Pre-Schwarzian and Schwarzian derivatives of harmonic mappings

Rodrigo Hernández; María J. Martín

Pre-Schwarzian and Schwarzian derivatives of harmonic mappings

Complex Variables 2012-10-09 v1

Authors: Rodrigo Hernández , María J. Martín

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 $\omega_f$ of $f$ . Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.

Keywords

harmonic mappings pre-Schwarzian derivative Schwarzian derivative univalence Becker criterion dilatation

Cite

@article{arxiv.1210.2330,
  title  = {Pre-Schwarzian and Schwarzian derivatives of harmonic mappings},
  author = {Rodrigo Hernández and María J. Martín},
  journal= {arXiv preprint arXiv:1210.2330},
  year   = {2012}
}

Comments

29 pages

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