So I have a question:
Let $U \subseteq \mathbb{C}$ be an open set and let $f_{j} : U \rightarrow \mathbb{C}$, $j \in \mathbb{N}$ be a sequence of holomorphic functions. Prove or Disprove: if $\sum_{j = 0}^{\infty} |f_{j}|$ converges normally on $U$, then $\prod_{j = 1}^{\infty} (1 + f_{j}(z))$ converges normally on $U$.
So I think this is false. I tried considering a sequence of holomorphic functions on the unit disc. The only functions I know is $f_{j}(z) = z^{j}$. Am I on the right track? Please give not hints!
Thank you very much
$\prod (1+f_n(z))$ converges uniformly on some set iff $\sum |f_n(z)|$ converges uniformly on that set. This is a well known fact and you can find a proof in Rudin's RCA. Applying this to compact subsets of $U$ we get the result.