Rational function which is bijective on unit disk

What is the general form of a rational function which is bijective on the unit disk?

I’m stuck on this problem. If I let $R(z) = \frac{a_0(z-a_n) \cdots (z-a_1)}{(z-b_m)\cdots (z-b_1)}$, then exactly one of the $a_i$ must fall in the unit disk and none of the $b_i$ can be in the unit disk.

Any hints on how to proceed? I see simple functions like $z$ work, but I don’t know how to show that higher combinations of them are bijective.

Solutions Collecting From Web of "Rational function which is bijective on unit disk"

The only bijective holomorphic functions from the unit disk onto itself are of the form $e^{i\theta}\psi_\alpha$, where $\theta$ is real and
$$
\psi_\alpha(z) = \frac{z – \alpha}{1-\bar{\alpha}z}
$$
with $|\alpha| < 1$. For a proof, consider any bijective holomorphic function $f$ from the unit disk onto itself. Choose an appropriate value of $\alpha$ in the disk, and then apply the Schwarz lemma to $f\circ \psi_\alpha$ and its inverse.


Added. Here’s a bit more detail, as requested by @Hope. We use the fact that a holomorphic bijection has a holomorphic inverse. First, a computation shows that $\psi_\alpha$ maps the unit disk $\mathbb D$ into itself, and since the inverse can be computed explicitly as $\psi_{-\alpha}$, it is bijective. Now assume $f : \mathbb D \to \mathbb D$ which is bijective, and choose the unique point $\alpha \in \mathbb D$ satisfying $f(\alpha) = 0$. Consider $g = f\circ \psi_{-\alpha}$, and note that $\psi_{-\alpha}(0) = \alpha$. By the preceding remarks, $g$ is a holomorphic bijection of $\mathbb D$ which fixes the origin. By the Schwarz lemma, one has $|g'(0)| \leq 1$ with equality if and only if $g$ is a rotation. The inverse map $h = g^{-1}$ also maps $\mathbb D$ into itself, and satisfies $h(0) = 0$, so we have $|h'(0)| \leq 1$ as well. But $h'(0) = 1/g'(0)$, so we find that $|g'(0)| = 1$ after all. By the case of equality in the Schwarz lemma, $g$ is a rotation, and therefore $f = e^{i\theta}\psi_{\alpha}$ for some real $\theta$.