$\mathbb{D}$ is the unit disc, $f: \mathbb{D} \rightarrow \mathbb{D}$ is analytic s.t $|f{'}(0)| = 1 - |f(0)|^2$, then $f \in $ Aut$(\mathbb{D})$

Question

Can someone please describe how to tackle this problem? I am lost for ideas. I do know that automorphisms of the unit disc are of the form $z \mapsto e^{i\phi}\frac{z - \alpha}{1 - \overline{\alpha}z}.$ So I'm guessing we are to show $f$ is of this form. Thanks for your help.

Answer 1

Let $\phi(z)=\frac {z-f(0)} {1-\overline {f(0)} z}$ and apply Schwarz Lemma the the function $\phi\circ f$. [Note that $(\phi\circ f)'(0)=1$].