Consider a domain $G\subseteq \mathbb{C}$ and $f_1,...,f_n: G\rightarrow\mathbb{C}$ holomorphic functions, such that the sum $|f_1|+...+|f_n|$ assumes a local maximum in $G$.
I am trying to show that $f_1,...,f_n$ are all constant in $G$.
As $|f_1|+...+|f_n|$ assumes a local maximum, there is a $z_0\in G$ such that $|f_1(z)|+...+|f_n(z)|\leq|f_1(z_0)|+...+|f_n(z_0)|:=M$.
So $|f_i|\leq M\ \forall i\in\{1,...,n\}$.
Now I tried to use the Maximum principle for holomorphic functions, which tells me that a holomorphic function $f:G\rightarrow \mathbb{C}$ is constant if $|f|$ assumes a local maximum for a $z_0\in G$.
The problem is, I know that $|f_i|$ are bounded, but I do not know if they assume a local maximum, so I doubt I can use the Maximum principle for holomorphic functions here.
Any hint would be highly appreciated. Many thanks in advance.
