Here, $A(\mathbb{D})$ represents the collection of all holomorphic functions on the unit disc. Let $\{f \in A(\mathbb{D}): f(z)=\sum_{n=0}^{\infty}a_nz^n , |a_n| \leq 1, \: \forall n \geq 1 \}$. Does the family normal or not?
If we can prove that the family is locally uniformly bounded, then by Montel theorem is normal family. These functions can be represented as power series at origin, so we conclude that
$|f(z)| \leq |f(0)| + \frac{r}{1-r}$, for all $z \in D(0,r)$ for some $r <1$.
This does not show the locally uniformly bounded for the family. But can I say that the family is not locally uniformly convergent and so it is not normal family? or it is insufficient work.