Prove that if $\mathcal{F}'$ is normal and $\sup_{f ∈ \mathcal{F}} \{ |f(z_0)| \} < +∞$, then $\mathcal{F}$ is normal

Question

The question

Let $Ω$ be a domain in $ℂ$. Let $\mathcal{F}$ be a family of holomorphic functions at $Ω$, and let $\mathcal{F}' = \left\{ f':\ f∈\mathcal{F} \right\}$ be the family of derivatives of $\mathcal{F}$.

1. Show that if $\mathcal{F}$ is a normal family, then $\mathcal{F}'$ is also normal.
2. Give an example to show that the reciprocal is incorrect.
3. Prove that if $\mathcal{F}'$ is normal and for some $z_0 ∈ Ω$ we have $\sup_{f ∈ \mathcal{F}} \{ |f(z_0)| \} < +∞$, then $\mathcal{F}$ is normal.

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What I can't solve

I've been able to solve 1. (as a Weierstrass consequence) and 2.

But I don't know how to even try 3.

Any help?

Answer 1

Here is what I've reached using the comment:

$$ f(z) = f(z_0) + \int_γ f'(w)\ dw, $$

where $γ$ is a path that connects $z_0$ and $z$. By Paul-Montel we know that $\mathcal{F}'$ is uniformly bounded over compacts of $Ω$, so if we take a compact such that $γ ⊂ K⊂Ω$, we can find a bound for the integral:

$$ |f(z)| ≤ |f(z_0)| + \int_γ \left|\ f'(w) \right|\ dw ≤ |f(z_0)| + M_K · \text{arc-length}(γ), $$

where $M_K$ is the bound for every $f'$ at $K$.

Then we have that every $f \in \mathcal{F}$ is uniformly bounded, and using Paul-Montel again we get that $\mathcal{F}$ is normal.