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Multipliers of Cauchy integrals of logarithmic potentials

Part of: Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  26 February 2010

D. J. Hallenbeck  and
K. Samotij
D. J. Hallenbeck
Affiliation:
Department Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.
K. Samotij
Affiliation:
Instytut Matematyki, Politechnika Wrocławska, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland.
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Extract

Let Δ = {z:|Z|<1}, Γ={z:|z|=1}, and ℳ denote the set of complex-valued Borel measures on Γ. Let Kα(z)=(1−z)−α for α>0 and K0(z)=log 1/(1−z). For α > 0 let ℱα denote the family of functions f on Δ having the property that there exists a measure μ∈ℳ such that

for |z|<1. When α=0, this condition is replaced by

MSC classification

Secondary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions
Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 397 - 405
Copyright
Copyright © University College London 1995

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References

1.Duren, P. L.. Theory of Hp Spaces (Academic Press, New York, 1970).Google Scholar
2.Hallenbeck, D. J., MacGregor, T. H. and Samotij, K.. Fractional Cauchy transforms, inner functions and multipliers. To appear in Proc. London Math. Soc.Google Scholar
3.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
4.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
5.Hibschweiler, R. A. and MacGregor, T. H.. Multipliers of families of Cauchy-Stieltjes transforms. Trans. Amer. Math. Soc, 331 (1992), 377–394.CrossRefGoogle Scholar
6.Pommerenke, Ch.. On the coefficients of close-to-convex functions. Michigan Math. J., 9 (1962), 259–269.CrossRefGoogle Scholar
7.Titchmarsh, E. C.. The Theory of Functions 2nd Edition (Oxford Univ. Press London, 1939).Google Scholar
8.Vinogradov, S. A.. Properties of multipliers of Cauchy-Stieltjes integrals and some factorization problems for analytic functions. Amer. Math. Soc. Transl. (2), 115 (1980), 1–32.Google Scholar