This is the problem:
Let $0\in\Omega\subset \mathbb{C}$ connected, open neighborhood of the origin. Let $f,g:\Omega\to\mathbb{C}$ holomorphic functions such that $f(0)\neq0\neq g(0).$ Let $h:\Omega\to\mathbb{C}$ to be $h(z)=f(z)\overline{g(z)}z^a\overline{z}^b$ with $a\neq b,$ positive integers. Take $U$ a neiberhood of the origin. show that $h(U)$ is a neighborhood of the origin.
Clearly, the only thing I have to prove is that $h(U)$ is open. $h$ is not holomorphic but I've tried to mimic the proof for the open map theorem with no succes. I've tried to some arguments with isolated zeros of $f$ and $g$ but no luck.