I have to decide if the following families are normal.
1) $\mathscr{F}=\{f\in \mathscr{H}(D): f(0)=0\ \& \ |f'(z)|(1-|z|^2)\leq1,\ \forall z\in D\}$
2) $\mathscr{F}=\{f\in \mathscr{H}(D): f(0)=0\ \& \ |f''(z)|(1-|z|^2)\leq1,\ \forall z\in D\}$
I want to use the Montel's theorem, so I have to see if the families are uniformly bounded, but I don't know how to use the conditions on the derivatives. Hints?