How to show that if $f:\mathbb{C}\to \mathbb{C}$ is an entire function with $f(\mathbb{C})\subset G:=\mathbb{C}\setminus [0, \infty)$, then $f$ is constant.
Approach: I showed that there is a conformal map $g$ that maps $G$ into the unit disc $\mathbb{D}$. So $g\circ f$ is an entire function that maps $\mathbb{C}$ into the unit disc. This function is bounded, so by Liouville's theorem it is constant. How can I conclude from this that $f$ should be constant? Thanks in advance.