I would like to show that "if holomorphic function $f,g$ satisfies $f(z)^d+g(z)^d=1$ for all $z\in\mathbb{C}$ (where $d\in\mathbb{Z}_{\geq3}$), then both $f$ and $g$ is constant." I know that when $d=2$, there are some counterexamples (e.g. $f(z)=\cos z,~g(z)=\sin z$), but I have no idea how the difference between $d=2$ and $d\geq 3$ works. I thought the max principle might have something to do with it, but I never figured it out.
Are there any good ideas?
Thank you.