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Schwarzian norms and
two-point distortion
Martin Chuaqui, Peter Duren, William Ma, Diego Mejía, David Minda and Brad Osgood |
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Vol. 254 (2011), No. 1, 101–116
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Abstract |
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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 ℝn 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
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Mathematical Subject Classification 2010
Primary: 30C55
Secondary: 31A05, 53A10
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Milestones
Received: 23 September 2010
Revised: 1 February 2011
Accepted: 4 February 2011
Published: 7 February 2012
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Authors |
| Martin Chuaqui | |
| Facultad de Matemáticas P. Universidad Católica de Chile Casilla 306 Santiago 22 Chile |
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| Peter Duren | |
| Department of Mathematics University of Michigan Ann Arbor MI 48109-1043 United States |
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| William Ma | |
| School of Integrated Studies Pennsylvania College of Technology Williamsport PA 17701 United States |
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| Diego Mejía | |
| Departamento de Matemáticas Universidad Nacional A.A. 3840 Medellín Colombia |
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| David Minda | |
| Department of Mathematical
Sciences University of Cincinnati Cincinnati OH 45221-0025 United States |
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| Brad Osgood | |
| Department of Electrical
Engineering Stanford University Stanford CA 94305 United States |
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