Let $F$ be the class of all holomorphic functions on the unit disc $D$ satisfying
$$\int_0^{2\pi}|f(re^{i\theta})|d\theta\le1$$
for each $r$ in $(0,1)$.
Prove that $F$ is a normal family in the unit disc $D$.
I have a hint : use Montel's theorem (If $F$ is bounded by for each cpt subset $K$ of $D$, then $F$ is a normal family in $D$)
I tried to show that $F$ is bounded directly, or start with an assumption that $F$ is not bounded in some compact subset to derive a contradiction. But they are not successful.
Any help will be appreciated. Thank you.
Best Answer
In order to bound $f$ on the closed ball centered at $0$ and radius $R$, pick $r\in (R,1)$ and use Cauchy's integral formula where $\gamma$ is the circle centered at the origin and with radius $r$.