Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

Rodrigo Hernández, María J. Martín

Research output: Contribution to journalArticlepeer-review

51 Scopus citations


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.

Original languageEnglish
Pages (from-to)64-91
Number of pages28
JournalJournal of Geometric Analysis
Issue number1
StatePublished - Jan 2013


  • Becker’s criterion
  • Convexity
  • Harmonic mappings
  • Pre-Schwarzian derivative
  • Schwarzian derivative
  • Univalence


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