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Omitted rays and wedges of fractional Cauchy transforms

Part of: Spaces and algebras of analytic functions Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  09 April 2009

R. A. Hibschweiler  and
T. H. Macgregor
R. A. Hibschweiler
Affiliation:
University of New Hampshire, Department of Mathematics and Statistics, Durham, NH 03824, USA, e-mail: [email protected]
T. H. Macgregor
Affiliation:
Bowdoin College, Department of Mathematics, Brunswick, ME 04011, USA
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Abstract

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For α > 0 let α denote the set of functions which can be expressed where μ is a complex-valued Borel measure on the unit circle. We show that if f is an analytic function in Δ = {z ∈ : |z| < 1} and there are two nonparallel rays in /f(Δ) which do not meet, then f ∈ α where απ denotes the largest of the two angles determined by the rays. Also if the range of a function analytic in Δ is contained in an angular wedge of opening απ and 1 < α < 2, then f ∈ α.

MSC classification

Secondary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions 30H05: Bounded analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 367 - 373
Copyright
Copyright © Australian Mathematical Society 2006

References

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