Cross‐ratio distortion and Douady–Earle extension: I. A new upper bound on quasiconformality

Let f be an orientation‐preserving circle homeomorphism and Φ ( f ) the Douady–Earle extension of f , and let || f || cr be the cross‐ratio distortion norm of f and K (Φ ( f )) be the maximal dilatation of Φ ( f ). As a consequence of results in [A. Douady and C. J. Earle, ‘Conformally natural extension of homeomorphisms of circle’, Acta Math . 157 (1986) 23–48 and M. Lehtinen, ‘The dilatation of Beurling–Ahlfors extensions of quasisymmetric functions’, Ann. Acad. Sci. Fenn. Ser. A I Math . 8 (1983) 187–191], ln K (Φ ( f )) has an upper bound depending on || f || cr exponentially. In this paper, we first show that ln K (Φ ( f )) has an upper bound depending on || f || cr linearly. Then we extensively study the Douady–Earle extension of a very simple map f λ which depends on a real non‐negative parameter λ. For this example, we show that, as λ→∞, (1) our new upper bound on ln K (Φ ( f λ )) is substantially smaller than the one given by Douady and Earle in terms of the maximal dilatation of the extremal extension of f λ , and (2) the Douady–Earle extension Φ ( f λ ) stays exponentially far away from being extremal. Finally, we show that, in general, our upper bound on ln K (Φ ( f )) implies the one of Douady and Earle when || f || cr is large enough.