A construction of trivial Beltrami coefficients

Abstract

A measurable function mu on the unit disk D of the complex plane with parallel to mu parallel to(infinity) 1 is sometimes called a Beltrami coefficient. We say that mu is trivial if it is the complex dilatation f(<(z) over bar)/f(z) of a quasiconformal automorphism f of D satisfying the trivial boundary condition f(z) = z, vertical bar z vertical bar = 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 Lowner chains.