Prove that if $|f| + |g| \equiv c$ ($c\in \Re$) in set $Ω\subseteq C$ where $f,g$ holomorphic in Ω then $f\equiv a$ , $g\equiv b$ where $a,b\in C$. Well the real problem was to show it for $\sum_{i=1}^{n}|f_{i}| \equiv c$ but I think i should starting by proving it for 2 of them.(This was once given to us on an exam as the only problem in it and it was valuated with 100% of the grade)
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I fixed the question. I had deleted a part by accident. also i found the answer https://math.stackexchange.com/questions/2082670/if-fg-is-constant-on-d-prove-that-holomorphic-functions-f-g-are-con – Pookaros Sep 19 '17 at 12:41
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What does stable mean? I am sorry that I do not know. – Sarvesh Ravichandran Iyer Sep 19 '17 at 12:42
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And the general case is answered here: Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.. – Martin R Sep 19 '17 at 12:52