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Inequalities for angular derivatives and boundary interpolation

Inequalities for angular derivatives and boundary interpolation The classical Julia–Wolff–Carathéodory theorem asserts that the angular derivative of a holomorphic self-mapping of the open unit disk (Schur function) at its boundary fixed point is a positive number. Cowen and Pommerenke (J Lond Math Soc 26:271–289, 1982) proved that if a Schur function has several boundary regular fixed (or mutual contact) points, then the angular derivatives at these points are subject to certain inequalities. We develop a unified approach to establish relations between angular derivatives of Schur functions with a prescribed (possibly, infinite) collection of either mutual contact points or boundary fixed points. This approach yields diverse inequalities improving both classical and more recent results. We apply them to study the Nevanlinna–Pick interpolation problem with boundary data. Our methods lead to fairly explicit formulas describing the set of solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Inequalities for angular derivatives and boundary interpolation

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Springer Basel
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-012-0050-5
Publisher site
See Article on Publisher Site

Abstract

The classical Julia–Wolff–Carathéodory theorem asserts that the angular derivative of a holomorphic self-mapping of the open unit disk (Schur function) at its boundary fixed point is a positive number. Cowen and Pommerenke (J Lond Math Soc 26:271–289, 1982) proved that if a Schur function has several boundary regular fixed (or mutual contact) points, then the angular derivatives at these points are subject to certain inequalities. We develop a unified approach to establish relations between angular derivatives of Schur functions with a prescribed (possibly, infinite) collection of either mutual contact points or boundary fixed points. This approach yields diverse inequalities improving both classical and more recent results. We apply them to study the Nevanlinna–Pick interpolation problem with boundary data. Our methods lead to fairly explicit formulas describing the set of solutions.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Nov 1, 2012

References