This is a question from an old qualifying exam that I was trying to solve for practice:
Prove or disprove that there exist nonconstant holomorphic functions $f$ and $g$ on the open unit disk $D=\{z\in \mathbb{C}: |z|<1\}$ such that $e^{f(z)}+e^{g(z)}=1$ for all $z\in D$?
My initial thought was to show it is true on the real line and then use the uniqueness principle. But the examples I came up with on the real line use logarithms, which will not be continuous when I consider them as functions on the unit disk.
Since this is for practice, I was hoping only for hints and suggestions instead of complete answers. Thank you!