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I am trying to show the following proposition:

if $\{ f \}$ is a locally uniformly bounded family of holomorphic functions on some domain, $D$. Then $\{ f' \}$ is also locally uniformly bounded.

I'm assuming that a locally uniformly bounded family of functions is actually a family of locally uniformly functions, and this so-called family is actually just a set - or is it not? If so, why not call it set?

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Yes, "family" is the same as "set". Saying it is locally uniformly bounded means that for every $z_0 \in D$ there exist $M$ and $r > 0$ such that $|f(z)| < M$ for all $f$ in the family and all $z$ with $|z - z_0| < r$.

Hint: express $f'$ in terms of a contour integral involving $f$.

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