Demonstrate inequality of holomorphic and non-constant functions

Question

Would you know how to demonstrate the following inequallity?

Every holomorphic and non-constant function $f:\mathbb D \to \mathbb D$ satisfies: $$ \frac{|f(0)| - |z|}{1+|f(0)||z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1-|f(0)||z|} $$ and $$ |f'(z)| \leq \frac{1-|f(z)|^2}{1-|z|^2} $$

Thanks.

Answer 1

Hint: First look at https://en.wikipedia.org/wiki/Schwarz_lemma. Also, the map $\phi_a: z\mapsto \frac{a-z}{1-\bar{a}z}$ is a biholomorphic map from $\mathbb{D}$ to itself. What do you get when you take $\phi_{f(0)}\circ f$?