Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

Rodrigo Hernández; María J. Martín

doi:10.1007/s12220-013-9413-x

Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

Rodrigo Hernández
, María J. Martín

Faculty of Engineering and Science

Research output: Contribution to journal › Article › peer-review

60 Scopus citations

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.

Original language	English
Pages (from-to)	64-91
Number of pages	28
Journal	Journal of Geometric Analysis
Volume	25
Issue number	1
DOIs	https://doi.org/10.1007/s12220-013-9413-x
State	Published - Jan 2013

Keywords

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

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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 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.",

keywords = "Becker{\textquoteright}s criterion, Convexity, Harmonic mappings, Pre-Schwarzian derivative, Schwarzian derivative, Univalence",

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AB - 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.

KW - Becker’s criterion

KW - Convexity

KW - Harmonic mappings

KW - Pre-Schwarzian derivative

KW - Schwarzian derivative

KW - Univalence

UR - https://www.scopus.com/pages/publications/84875925457

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EP - 91

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JF - Journal of Geometric Analysis

IS - 1

ER -

Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

Abstract

Keywords

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