Let $f_1, f_2, \ldots ,f_n$ be holomorphic functions on a region $\Omega$. Show that if $\phi (z)=|f_1(z)|+|f_2(z)|+\ldots +|f_n(z)|$ attains a maximum value on $\Omega$, then $f_i$ is constant for each $i=1,\ldots ,n$.
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3Welcome to M.SE! Please try to include explanations of what you have tried so far and where you are having trouble with your questions. Among other things, this makes it easier for us to give you useful answers. You can edit your post to include this information. – vociferous_rutabaga Jul 23 '14 at 21:47
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3Not really a duplicate of that. Here, the assertion is that all $f_i$ are constant, in the alleged duplicate, only that $\phi$ is constant. – Daniel Fischer Jul 23 '14 at 22:33