Reference on boundary behavior of conformal maps

Question

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 $\mathbb{C}-\Gamma$?

Answer 1

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.