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On Generalized Schwarz-Pick Estimates

Part of: General properties Special classes of linear operators

Published online by Cambridge University Press:  21 December 2009

J. M. Anderson  and
J. Rovnyak
J. M. Anderson
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K.
J. Rovnyak
Affiliation:
Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904–4137, U.S.A. E-mail: rovnyak@virginia.edu
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Extract

By Pick's invariant form of Schwarz's lemma, an analytic function B (z) which is bounded by one in the unit disk D = {z: |z| < 1} satisfies the inequality

at each point α of D. Recently, several authors [2, 10, 11] have obtained more general estimates for higher order derivatives. Best possible estimates are due to Ruscheweyh [12]. Below in §2 we use a Hilbert space method to derive Ruscheweyh's results. The operator method applies equally well to operator-valued functions, and this generalization is outlined in §3.

MSC classification

Secondary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 30A10: Inequalities in the complex domain

Information

Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 161 - 168
Copyright
Copyright © University College London 2006

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References

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