Boundedness of Complex Functions

Definition: Boundedness

Let $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function.

We say that $B>0$ is a bound of $f$ on $S\subseteq \mathcal{D}$ if

$$|f(z)|\le B\forall z\in S$$

We say that $f$ is bounded on $S$ if it has a bound on $S$.