Let $f,g: U \rightarrow \mathbb{C}$ be holomorphic on the open and connected subset $U$. If $|f| + |g|$ is constant on $U$ show that $f, g$ are constant on $U$.
What can we say about finite or countable number of holomorphic functions?
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1What have you tried? Where have you got stuck? Any kind of context for this problem? – Some Math Student Dec 21 '14 at 14:29
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your second sentence doesn't make any sense – mercio Dec 21 '14 at 14:30
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It seems to fail already for $g=0$. – Urgje Dec 21 '14 at 15:06
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I do not get the @Urgje comment. I think this question is begging you to apply some theorem you know, such as the Liouville theorem http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis). – Michael Dec 22 '14 at 01:59
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This is not a duplicate of the linked question. – copper.hat Jan 03 '17 at 23:45