When does the normality of the family of derivatives imply that the family itself is normal? Let $\mathcal{F}$ be a family of analytic functions in a domain $G$. Denote by $\mathcal{F}':=\{ f' \mid f \in \mathcal{F}\}$. It does not suffice to know that $\mathcal{F}'$ is normal to ensure that $\mathcal{F}$ is normal (namely take $\mathcal{F} = \{ n \mid n \in \mathbb{N}\}$ as a counterexample) but someone assured me that if you assume at some fixed point $z_0 \in G$, if you know that $f(z_0) = 0$ for all $f \in \mathcal{F}$ and that $\mathcal{F}'$ is normal, then $\mathcal{F}$ is normal. But I can't see how to do it. 
A friend tried something with line integrals to show that the $f$ are locally bounded on compact subsets, but he only considered compact sets that were the image of a path, which I don't think is sufficient. Which characterization of normality should I try (local boundedness? convergence in the space of functions?)? Does anyone have an idea? 

I may have worked it out. If anyone finds an issue, let me know!
Let $\{f_n\}$ be a sequence in $\mathcal F$. It suffices to show that $\{ f_n\}$ has a subsequence which converges uniformly on compact subsets of the domain $G$. Let $\{f_n'\}$ be the corresponding sequence of derivatives. Since $\mathcal F'$ is a normal family, we have that $\{ f_n'\}$ has a convergent subsequence, $\{f_{n_k}'\}$ which converges uniformly on compact subsets to a function $g \in \mathcal F'$. By the definition of $\mathcal F'$, we must have that $g = f'$ for some $f \in \mathcal F$.
Now we'd like to show that $\{f_{n_k}\}$ converges uniformly on compact sets to $f$. But since the $f_{n_k}' \rightarrow f'$, applying the Fundamental Theorem of Calculus and continuity of the integral we have $\int f_{n_k}' \rightarrow \int f'$ so $f_{n_k}(z) + C_{n_k} \rightarrow f(z) + C$. But since each $f_{n_k}$ and $f$ is in $\mathcal F$, we have that $f(0) = f_{n_k}(0) = 0$ and so the $C_{n_k} \rightarrow C$. Thus $f_{n_k} \rightarrow f$ uniformly on compact subsets, and so since the sequence $\{f_n\}$ was arbitrary, $\mathcal F$ is a normal family.
I'm worried about playing with the constants of integration (basically the FTC step in general) but I think it should work.
