Do there exist some non-constant holomorphic functions $f_1,f_2,\ldots,f_n$such that $$\sum_{k=1}^{n}\left|\,f_k\right|$$ is a constant? Can you give an example? Thanks very much
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holomorhpic on which region? – Ittay Weiss May 15 '13 at 08:26
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@IttayWeiss any region $\Omega$ – Laura May 15 '13 at 08:34
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1Closely related: http://math.stackexchange.com/questions/96257/maximum-of-sum-of-finite-modulus-of-analytic-function?rq=1 – Caleb Stanford May 15 '13 at 08:43
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This question has been asked before but without an answer http://math.stackexchange.com/questions/289114/show-that-holomorphic-f-1-f-n-are-constant-if-sum-k-1n-left-f?rq=1 – clark May 15 '13 at 10:43
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NO. Suppose $f, g$ are holomorphic functions on the unite disc. $$ 2\pi r M=2\pi r( |f(z_0)|+|g(z_0)|)=|\int_{|z-z_0|=r} fdz|+|\int_{|z-z_0|=r} gdz|\le \int_{|z-z_0|=r} (|f|+|g|)|dz|=2\pi r M $$ so all equal sign hold, then $f, g$ are constants.
Ma Ming
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2if f is holomorphic on unit disk isn't $\int f dz = 0 ?$ or do you mean to say $\int f/(z-z_0)?$ – rohit May 15 '13 at 09:48
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1as Goos Hinted, pick up and interior point z_0 and unit wi s.t. $f_i(z_0) = |f_i(z_0)|wi$, define f = $\sum conj(wi) f_i$, it will attain its maximum at $z_0$ will imply that f is constant, do it for varius z_0 and you will get n equations in n unknowns solve it to get $f_i= constants$ – rohit May 15 '13 at 10:10
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I am not able to get why inequalities are actually equalities implies that $f$ and $g$ are constants. Please elaborate @MaMing. – MathRookie2204 Oct 15 '23 at 11:21