# Thread: Analytic Function in the Unit Disk

1. ## Analytic Function in the Unit Disk

Dear Colleagues,

Could you please help me in solving the following function:

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

Best Regards,
Raed.

2. ## 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))