Let $\mathbb D$ be the unit disk and let $a\in \mathbb D$. Find a holomorphic map $f: \mathbb D \rightarrow \mathbb D$ that interchanges $0$ and $a$.
Attempt:
Define $f: \mathbb D \rightarrow \mathbb D$ as $f(z)= \frac{a-z}{1-\overline a z}$
will take $0 \to a \;\& \;a\to 0$.
It will be holomorphic as it's a linear fractional transformation.
To show $f: \mathbb D \rightarrow \mathbb D$, we need to show $\frac{a-z}{1-\overline a z} \in \mathbb D$.
How to do that ?
