Continuity of the barycentric extension of circle diffeomorphisms with Hölder continuous derivative

The barycentric extension due to Douady and Earle yields a conformally natural extension of a quasisymmetric self‐homeomorphism of the unit circle to a quasiconformal self‐homeomorphism of the unit disk. We consider such extensions for circle diffeomorphisms with Hölder continuous derivative and show that this operation is continuous with respect to an appropriate topology for the space of the corresponding Beltrami coefficients.