Holomorphic $f$ and $g$ such that $e^{f(z)}+e^{g(z)}=1$ on $\mathbb C$

Question

I have a holomorphic function $f,g: \mathbb C \to \mathbb C$ such that $e^{f(z)} + e^{g(z)} = 1$ for any $z$ in $\mathbb C$. Does anyone have any tips that to show that this is bounded? Or check whether it is bounded?

Answer 1

Use Picard's theorem. If an entire function omits two points it is constant.

Firstly $e^{g(z)}$ and $e^{f(z)}$ can't equal zero, because $e^z$ never equals zero. But just as well, $e^{f(z)} = 1 - e^{g(z)}$ so $e^{g(z)}$ can't equal $1$. Therefore $e^{g(z)}$ is an entire function that omits two points, it must be constant by Picard. Therefore $e^{f(z)}$ is constant. Both are obviously bounded.