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