Schwarzian norms and two-point distortion

Abstract

An analytic function f with Schwarzian norm f ≤ 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 f ≤ 2(1 + δ 2 ). Analogues of the lower bound are also developed for curves in ‫ޒ‬ n and for canonical lifts of harmonic mappings to minimal surfaces.

Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions

“…Chuaqui and Pommerenke in [4] as well as Chuaqui et.al. in [2] established the sharp bounds on the distortion |f ′ (z 1 )f ′ (z 2 )| for the analytic and locally univalent functions in U under certain restrictions imposed on the Schwarzian norm defined by…”