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A simple proof of Livingston's inequality for Carathéodory functions

Authors: Philippe Delsarte and Yves Genin
Journal: Proc. Amer. Math. Soc. 107 (1989), 1017-1020
MSC: Primary 30C50; Secondary 30D50
MathSciNet review: 984785
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Abstract: The Livingston determinant inequality involving the Maclaurin coefficients of a Carathéodory function are derived in a straightforward manner by use of the Riesz-Herglotz representation and the Schwarz inequality. The result is extended to the case of matrix-valued functions.

References [Enhancements On Off] (What's this?)

  • [1] N. I. Akhiezer, The classical moment problem, Oliver and Boyd, London, 1965.
  • [2] P. Delsarte, Y. Genin and Y. Kamp, Orthogonal polynomial matrices on the unit circle, IEEE Trans. Circuits and Systems CAS-25 (1978), 149-160. MR 0481886 (58:1981)
  • [3] P. L. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [4] A. E. Livingston, A coefficient inequality for functions of positive real part with an application to multivalent functions, Pacific J. Math. 120 (1985), 139-151. MR 808934 (87b:30024)

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Keywords: Carathéodory functions, coefficient inequalities, Riesz-Herglotz representation
Article copyright: © Copyright 1989 American Mathematical Society

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