I am trying to prove that if $f: B(0,1) \rightarrow B(0,1)$ is an analytic bijection then there exists $a\in\mathbb{C}$ and $t\in\mathbb{R}$ such that $\forall z\in B(0,1)$ $$f(z)=e^{it}\frac{z-a}{1-\bar{a}z}$$
I can see that the required form is a Möbius map, so I'm guessing it is necessary to prove it is a Möbius map somehow. I have tried using the Riemann Mapping Theorem, but this doesn't seem to be getting me anywhere.
Any help with this would be much appreciated.