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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

[1] Bagemihl, F.: Curvilinear cluster sets of arbitrary functions, Proc. Nat. Acad. Sci. (U.S.A.), 41, No. 6 (1955), pp. 379–382.CrossRefGoogle ScholarPubMed
[2] Heins, M.: On the Lindelöf principle, Ann. of Math., 61 (1955), pp. 440–473.Google Scholar
[3] Heins, M.: Lindelöfian maps, Ann. of Math., 62 (1955), pp. 418–446.CrossRefGoogle Scholar
[4] Lohwater, A. J.: The boundary values of a class of meromorphic functions, Duke Math. J., 19 (1952), pp. 243–252.Google Scholar
[5] Lohwater, A. J.: The reflection principle and the distribution of values of functions defined in a circle, Ann. Acad. Scient. Fennicae, AI 229 (1956), pp. 1–18.Google Scholar
[6] Lohwater, A. J.: The boundary behavior of meromorphic functions, Ann. Acad. Scient. Fennicae, 250/22 (1958), pp. 1–6.Google Scholar
[7] Noshiro, K.: Contributions to the theory of the singularities of analytic functions, Jap. J. Math., 19 (1958), pp. 299–327.Google Scholar
[8] Noshiro, K.: Cluster sets, Springer-Verlag, Berlin (1960).Google Scholar
[9] Pfluger, A.: Extremallängen und Kapazität, Comment. Math. Helvet., 29 (1955), pp.120–131.Google Scholar
[10] Storvick, D. A.: On pseudo-analytic functions, Nagoya Math. J., 12 (1957), pp. 131–138.Google Scholar
[11] Tsuji, M.: Potential theory in modern function theory, Maruzen, Tokyo (1959).Google Scholar
[12] Whyburn, G. T.: Analytic topology, Amer. Math. Soc. Colloq. Pub., 28 (1942).Google Scholar