Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$.
Then is it true that $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly on compact subsets (avoiding zeroes of all $f_{n}$ and of $f$) to $\mathrm{Log}(f)$ for appropriate branches of the logarithm?
Are there any books which analyze this questions ?