Proving two entire functions are constant.

Solution 1:

Here's a proof adapted from Remmert's book Classical Topics in Complex Function Theory, page 236.

Suppose $g\neq 0$. Since $f$ and $g$ cannot have common zeros, $f/g$ is a meromorphic function that takes the value $w$ at $z$ if and only if $f(z)=wg(z)$.

We can factor the given equation as

$$1=\prod_1^n (f-\zeta_ig),$$

where the $\zeta_i$ are roots of $x^n+1$. Dividing through by $g$, we see $f/g$ cannot take the (distinct) values $\zeta_i$. By Picard's theorem for meromorphic functions, a meromorphic function that omits $3$ values is constant. So $f/g$ is constant, $f=cg$ for a constant $c$, and the rest follows easily.