scholarly journals Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative

2014 ◽  
Vol 104 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Rodrigo Hernández ◽  
María J. Martín
Keyword(s):  
Schwarzian Derivative ◽  
Quasiconformal Extension ◽  
Harmonic Mappings

The Schwarzian derivative for harmonic mappings

2003 ◽  
Vol 91 (1) ◽  
pp. 329-351 ◽  
Author(s):  
Martin Chuaqui ◽  
Peter Duren ◽  
Brad Osgood
Keyword(s):  
Schwarzian Derivative ◽  
Harmonic Mappings

scholarly journals Schwarzian derivative criteria for valence of analytic and harmonic mappings

2007 ◽  
Vol 143 (2) ◽  
pp. 473-486 ◽  
Author(s):  
MARTIN CHUAQUI ◽  
PETER DUREN ◽  
BRAD OSGOOD
Keyword(s):  
Unit Disk ◽  
Minimal Surfaces ◽  
Analytic Functions ◽  
Schwarzian Derivative ◽  
Harmonic Mappings ◽  
Planar Harmonic Mappings

AbstractFor analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass–Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally, certain classes of harmonic mappings are shown to have finite Schwarzian norm.

QUASICONFORMAL EXTENSIONS OF HARMONIC MAPPINGS WITH A COMPLEX PARAMETER

2016 ◽  
Vol 102 (3) ◽  
pp. 307-315 ◽  
Author(s):  
XINGDI CHEN ◽  
YUQIN QUE
Keyword(s):  
Analytic Functions ◽  
Extension Theorem ◽  
Sufficient Conditions ◽  
Complex Parameter ◽  
Quasiconformal Extension ◽  
Harmonic Mappings ◽  
Maximal Dilatation ◽  
The One ◽  
Teichmüller Mapping

In this paper, we study quasiconformal extensions of harmonic mappings. Utilizing a complex parameter, we build a bridge between the quasiconformal extension theorem for locally analytic functions given by Ahlfors [‘Sufficient conditions for quasiconformal extension’, Ann. of Math. Stud.79 (1974), 23–29] and the one for harmonic mappings recently given by Hernández and Martín [‘Quasiconformal extension of harmonic mappings in the plane’, Ann. Acad. Sci. Fenn. Math.38 (2) (2013), 617–630]. We also give a quasiconformal extension of a harmonic Teichmüller mapping, whose maximal dilatation estimate is asymptotically sharp.

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