Holomorphic Function and Cauchy Schwartz

Question

Let $f(z)$ be a holomorphic function in the disk $D = {|z| < 1}$, such that $|f(z)| ≤ M$ in D. Let ${a_i}^n_{i=1}$ be the zeroes of $f$, counted with their multiplicities. Show that

$|f(z)|≤ M ∏^n_{i=1} (z−a_k)/(1 − \overline{a}_kz) $

I thought this could be done by induction, but i got stuck. Can someone help out on how to solve this?

Answer 1

You cannot do this by induction. The product on the right is an analytic function which is 1 on the boundary. You have to apply Maximum Modulus Principle to the ratio of f and this function. There are many such inequalities in Complex Analysis which follow from Maximum Modulus Principle.

Answer 2

Hint：For n=1，we can use a auto-isomorphism $\alpha$ of $\mathbb{D}$ which can sent $a_1$ to 0.

Then considering $\frac{1}{M}f\circ \alpha^{-1}$，use Schwartz lemma.