Analytic Function in the Unit Disk

Dear Colleagues,

Could you please help me in solving the following function:

Let $\displaystyle f$ be analytic function in the unit disk $\displaystyle D=\{z \in C:|z|<1\}$ with $\displaystyle |f(z)|<1$, for any $\displaystyle z \in D$. If $\displaystyle f(z_{1})=z_{1}, f(z_{2})=z_{2}$ where $\displaystyle z_{1}$ and $\displaystyle z_{2}$ are distinct, show that $\displaystyle f(z)=z$ for any $\displaystyle z \in D$.

Best Regards,

Raed.

Re: Analytic Function in the Unit Disk

Use mobius transform g that maps 0 to z1, then apply Schwarz lemma - Wikipedia, the free encyclopedia to the composed function g^{-1}(f(g))