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Quasihomography is a useful notion to represent a sense-preserving automorphism of the unit circle T which admits a quasiconformal extension to the unit disc. For K ≥ 1 let AT (K) denote the family of all K-quasihomographies of T . With any f ∈ AT (K) we associate the Douady–Earle extension Ef and give an explicit and asymptotically sharp estimate of the L∞ norm of the complex dilatation of Ef . Introduction. Let AT denote the family of all sense-preserving automorphisms of the unit circle T . With any f ∈ AT we associate the Douady–Earle extension Ef which is a homeomorphic automorphism of the unit disc ∆ and has a continuous extension to f on the boundary T = ∂∆ (see [DE] and [LP]). If z ∈ ∆ and f ∈ AT , then Ef (z) is the unique w ∈ ∆ such that

The Douady-Earle extension of quasihomographies