nth derivative of an exterior conformal mapping in complex analysis

Question

I would like to ask a question about an exterior conformal mapping. Could you please help me?

Let $L$ be a Jordan curve in $\mathbb{C}.$ Let $G$ be the exterior of $L.$ Then, there exists a conformal mapping $\Phi:G\rightarrow \overline{\mathbb{C}}\setminus \{z:|z|\leq 1\}$ such that $\Phi(\infty)=\infty$ and $\Phi'(\infty)>0.$ My question is

"Is the $n$-th derivative $\Phi^{(n)}(z)\not=0$ for all $z\in G$?"

Thank you so much.

Mike

Answer 1

Take $L$ the unit circle and $\Phi$ the identity.Obviously $\Phi'' = 0$