Vol. 254, No. 1, 2011

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Schwarzian norms and two-point distortion

Martin Chuaqui, Peter Duren, William Ma, Diego Mejía, David Minda and Brad Osgood

Vol. 254 (2011), No. 1, 101–116
Abstract

An analytic function f with Schwarzian norm ∥𝒮f∥≤ 2(1 + δ2) is shown to satisfy a pair of two-point distortion conditions, one giving a lower bound and the other an upper bound for the deviation. Conversely, each of these conditions is found to imply that ∥𝒮f∥≤ 2(1 + δ2). Analogues of the lower bound are also developed for curves in n and for canonical lifts of harmonic mappings to minimal surfaces.

Keywords
univalent function, Schwarzian derivative, Schwarzian norm, hyperbolic metric, two-point distortion, harmonic mapping, minimal surface
Mathematical Subject Classification 2010
Primary: 30C55
Secondary: 31A05, 53A10
Milestones
Received: 23 September 2010
Revised: 1 February 2011
Accepted: 4 February 2011
Published: 7 February 2012
Authors
Martin Chuaqui
Facultad de Matemáticas
P. Universidad Católica de Chile
Casilla 306
Santiago 22
Chile
Peter Duren
Department of Mathematics
University of Michigan
Ann Arbor MI 48109-1043
United States
William Ma
School of Integrated Studies
Pennsylvania College of Technology
Williamsport PA 17701
United States
Diego Mejía
Departamento de Matemáticas
Universidad Nacional
A.A. 3840 Medellín
Colombia
David Minda
Department of Mathematical Sciences
University of Cincinnati
Cincinnati OH 45221-0025
United States
Brad Osgood
Department of Electrical Engineering
Stanford University
Stanford CA 94305
United States