$f :\mathbb{D}\rightarrow \mathbb{D}$ is a holomorphic function and $z_0$ is a root of it, How can I prove $|z_0| \geq |f(0)|$

Question

How can I prove this? I don't have any idea about its proof.

Suppose $f:\mathbb{D}\rightarrow \mathbb{D}$ be holomorphic function and $f(z_0) = 0$. Prove $|z_0| \geq |f(0)|$.

Answer 1

When you see functions $f: \mathbb{D} \to \mathbb{D}$ usually the inclination is to use Schwarz's Lemma. Since $0$ is not fixed by the function, it is convenient to introduce the unit disk automorphisms of the form $\varphi_c := \frac{z+c}{1+\overline{c}z}$. These allow you to manipulate the function values to apply Schwarz's Lemma. So, since $f(z_0) = 0$ we want to construct a function that sends $0$ to $0$. The function $\varphi_{-f(z_0)} \circ f \circ \varphi_{z_0}$ will do the job. Now, we can apply Schwarz's Lemma to this and all that needs to be done from there is calculation. This technique is actually how you prove the Schwarz-Pick Theorem, albeit with some modifications since it does not presuppose a root in the hypothesis.