PROBLEM: Let $\{f_n\}_{n=1}^\infty$ be a sequence of entire functions such that for all $z = x+iy \in \mathbb{C}$ there holds $$\sum_{n = 1}^\infty |f_n(x+iy)|^{1/n} \le e^x.$$ Let $f(z) = \sum_{n = 1}^\infty f_n(z).$ Show that $f$ is analytic on $\{z : \text{Re}(z) < 0\}$ and prove that $f$ has period $2\pi i$.
SOLUTION PROGRESS: Let $K \subseteq \{z : \text{Re}(z) < 0\}$ be compact. Then, there exists $\epsilon > 0$ such that $\text{Re}(z) < -\epsilon$ for all $z \in K$. Now, the assumption of the problem implies that $|f_n(x+iy)| \le (e^{x})^n$ for all $n$ and $x+ iy$. Since $e^{-\epsilon} < 1$, by comparing with a geometric series it follows that the series for $f$ converges uniformly on $K$. We see then that on $\{z : \text{Re}(z) < 0\}$ $f$ is locally the uniform limit of analytic functions and hence is analytic.
I'm stuck on the part about proving that the function is periodic. Setting $f_n(z) = 0$ for all $n > 1$ the problem reduces to showing that an entire function satisfying $|f(z)| \le e^{\text{Re}(z)}$ for all $z$ must be periodic with period $2\pi i$. This has been the first simple case that I've been looking at. Any hints on how to proceed?