Let $\mathcal{F}$ be a family of holomorphic functions on a common domain $U \subset \mathbb C$. Suppose that $\mathcal{F}$ is pointwise bounded in the sense that for each $z \in U$, there is a constant $b > 0$ (potentially dependent on $z$) such that $|f(z)| \leq b$ for all $f \in \mathcal{F}$. Must it hold that $\mathcal{F}$ is uniformly bounded on each compact subset of $U$?
Using a covering argument, I can show that this holds provided $\mathcal{F}$ is equicontinuous. Moreover, the set $\mathcal{F} = \{f(z) =1/z\}$ is a counterexample if we don't require the sets on which the functions are bounded be compact. However, I am unsure about whether this is true as stated.