Prove that a family F of analytic functions in closed unit disc D is locally uniformly bounded in D if and only if F is uniformly bounded in D
Prove that a family F of analytic functions in closed unit disc D, F is locally uniformly bounded in D if and only if F is uniformly bounded in D
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complex-analysis
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1 Answers
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Uniformly bounded $\implies$ locally uniformly bounded is trivial.
For the other direction, for each $z\in D$ let $U_{z}$ be an open neighborhood of $z$ that is both small enough to be contained in $D$ and such that $F$ is uniformly bounded on it, say by $M_{z}$. Then $D$ is covered by $\{ U_{z} | z\in D \}$, hence, since $D$ is compact, there are finitely many $z$'s, say, $z_{1},...,z_{n}$, such that $D$ is covered by $\{U_{z_{1}},...,U_{z_{n}}\}$. Let $M = max\{M_{z_{1}},...,M_{z_{n}}\}$. Then $F$ is uniformly bounded by $M$ on $D$.
You might also want to say something about analytic continuation or you might not.