Check whether the map $f(z)={z\over e^z-1}$ is open or not in $\{z|$Im$(z)>0\}$

Question

As per open mapping theorem, a holomorphic, non-constant function on a region is an open map.
Here the domain $\Omega=\{z\in\Bbb{C}|\operatorname{Im}(z)>0\}$ is a region, $f$ is non-constant.
Here $f$ has removable singularities at the points $2k\pi i$, $k\in\Bbb{Z}$, since it can be easily shown that $\lim_{z\to2k\pi i}(z-2k\pi i)f(z)=0\ \forall k\in\Bbb{Z}$.
So, can I use open mapping theorem here? Thanks for the assistance in advance.