I'm studying for an exam an I came across a problem that I am having difficultly solving.
Let $\mathcal{F}$ is a family of analytic functions on the closed unit disc, $D$.
Suppose
$\int_{D} |f|^{2} dA \le 1$
for all $f \in \mathcal{F}$.
Can I conclude that $\mathcal{F}$ is locally bounded?