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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.

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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 $r$.

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  • $\begingroup$ Is this equivalent to the statement " $|f|$ does not exceed the mean value of $|f|$ on a small disc centered at 0"? $\endgroup$ – The Substitute May 13 '15 at 4:11

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