TITLE: $f,g,h$ be a holomorphic functions such that $|f(z)|+|g(z)|+|h(z)|=1$
QUESTION [5 upvotes]: Let $U\subseteq \mathbb{C}$ be a connected open set, and let $f,g,h:U\to\mathbb{C}$ be holomorphic functions such that $$|f(z)|+|g(z)|+|h(z)|=1$$ for all $z\in U$. How does one prove that $f,g,h$ are constant functions?
Any hints would be appreciated.

REPLY [7 votes]: We recall the following identity for harmonic functions
$$\Delta (u^2 + v^2) = 2(|\nabla u|^2 + |\nabla v|^2).$$
Considering a more general problem, suppose $$\sum_{i=1}^n |f_i(z)| =1$$
for all $z\in U$ for some set of holomorphic functions $f_i$.
Working locally, we may assume that $f_i= g_i^2$ for some holomorphic functions $g_i=u_i+iv_i$. Then taking the Laplacian of both sides yields
$$\sum_{i=1}^n |\nabla u_i|^2 + |\nabla v_i|^2 = 0.$$
Hence the gradients of the real and imaginary parts of $u_i$ and $v_i$ vanish for all $i$, so each $f_i$ must be constant.