Let $f$ be a nonvanishing entire function of finite order. It follows immediately from Hadamard's factorization theorem that $f(z)=e^{P(z)}$ for some polynomial $P(z)$ whose degree is at most the order of $f$. However, I'm wondering if I can prove this in a more elementary way.
Let's say for simplicity that I know $|f(z)|\leq e^{|z|^5}$ for all $z\in\mathbb C$. Then I know I can write $f(z)=e^{g(z)}$ for some entire function $g$. The inequality tells me that $\Re(g(z))\leq |z|^5$ for all $z$. At this point I'm not sure how I can deduce that $g$ must be a polynomial of degree at most $5$.