Schwarzian Derivatives and Uniform Local Univalence

  • Martin Chuaqui
  • Peter Duren
  • Brad Osgood


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.


Analytic function valence harmonic mapping Schwarzian derivative uniform local univalence Schwarzian norm minimal surface harmonic lift 

2000 MSC

30C99 31A05 30C55 


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Copyright information

© Heldermann  Verlag 2008

Authors and Affiliations

  1. 1.Facultad de MatemáticasP. Universidad Católica de ChileSantiago 22Chile
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Department of Electrical EngineeringStanford UniversityStanfordUSA

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