Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T12:02:59.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Functions of bounded expansion: normal and Bloch functions

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

P. M. Gauthier  and
J. Xiao
P. M. Gauthier
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, Montréal H3C3J7, Canada e-mail: gauthier@dms.umontreal.ca
J. Xiao
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China e-mail: jxiao@sxxxo.math.pku.edu.cn
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.

Normal functions and Bloch functions are respectively functions of bounded spherical expansion and bounded Euclidean expansion. In this paper we discuss the behaviour of normal functions and of Bloch functions in terms of the maximal ideal space of H∞, the Bergman projection and the Ahlfors-Shimizu characteristic.

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 2 , April 1999 , pp. 168 - 188
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Aulaskari, R., ‘On the boundary behaviour of Bloch and normal functions’, Bull. Austral. Math. Soc. 30 (1984), 299–305.CrossRefGoogle Scholar
[2]Aulaskari, R. and Lappan, P., ‘Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal’, in: Complex analysis and its applications, Pitman Res. Notes Math. 305 (Longman Sci. Tech., Harlow, 1994) pp. 136–146.Google Scholar
[3]Aulaskari, R., Lappan, P., Xiao, J. and Zhao, R., ‘On α-Bloch spaces and mulitipliers of Dirichlet spaces’, J. Math. Anal. Appl. 209 (1997), 103–121.CrossRefGoogle Scholar
[4]Axler, S., ‘The Bergman space, the Bloch space, and commutators of multiplication operators’, Duke Math. J. 53 (1986), 315–332.CrossRefGoogle Scholar
[5]Axler, S. and Zhu, K., ‘Boundary behavior of derivatives of analytic functions’, Michigan Math. J. 39 (1992), 129–143.CrossRefGoogle Scholar
[6]Baernstein, A. II, ‘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
[7]Brezis, H. and Nirenberg, L., ‘Degree theory and BMO: Part II: Compact manifolds with boundaries’, Selecta Math., Soviet New Ser. 2 (1996), 309–368.CrossRefGoogle Scholar
[8]Brown, L. and Gauthier, P. M., ‘Behavior of normal meromorphic functions on the maximal ideal space of H∞’, Michigan Math. J. 18 (1971), 365–371.CrossRefGoogle Scholar
[9]Carleson, L., ‘Interpolation by bounded analytic functions and the corona problem’, Ann. of Math. 76 (1962), 547–559.CrossRefGoogle Scholar
[10]Coifman, R., Rochberg, R. and Weiss, G., ‘Factorization theorems for Hardy spaces in several complex variables’, Ann. of Math. 103 (1976), 611–635.CrossRefGoogle Scholar
[11]Garnett, J., Bounded analytic functions (Academic Press, New York, 1981).Google Scholar
[12]Hoffman, K., ‘Bounded analytic functions and Gleason parts’, Ann. of Math. 86 (1969), 74–111.CrossRefGoogle Scholar
[13]Lappan, P., ‘Non-normal sums and products of unbounded normal functions’, Michigan Math. J. 8 (1961), 187–192.CrossRefGoogle Scholar
[14]Lappan, P., ‘Some results on harmonic normal functions’, Math. Z. 90 (1965), 155–159.CrossRefGoogle Scholar
[15]Lehto, O. and Virtanen, K. I., ‘Boundary behaviour and normal meromorphic functions’, Acta Math. 97 (1951), 47–65.CrossRefGoogle Scholar
[16]Noshiro, K., ‘On the theory of cluster sets of analytic functions’, J. Fac. Sci. Hokkaido Univ. 7 (1938), 149–159.Google Scholar
[17]Pommerenke, Ch., ‘Estimates for normal functions’, Ann. Acad. Sci. Fenn. Ser A I Math. 476 (1970), 1–10.Google Scholar
[18]Pommerenke, Ch., Univalent functions (Vandenhoeck and Ruprecht, Göttingen, 1975).Google Scholar
[19]Reimann, H. M., ‘Functions of bounded mean oscillation and quasiconformal mappings’, Comm. Math. Helv. 49 (1974), 260–276.CrossRefGoogle Scholar
[20]Reimann, H. M. and Rychener, T., Funktionen beschrankter mittlerer Ozillation, Lect. Notes in Math. 487 (Springer, Heidelberg, 1975).CrossRefGoogle Scholar
[21]Schiff, J. L., Normal families (Springer, Heidelberg, 1993).CrossRefGoogle Scholar
[22]Stroethoff, K., ‘Nevanlinna-type characterizations for theBloch space and related spaces’, Proc. Edinburgh Math. Soc. 33 (1990), 123–141.CrossRefGoogle Scholar
[23]Wirths, K. and Xiao, J., ‘Image areas of functions in the Dinchiet type spaces and their Möbius invariant subspaces’, Maria Curie-Slowdkowska, Sect. A 22 (1996), 239–245.Google Scholar
[24]Xiao, J., ‘Carleson measure, atomic decomposition and free interpolation from Bloch space’, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 35–46.Google Scholar
[25]Xiao, J. and Zhong, L., ‘On little Bloch space and its Carleson measure, atomic decomposition and free interpolation’, Complex Variables 27 (1995), 175–184.Google Scholar
[26]Yamashita, S., ‘Functions of uniformly bounded characteristic’, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 349–367.CrossRefGoogle Scholar
[27]Yamashita, S., ‘Area criteria for functions to be Bloch, normal and Yosida’, Proc. Japan Academy 59 (1983), 462–464.Google Scholar
[28]Zhu, K., Operator theory in function spaces, Pure and Applied Math. 139 (Marcel Dekker, New York, 1990).Google Scholar