Montel theorem - Knowlpedia

Statement

Let $D\subseteq\mathbb C$ be a domain, and let $\mathcal F$ be a family of holomorphic functions $D\to\mathbb C$ . Suppose that $\mathcal F$ is locally uniformly bounded: for every compact set $K\subset D$ , there is a constant $M_K$ such that

|f(z)|\le M_K \qquad(f\in\mathcal F,\ z\in K).

Then $\mathcal F$ is a normal family.

Sequential conclusion

Every sequence in $\mathcal F$ has a subsequence converging uniformly on compact subsets of $D$ to a holomorphic function. The locally uniform boundedness prevents the subsequence from escaping locally uniformly to $\infty$ .

Proof idea

On every relatively compact disc, the Cauchy estimates give common derivative bounds. The family is therefore equicontinuous there. The Arzelà–Ascoli theorem gives convergent subsequences on an exhaustion by compact sets, and a diagonal argument produces one subsequence converging locally uniformly throughout $D$ .

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter VII.