Reference on boundary behavior of conformal maps

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 2.6 Let $$f$$ 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 $$\mathbb{C}-\Gamma$$?

1. You stated Caratheodory's theorem incorrectly. Conformal map of $$C\backslash\Gamma$$, where $$\Gamma$$ is a Jordan arc, onto the exterior of the unit disk NEVER extends to a continuous map between closures. It the INVERSE map that extends.
2. The more general statement of Caratheodory's theorem says that a conformal map of the unit disk onto a region $$D$$ extends to a continuous map of the closure of the disk to the closure of $$D$$ if and only if the boundary of $$D$$ is locally connected. This contains the case you are asking about. The good reference for this more general theorem is J. Milnor, Dynamics in one complex variable: introductory lectures (available free on the Internet).
3. To prove smoothness, you "open" your arc. Let $$a,b$$ be endpoints. Suppose for simplicity that $$a=-1$$, $$b=1$$, which does not restrict generality. The composition $$g(z)=J^{-1}(f(z))$$, where $$J(z)=1/2(z+1/z)$$ is the Joukowski function, is well defined and analytic in the unit disk for any fixed branch of the $$J^{-1}$$ by the Monodromy theorem. Then $$g(z)$$ maps the unit disk onto a Jordan region, and Pommerenke's statement applies to it. Then see what this implies for $$f$$ by differentiating the composite function. It implies the same except at the preimages of $$a$$ and $$b$$ on the unit circle. But the boundary of your region is not smooth at these points.
• Thanks. Sorry for the mistake, but I understand the closure of $\mathbb{C}-\Gamma$ taking the limits in the upper and lower banks of the arc. I just search in Milnor's book but there is no the result I am looking for. I need a stronger result than Carathéodory's, because I am looking for smoothness on the boundary of the conformal map. – Luis Giraldo Gonzalez Apr 10 '20 at 0:05
• In this case, what is $f$? – Luis Giraldo Gonzalez Apr 10 '20 at 8:24
• $f$ is the function mapping the unit disk on $\overline{C}\backslash\Gamma$. – Alexandre Eremenko Apr 10 '20 at 11:28