Biholomorphic functions on unit disc

Question

Is it possible to classify all biholomorphic functions $f:\mathbb{D}\backslash[-1,0]\to\mathbb D$ where $\mathbb D$ is the unit disk?

My repertoire of tools is very limited in this regard, especially since $\mathbb{D}\backslash[-1,0]$ is not a connected space, so Riemanns mapping theorem doesn't guarantee that there is one. However, I know that the automorphism group $Aut(\mathbb D)$ contains only elements of the form $\displaystyle f(z)=e^{i\theta}\frac{z-a}{\bar az-1}$. How do I manipulate this information to get the desired solution?

Answer 1

My approach would be to find a biholomorphic conformal mapping $g$ between $\mathbb D \ \backslash \ [-1, 0]$ and $\mathbb D$. For example:

• go from $\mathbb D \ \backslash \ [-1, 0]$ to a half-disk using the $z \mapsto \sqrt{z}$ map
• go from a half-disk to the full disk $\mathbb D$ by following the excellent answer here.

Once we have written down a conformal mapping $g : \mathbb D \ \backslash \ [-1, 0] \to \mathbb D$, we can use your knowledge about $Aut(\mathbb D)$ to infer that the biholomorphic maps $\mathbb D \ \backslash \ [-1, 0] \to \mathbb D$ are precisely the functions of the form $f \circ g$, where $f(z) = e^{i\theta} \frac{z - a}{\bar a z - 1}$ for some $\theta \in [0, 2\pi)$ and $a \in \mathbb D$.

Your strategy of "restricting the functions $f$ given above" (from your comment) won't work, I'm afraid, because the restrictions of these functions to $\mathbb D \ \backslash \ [-1, 0]$ won't be surjective maps to $\mathbb D$.