A construction of trivial Beltrami coefficients

Abstract

A measurable function µ on the unit disk D of the complex plane with µ ∞ < 1 is sometimes called a Beltrami coefficient. We say that µ is trivial if it is the complex dilatation f z /f z of a quasiconformal automorphism f of D satisfying the trivial boundary condition f (z) = z, |z| = 1. Since it is not easy to solve the Beltrami equation explicitly, to detect triviality of a given Beltrami coefficient is a hard problem, in general. In the present article, we offer a sufficient condition for a Beltrami coefficient to be trivial. Our proof is based on Betker's theorem on Löwner chains.