Let $\Omega \subset \Bbb C$ an open and connected subset. Let $f,f_i \in H(\Omega)$ for all $i \in \{1,2,...,n\}$ For all $n \in \Bbb N$, characterise $f$ knowing that, $$|f_1(z)|+|f_2(z)|+...+|f_n(z)| = |f(z)|$$ for all $z \in \Bbb C$
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1Divide by $f$ and then apply Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.. – Martin R Jan 09 '20 at 20:12