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Nevanlinna-type characterizations for the Bloch space and related spaces

Published online by Cambridge University Press:  20 January 2009

Karel Stroethoff
Karel Stroethoff
Affiliation:
Department of Mathematical Sciences, University of Montana, MISSOULA, Montana 59812USA
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Abstract

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We give a characterisation of the Bloch space in terms of an area version of the Nevanlinna characteristic, analogous to Baernstein's description of the space BMOA in terms of the usual Nevanlinna characteristic. We prove analogous results for the little Bloch space and the space VMOA, and give value distribution characterizations for all these spaces. Finally we give valence conditions on a Bloch or little Bloch function for containment in BMOA or VMOA.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 1 , February 1990 , pp. 123 - 141
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

1.Anderson, J. M., Clunie, J. and Pommerenke, CH., On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974).Google Scholar
2.Axler, S., The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), 315–332.CrossRefGoogle Scholar
3.Baernstein, A., Analytic functions of bounded mean oscillation, in Aspects of Contemporary Complex Analysis (Eds. Brannan, D. A. and Clunie, J. G., Academic Press, London, 1980), pp. 3–36.Google Scholar
4.Duren, P. L., Theory of Hp Spaces (Academic Press, New York, 1970).Google Scholar
5.Garnett, J. B., Bounded Analytic Functions (Academic Press, New York, 1981).Google Scholar
6.Hille, E., Analytic Function Theory, Vol. II (Ginn and Company, New York, 1962).Google Scholar
7.John, F. and Nirenberg, L., On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.Google Scholar
8.Lehto, O., A majorant principle in the theory of functions, Math. Scand. 1 (1953), 1–17.CrossRefGoogle Scholar
9.Pommerenke, CH., Schlichte Funktionen und analytische Funktionen von beschrankter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), 591–602.CrossRefGoogle Scholar
10.Pommerenke, CH., On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), 199–208.Google Scholar
11.Rubel, L. A. and Timoney, R. M., An extremal property of the Bloch space, Proc. Amer. Math.Soc. 75 (1979), 45–49.Google Scholar
12.Sarason, D., Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405.Google Scholar
13.Sarason, D., Function Theory on the Unit Circle (Lecture notes of Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1978).Google Scholar