What does bounded in module mean?

Question

I'm trying to understand Liouville's theorem and in the book I'm reading it's stated as Let $f$ be a holomorphic function on the complex plane, which is bounded in module in a neighborhood of infinity (i.e. bounded in module in all of C), then $f$ is a constant function.

What exactly does "bounded in module" mean?

Answer 1

Bounded in absolute value. That is, there exists $M > 0$ such that $|f(z)| \leq M$ for all $z \in \mathbb{C}$.

Answer 2

It means that $|f|$ is bounded in a neighbourhood of infinity. Since we don't have $\leq$ available in $\Bbb C$, to talk about boundedness in $\Bbb C$ we must appeal to $|\cdot|$.