Prove if $f$ and $g$ are entire and $e^f+e^g=1$, then $f$ and $g$ are constant.
I believe the simplest way would be to use Louiville’s theorem by using Pick’s theorem but I am not sure on how to go about this.
Use the little Picard theorem:
If a function $f : \mathbb C\rightarrow \mathbb C$ is entire and non-constant, then the set of values that $f(z)$ assumes is either the whole complex plane or the plane minus a single point.
The range of $e^f$ doesn’t contain $0$. It also doesn’t contain $1$ (otherwise we would have: $1 + e^g = 1 \implies e^g = 0$, a contradiction). Hence, it’s constant.