I am having difficulties with the following problem:
$\bf Given$: $ f$ is an analytic map from unit disk $D$ to itself and: $f(0)=0$. $\bf To \; prove:$ $|f(z)|\leq |z|$ for $z\in D$; and: $|f'(0)|\leq 1$
What I thought is: $$|f(z)|\leq 1,$$ because $f$ maps to unit disk.
This is apparently wrong. But how then should I approach this problem?