Complex Analysis – Analytic Functions in Open Unit Disk

Question

Let $f$ be analytic in open unit disk, we need to show there exist $\{z_n\}$ with $|z_n|<1$ and $|z_n|\rightarrow 1$ then $f(z_n)$ is bounded.

could any one give me Hints for this one?

Answer 1

Hint: If no such sequence exists, $f$ has only finitely many zeros in the open unit disk. Use the Maximum Modulus principle on $p(z)/f(z)$ for a suitable polynomial $p$.