I need some help with this problem:
Let $f\colon D \to D$ analytic and $f(z_1)=0, f(z_2)=0, \ldots, f(z_n)=0$ where $z_1, z_2, \ldots, z_n \in D= \{z:|z|<1\}$. I want to show that $|f(z)| \leq \prod_{k=1}^n \left| \frac{z-z_k}{1-\overline{z_k}\, z} \right|$ for all $z \in D$.
It seems that I need to use Schwarz-Pick Lemma but it seems that the problem doesn't satisfy the conditions. Another lemma that I can use is that of Lindelöf saying: Let $f:D \to D$ analytic, then $|f(z)|\leq \frac{|f(0)|+|z|}{1+|f(0)| \cdot |z|}$ for all $ z \in D$.
It seems to be an easy problem but I couldn't succeed in solving it.