Proposition

Suppose $f$ is entire and there exists $M > 0$ such that $\abs{f(z)} > M$ for all $z \in \mathbb{C}$. Prove that $f$ is constant.

Solution

Since $f(z) \ne 0$ for any $z \in \mathbb{C}$, $1/f(z)$ is an entire function. Since $\abs{1/f(z)} < 1/M$, $1/f(z)$ is a bounded entire function. By Liouville’s theorem, $1/f(z)$ is constant. Therefore, $f(z)$ must be constant.