Consider a conformal mapping of the form

$$f(z)=\frac{a+b z}{c+d z}, \quad z \in \mathbb{C}$$

where $a, b, c, d \in \mathbb{C}$, and $a d \neq b c$. You may assume $b \neq 0$. Show that any such $f(z)$ which maps the unit circle onto itself is necessarily of the form

$$f(z)=e^{i \psi} \frac{a+z}{1+\bar{a} z} .$$

[Hint: Show that it is always possible to choose $b=1$.]