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Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

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Abstract

In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping f in the complex plane without assuming any additional condition on the (second complex) dilatation ω f of f. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove theorems analogous to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.

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Acknowledgements

We would like to thank Professors Martin Chuaqui and Dragan Vukotić for their helpful comments that improved the clarity of the exposition in this paper. We are also indebted to the anonymous referee for suggesting some topics for further work.

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Authors and Affiliations

  1. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Avda. Padre Hurtado 750, Viña del Mar, Chile

    Rodrigo Hernández

  2. Departamento de Matemáticas (Módulo 17, Edificio de Ciencias), Universidad Autónoma de Madrid, 28049, Madrid, Spain

    María J. Martín

Authors
  1. Rodrigo Hernández
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  2. María J. Martín
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Correspondence to María J. Martín.

Additional information

Communicated by Steven G. Krantz.

The authors were partially supported by Fondecyt Grant # 1110160, Chile, and Grants MTM2009-14694-C02-01 (MICINN) and MTM2012-37436-C02-02 (MINECO), Spain.

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Hernández, R., Martín, M.J. Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings. J Geom Anal 25, 64–91 (2015). https://doi.org/10.1007/s12220-013-9413-x

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  • DOI: https://doi.org/10.1007/s12220-013-9413-x

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