An extension for quasisymmetric homeomorphisms had previously been given by Lars Ahlfors and Arne Beurling; a different equivariant construction had been given in 1985 by Pekka Tukia.
By the Radó–Kneser–Choquet theorem, the Poisson integral of a homeomorphism f of the circle defines a harmonic diffeomorphism of the unit disk extending f. If f is quasisymmetric, the extension is not necessarily quasiconformal, i.e. the complex dilatation does not necessarily satisfy However F can be used to define another analytic extension Hf of f−1 which does satisfy this condition.
If g is any Möbius transformation preserving the unit circle and disk, then For |a| < 1 define to be the unique w with |w| < 1 and For |a| =1 set In this section it is established that the extension of a quasisymmetric homeomorphism is quasiconformal.
The complex dilatation μ of a diffeomorphism F of the unit disk is defined by If F and G are diffeomorphisms of the disk, then In particular if G is holomorphic, then When F = Hf, where To prove that F = Hf is quasiconformal amounts to showing that Since f is a quasi-Möbius homeomorphism the compositions g1 ∘ f ∘ g2 with gi Möbius transformations satisfy exactly the same estimates, since Möbius transformations preserve the cross ratio.
On the other hand, if f is quasi-Möbius and fixes 1, i and −i, then f satisfies a Hölder continuity condition: for another positive constant C independent of f. The same is true for the f−1's.