Douady–Earle Extensions of Circle Homeomorphisms with One-Point Differentiability at a Hölder Convergence Rate

Let h be a sense-preserving homeomorphism of the unit circle \({\mathbb {S}}\) and \(\Phi (h)\) the Douady–Earle extension of h to the closure of the open disk \({\mathbb {D}}\). In this paper, assuming that h is differentiable at a point \(\xi \in {\mathbb {S}}\) with \(\alpha \)-Hölder convergence rate for some \(0<\alpha <1\), we prove a similar regularity for \(\Phi (h)\) near \(\xi \) on \({\mathbb {D}}\) in any non-tangential direction towards \(\xi \).