Problem on holomorphic function (Schwarz's Lemma)

Question

Let $f$ be a holomorphic function on unit disc $D:|z|<1$. (a) $z_1, z_2, ..., z_n$ are the zeros of $f$ on $D$. Suppose that $|f(z)|\leq M$. Show that $$|f(z)|\leq M \prod_{i=1}^{n} \frac{|z-z_i|}{|1-\bar{z_i}z|}$$ I know that $\frac{z-z_i}{1-\bar{z_i}z}$ maps the unit disc to itself, and maps $z_i$ to 0. So $\frac{|z-z_i|}{|1-\bar{z_i}z|}$ should be no greater than 1. But it's the wrong direction. I think that Schwarz's lemma should be used(but I don't know how). Thanks for help.

Answer 1

WLOG take $M = 1$. Let us consider (a finite Blaschke product) $\displaystyle g(z) = e^{i\phi}\prod_{i=1}^n\frac{z - z_i}{1-\overline{z_i}z}$, $z_i \in \mathbb{D}$ with the same roots as $f$. The functions $\displaystyle \left\vert\frac{g}{f}\right\vert$ and $\displaystyle \left\vert\frac{f}{g}\right\vert$ are both holomorphic in $\mathbb{D}$, with the property that $\displaystyle \left\vert\frac{f}{g}\right\vert \leq 1$ on the unit circle ($\partial \mathbb{D}$). Thus, $\displaystyle \left\vert\frac{f}{g}\right\vert \leq 1$ in the unit circle by the maximum modulus principle. Thus, $\displaystyle \vert f(z) \vert \leq \vert g(z) \vert$.