Choquet also generalized the result to the Poisson integral of a homeomorphism from the unit circle to a simple Jordan curve bounding a convex region.
Standard properties of the Poisson integral show that Ff is a harmonic function on |z| < 1 which extends by continuity to f on |z| = 1.
With the additional assumption that f is orientation-preserving homeomorphism of this circle, Ff is an orientation preserving diffeomorphism of the open unit disk.
To prove that Ff is locally an orientation-preserving diffeomorphism, it suffices to show that the Jacobian at a point a in the unit disk is positive.
In that case where the an are the Fourier coefficients of f: Following Douady & Earle (1986), the Jacobian at 0 can be expressed as a double integral Writing where h is a strictly increasing continuous function satisfying the double integral can be rewritten as Hence where This formula gives R as the sum of the sines of four non-negative angles with sum 2π, so it is always non-negative.