Example of holomorphic function from unit disc to itself

Question

let $f:\mathbb{D} \to \mathbb{D}$ be analytic function with $f(0)=0$,where $\mathbb{D}$ is the open disc $\{z \in \mathbb{C}:|z|<1 \}$ then

$1.|f'(0)|=1$

$2.|f(\frac{1}{2})|\leq \frac{1}{2}$

$3.|f(\frac{1}{2})| \leq \frac{1}{4}$

$4.|f'(0)| \leq \frac{1}{2}$

Can I use here Cauchy integral formula? please give me some hint i have no idea about it.Thanks in advance.

Answer 1

The answer is 2. There is a result that if let f:D→D be analytic function with f(0)=0,where D is the open disc {z∈C:|z|<1} then |f(z)|<=|z|. this is where it comes from.