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Schwarzian Derivatives and Uniform Local Univalence

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Abstract

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.

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Authors and Affiliations

  1. Facultad de Matemáticas, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

    Martin Chuaqui

  2. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 48109-1043, USA

    Peter Duren

  3. Department of Electrical Engineering, Stanford University, Stanford, California, 94305, USA

    Brad Osgood

Authors
  1. Martin Chuaqui
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  2. Peter Duren
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  3. Brad Osgood
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Corresponding author

Correspondence to Martin Chuaqui.

Additional information

Dedicated to Walter Hayman on the occasion of his 80th birthday

The authors are supported by Fondecyt Grant # 1030589.

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Chuaqui, M., Duren, P. & Osgood, B. Schwarzian Derivatives and Uniform Local Univalence. Comput. Methods Funct. Theory 8, 21–34 (2008). https://doi.org/10.1007/BF03321667

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  • DOI: https://doi.org/10.1007/BF03321667

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