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Radial growth and variation of bounded analytic functions

Published online by Cambridge University Press:  13 July 2011

D. J. Hallenbeck  and
T. H. MacGregor
D. J. Hallenbeck
Affiliation:
Department of Mathematical SciencesUniversity of DelawareNewark, Delaware 19716, U.S.A.
T. H. MacGregor
Affiliation:
Department of Mathematics and StatisticsSUNY at Albany1400 Washington AvenueAlbany, New York 12222, U.S.A.
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If a function f analytic in Δ = {z∈ℂ:|z|<1} has a nontangential limit as z→eiθ, then limr→1−(1−r)f′(reiθ)=0 [7, p. 181). It follows that this limit is zero for almost all θ for a number of classes of functions including the set H∞ of bounded analytic functions. In this paper we prove that this result for H∞ is sharp in a strong sense.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 3 , October 1988 , pp. 489 - 498
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

REFERENCES

1.Clunie, J. G. and MacGregor, T. H., Radial growth of the derivative of univalent functions, Comment. Math. Helv. 59 (1984), 362–375.CrossRefGoogle Scholar
2.Duren, P. L., Theory of Hp spaces (Academic Press, New York, 1970).Google Scholar
3.Goluzin, G. M., Geometric theory of functions of a complex variable (Amer. Math. Soc., Providence, 1969).Google Scholar
4.MacGregor, T. H., Growth of the derivatives of univalent and bounded functions, Ann. Univ. Mariae Curie Sklodowska, Sect. A 36/37 (1982–1983), 101–113.Google Scholar
5.Makarov, N. G., On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. 51 (1985), 369–384.Google Scholar
6.Ruscheweyh, St., Über einige klassen im einheitskreis holomorpher funktionen (Math. Stat. Sektion Forschungszentrum Graz, Berich Nr. 7 1974).Google Scholar
7.Zygmund, A., On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170–204.CrossRefGoogle Scholar