Schwarzian norms and two-point distortion

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.