Two-point distortion theorems for harmonic mappings

Abstract

In earlier work the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in R n are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form. §1. Introduction.The classical Koebe distortion theorem gives sharp bounds on the derivative of a normalized analytic univalent function. Another measure of distortion is the distance |f (z 1 ) − f (z 2 )| between the images of two arbitrary points in the disk. Some years ago, Blatter [3] gave a sharp lower bound for this distance in terms of the hyperbolic distance between z 1 and z 2 . More recently, Chuaqui and Pommerenke [10] found a sharp two-point distortion theorem for functions whose Schwarzian derivative satisfies Nehari's condition |Sf (z)| ≤ 2(1 − |z| 2 ) −2 . Their result may be viewed as a quantitative form of Nehari's univalence criterion. The main purpose of the present paper is to carry out a similar analysis for harmonic mappings, or rather for their canonical lifts to minimal surfaces. Along the way we obtain distortion theorems for curves in R n , thereby recasting an injectivity criterion of Chuaqui and Gevirtz [7] in quantitative form.An important tool throughout the paper is the classical Sturm comparison theorem for solutions of linear differential equations of second order. A good reference for this topic is the book of Birkhoff and Rota [2].