Let $\gamma:[0,2\pi] \to \mathbb C$ be a $C^{1,\alpha}$ Jordan curve. ($0<\alpha<1$)
More precisely, $\gamma'$ is nonvanishing and is of class $C^{\alpha}$ (as a periodic function on $\mathbb R$).
By the Riemann mapping theorem, there exists a conformal map $f$ onto the interior component of $\gamma$, defined on the open unit disk.
I want to see a proof of the following theorem. (or, at least a reference)
$f$ can be extended to a (non-analytic) homeomorphism of the complex plane onto itself, with non-vanishing Hölder-continuous gradient.
For anyone's interest, this is an exercise from the book "Pommerenke - boundary behaviour of conformal maps", page 50.