle, Acta Math. 87 (1952), 83–146.Google Scholar
[2] Collingwood, E. F. and Lohwater, A. J., The Theory of Cluster Sets, Cambridge University Press, Cambridge, 1966.CrossRefGoogle Scholar
[3] Csordas, G., The Šilov boundary and a class of functions in H∞ Ph.D. Dissertation, Case Western Reserve University, Cleveland, Ohio, 1969.Google Scholar
[4] Csordas, G., A note on the Šilov boundary and the cluster sets of a class of functions H∞ , Acta Math. Acad. Sci. Hungar., 24 (1973), 5–11.Google Scholar
[5] Hoffman, K., Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N. J., 1962.Google Scholar
[6] Lohwater, A. J., Some function-theoretic results involving Baire category, Proc. Internat. Conf. Analysis, Jyväskylä Finland, 1970, Springer-Verlag, 1972.Google Scholar
[7] Lohwater, A. J. and Piranian, George, Bounded analytic functions with large cluster sets, Ann. Acad. Sci. Fenn. Ser. AI, no. 499 (1971), 1–7.Google Scholar
[8] Ohtsuka, M., On the asymptotic values of functions analytic in a circle, Trans. Amer. Math. Soc. 78 (1955), 294–304.Google Scholar
[9] Weiss, M. L., Cluster sets of bounded analytic functions from a Banach algebraic viewpoint, Ann. Acad. Sci. Fenn. Ser. AI, no. 367 (1965), pp. 1–14.Google Scholar