SCHWARZIAN NORMS AND TWO-POINT DISTORTION
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Date
2011
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Abstract
An analytic function f with Schwarzian norm parallel to gf parallel to <= 2(1 + delta(2)) 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 parallel to gf parallel to <= 2(1 + delta(2)). Analogues of the lower bound are also developed for curves in R-n and for canonical lifts of harmonic mappings to minimal surfaces.
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Keywords
univalent function, Schwarzian derivative, Schwarzian norm, hyperbolic metric, two-point distortion, harmonic mapping, minimal surface