Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

By V. Bravo, R. Hernández, S. Ponnusamy, O. Venegas. Monatshefte für Mathematik (Print), 2012

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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 ωf 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.

Subfields: Applied Mathematics, Geometry and Topology

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