Let $f$ be holomorphic on the unit disc and continuous on the unit circle. Suppose there is an $M \in \mathbb{R}$ such that $|f(z)| \leq M$ on the unit circle and let $\alpha_1, \alpha_2, ..., \alpha_n$ be zeros of $f$ in the unit disc listed according to multiplicity. Show that $|f(z)| \leq M \frac{|z-\alpha_1| \cdots |z- \alpha_n|}{|1-z \overline{\alpha_1}| \cdots |1-z \overline{\alpha_n}|}$.
Why can't I apply the Maximum Modulus theorem to $f$ directly? Is there something I am missing?