I am looking for some results on the boundary behavior of conformal maps between simply connected domains. In particular I am interested in conformal maps between $\mathbb{C}-\Delta$, where $\Delta$ is an interval (in general, $\mathbb{C}-\Gamma$, where $\Gamma$ is a Jordan arc) onto the exterior of the unit disc.
It is well known that every conformal map from $\mathbb{C}-\Gamma$ onto the exterior of the unit disc can be extended continuously to the boundary (Carathéodory's theorem). In the book *Boundary Behavior of Conformal Maps*, by Christian Pommerenke I found the following theorem:

Theorem 2.6Let $f$ map $\mathbb{D}$ conformally onto the inner domain of the Jordan curve $C$ of class $\mathcal{C}^{n,\alpha}$ where $n=1,2,3,\ldots$ and $0<\alpha<1$. Then $f^{(n)}$ has a continuous extension to $\overline{\mathbb{D}}$ and $$|f^{(n)}(z_1)-f^{(n)}(z_2)|\leq M|z_2-z_2|^\alpha,\;\; z_1,z_2\in\overline{\mathbb{D}}$$

Is there any reference to an analogue theorem from $\mathbb{D}$ onto $\mathbb{C}-\Gamma$?