Analyticity of Products

Question

Assume we have two functions $f,g:\Omega\rightarrow\mathbb{C}$ that are analytic and a third function $h:\Omega\rightarrow\mathbb{C}$ with $f=g\cdot h$. Can one now show that $h$ is analytic as well?

Of course $\Omega\subset\mathbb{C}$ is open.

Answer 1

No, you can't: try $f(z) = z$, $g(z) = z^2$, $h(z) = 1/z$ for $z \ne 0$ with $h(0)$ arbitrary.