TITLE: Entire function - exponent?
QUESTION [5 upvotes]: Suppose that $f(z)$ is an entire function and $ |f(z)| \le e^x \ (z = x + iy) $ throughout the complex plane. What can be said about $ f(z) $?

REPLY [6 votes]: Since $\left|e^z\right| = e^x$, we have:
$$
\left|\frac{f(z)}{e^z}\right| \le 1
$$
It follows that $f(z)/e^z$ is entire and bounded, hence constant by Liouville's theorem. Therefore, $f(z) = \lambda e^z$ for some $\lambda \in \mathbb C$, $|\lambda| \le 1$.