Publication: Schwarzian norms and two-point distortion
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Date
2011-11
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Abstract
An analytic function f with Schwarzian norm
φƒ
≤ 2(1+δ^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
φƒ
≤ 2(1+δ^2). Analogues of the lower bound are also developed for curves in Rn and for canonical lifts of harmonic mappings to minimal surfaces.
φƒ
≤ 2(1+δ^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
φƒ
≤ 2(1+δ^2). Analogues of the lower bound are also developed for curves in Rn and for canonical lifts of harmonic mappings to minimal surfaces.