Let \(\Omega\subseteq\CC\) be open and let \(\mathcal F\) be a family of holomorphic functions on \(\Omega\). The family \(\mathcal F\) is uniformly bounded on compact subsets of \(\Omega\) if for every compact set \(K\subseteq\Omega\) there exists \(M>0\) such that \[ \abs{f(z)}\le M\qquad\text{for all } z\in K \text{ and all } f\in\mathcal F. \]
Families uniformly bounded on compact subsets
D-MBDTR