Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T20:27:02.587Z Has data issue: false hasContentIssue false
  • English
  • Français

On families of Cauchy transforms and BMOA

Part of: Miscellaneous topics of analysis in the complex domain Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

Yusuf Abu Muhanna  and
El-Bachir Yallaoui
Yusuf Abu Muhanna
Affiliation:
Department of Mathematics American University of SharjhaPO Box 2666 SharjhaUnited Arab Emirates e-mail: ymuhanna@aus.ac.ae
El-Bachir Yallaoui
Affiliation:
Department of Mathematics and Statistics Sultan Qaboos UniversityPO Box 36 Al-Khod P.C. 123 Muscat Sultanate of Oman e-mail: bachiry@squ.edu.om
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.

In this paper we prove a number of results on Cauchy transforms of generalized type given by Borel measures supported on the class of analytic functions mapping the unit disc into the unit disk. We also give a BMOA characterization using these families.

Keywords

Cauchy transformsBMOA

MSC classification

Secondary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions 30D99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 3 , June 2000 , pp. 357 - 364
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Brannan, D. A., Clunie, J. G. and Kirwan, W. E., ‘On the coefficient problem for functions of bounded boundary rotation’, Ann. Acad. Sci. Finn. Ser. AI Math. Dissertationes 523 (1973).Google Scholar
[2]Garnett, J., Bounded analytic functions (Academic Press, New York, 1980).Google Scholar
[3]Hallenbeck, D. J., MacGregor, T. H. and Samotij, K., ‘Fractional Cauchy transforms, inner functions and multipliers’, Proc. London Math. Soc. (3) 72 (1996), 157–187.CrossRefGoogle Scholar
[4]Hallenbeck, D. J. and Samotij, K., ‘On Cauchy integrals of logarithmic potentials and their multipliers’, J. Math. Anal. Appl. 174 (1993), 614–634.CrossRefGoogle Scholar
[5]Hibschweiler, R. A. and MacGregor, T. H., ‘Closure properties of families of Cauchy-Stieltjes transforms’, Proc. Amer. Math. Soc. 105 (1989), 615–621.CrossRefGoogle Scholar
[6]Hibschweiler, R. A. and MacGregor, T. H., ‘Multipliers of families of Cauchy-Stieltjes transforms’, Trans. Amer. Math. Soc. 331 (1992), 377–394.CrossRefGoogle Scholar
[7]Hibschweiler, R. A. and Nordgren, E., ‘Cauchy transforms of measures and weighted shift operators on the disk algebra’, Rocky Mount. J. Math. 26 (1996), 627–653.Google Scholar
[8]Hruscev, S. V. and Vinogradov, S. A., ‘Inner functions and multipliers of Cauchy type integrals’, Arkiv Mat. 19 (1981), 23–42.CrossRefGoogle Scholar
[9]MacGregor, T. H., ‘Analytic and univalent functions with integral representations involving complex measure’, Indiana Univ. Math. J. 36 (1987), 109–130.CrossRefGoogle Scholar
[10]Vinogradov, S. A., ‘Properties of multipliers of Cauchy-Stieltjes integrals and some factorization problems for analytic functions’, Amer. Math. Soc. Trans. 115 (1980), 1–32.Google Scholar