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Prove that a family $F$ of analytic functions in $D(0,1)$ is locally uniformly bounded on closed unit disk $D(0,1)$ if and only if $F$ is uniformly bounded in $D(0,1)$.

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This is false as written. For instance, let $F$ be the family consisting only of the function $f(z) = 1/(z-1)$. Then $F$ is locally uniformly bounded, but not uniformly bounded. –  froggie Oct 17 '12 at 12:47
Do you mean functions in the closed unit disk? (Then it would be true, since anything that is locally uniformly bounded on a compact set $K$ is uniformly bounded on $K$. This is a topological fact, you don't need analyticity or anything really.) –  Lukas Geyer Oct 17 '12 at 14:58

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