Let f1, f2, ..., fn be holomorphic functions in a region \(\displaystyle \Omega\) and define f(z) by f(z)=|f1(z)|2+|f2(z)|2+...+|fn(z)|2. Show that f has no local maximum in \(\displaystyle \Omega\) unless all the functions fj, j=1, 2, ..., n, reduce to constant functions.
So far I was able to show that if f has a local maximum in \(\displaystyle \Omega\), then f must be constant on \(\displaystyle \Omega\). But how do I go about showing that each fj must be constant?
So far I was able to show that if f has a local maximum in \(\displaystyle \Omega\), then f must be constant on \(\displaystyle \Omega\). But how do I go about showing that each fj must be constant?