Show all holomorphic function $f: \Omega\to\mathbb{C}$ s.t. $|f'(z)|\leq e^{|f(z)|}$ forms a normal family, where $\Omega$ is a plane region.

Question

Here I mean normal family of meromorphic functions, which means every sequence ${f_k}$ has a subsequence that converges uniformly on every compact subset of $\Omega$, and the limit is holomorphic or identically equals $\infty$

I tried to show all such functions $f$ are uniformly bounded on every compact subset of $\Omega$ and use Montel theorem. But I find it not so easy… So what else could I try? (Thanks to Theo, now I know that this approach will never work…)