Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T04:49:10.670Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the radial growth of Bloch functions

Published online by Cambridge University Press:  17 April 2009

Daniel Girela
Daniel Girela
Affiliation:
Análisis Matemático Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
Rights & Permissions [Opens in a new window]

Abstract

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.

The radial growth of Bloch functions has been extensively studied. Using integral means estimates and the Hardy Littlewood theorem, Makarov proved the so called law of iterated logarithm for Bloch functions. This result has also been obtained using probabilistic arguments. In this paper we present another method of studying the radial growth of Bloch functions, having the integral means estimates as starting point and using certain results about normal functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 1 , February 1992 , pp. 143 - 149
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Ban¯uelos, R., ‘Brownian motion and area functions’, Indiana Univ. Math. J. 35 (1986), 643–668.CrossRefGoogle Scholar
[2]Bagemihl, F. and Seidel, W., ‘Sequential and continuous limits of meromorphic functions’, Ann. Acad. Sci. Fenn. Ser. A I Math. 1280 (1960), 1–17.Google Scholar
[3]Chang, S.Y.A., Wilson, J.M. and Wolff, T.H., ‘Some weighted norm inequalities concerning the Schrödinger operators’, Comment. Math. Helv. 60 (1985), 217–246.CrossRefGoogle Scholar
[4]Clunie, J.G. and MacGregor, T.H., ‘Radial growth of the derivative of univalent functions’, Comment. Math. Helv. 59 (1984), 362–375.CrossRefGoogle Scholar
[5]Duren, P.L., Theory of HP spaces (Academic Press, New York, 1970).Google Scholar
[6]Girela, D., ‘Integral means and radial growth of Bloch functions’, Math. Z. 195 (1987), 37–50.CrossRefGoogle Scholar
[7]John, F. and Nirenberg, L., ‘On functions of bounded mean oscillation’, Comm. Pure Appl. Math. 14 (1961), 415–426.CrossRefGoogle Scholar
[8]Korenblum, B., ‘BMO estimates and radial growth of Bloch functions’, Bull. Amer. Math. Soc. 12 (1985), 99–102.CrossRefGoogle Scholar
[9]Lehto, O. and Virtanen, K.I., ‘Boundary behaviour and normal meromorphic functions’, Acta Math. 97 (1957), 47–65.CrossRefGoogle Scholar
[10]Makarov, N.G., ‘On the distortion of boundary sets under conformal mappings’, Proc. London Math. Soc. (3) 51 (1986), 369–384.Google Scholar
[11]Pommerenke, Ch., Univalent functions (Vandenhoeck und Ruprecht, Göttingen, 1975).Google Scholar
[12]Pommerenke, Ch., ‘The growth of the derivative of a univalent function’, in The Bieberbach conjecture, Proceedings of the symposium on the occasion of the proof, Editors Baernstein, A., Drasin, D., Duren, P.L. and Marden, A., pp. 143–152 (Math. Surveys Monographs 21, Amer. Math. Soc., Providence, R.I., 1986).Google Scholar
[13]Przytycki, F., ‘On the law of iterated logrithm for Bloch functions’, Studia Math. 93 (1989), 145–154.CrossRefGoogle Scholar