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Inequalities for angular derivatives and boundary interpolation

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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.

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Acknowledgments

The authors are very grateful to N. Tarkhanov for careful reading the manuscript and several valuable suggestions that improved the presentation.

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Authors and Affiliations

  1. Department of Mathematics, The College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Vladimir Bolotnikov

  2. Department of Mathematics, Ort Braude College, 21982 , Karmiel, Israel

    Mark Elin & David Shoikhet

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  1. Vladimir Bolotnikov
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  2. Mark Elin
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  3. David Shoikhet
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Correspondence to Mark Elin.

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Bolotnikov, V., Elin, M. & Shoikhet, D. Inequalities for angular derivatives and boundary interpolation. Anal.Math.Phys. 3, 63–96 (2013). https://doi.org/10.1007/s13324-012-0050-5

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  • DOI: https://doi.org/10.1007/s13324-012-0050-5

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