I'm having trouble with an exercise about holomorhpic functions. Let $\Omega \subset \mathbb{C}$ be an open convex set and let $F \subset H(\Omega)$ be a relatively compact set in the open-compact topology. Show that $F'=\{f':f \in F\}$ is relatively compact. Give a counter example that the reciprocal is not true and give an additional condition to ensure that if $F'$ is relatively compact then $F$ is relatively compact.
I tried using sequentially compactness of $F$ to prove sequentially compactness of $F'$, but I asked my professor and he said that we don't know if $H(\Omega)$ is sequentially compact. He told me that I should try using Ascoli-Arzelá Theorem, but I need uniform boundedness of $F'$ and I don't know how to prove that.
I have the counterexample for the second part,I defined $F=\{f_n : n\in \mathbb{N}\}$ with $f_n(z)=n$, so that $F'=\{0\}$ that is compact but $F$ is not pointwise bounded, so by Ascoli-Arzlá Theorem it can't be relatively compact. But I have no clue in what condition should I give to have the recyprocal.
Can anyone help me or give me a hint on how to do this exercise?