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I have the following problem:

Let $f,g,h$ be holomorphic functions (non-constant) in some domain $D$. Show that the function $F(z):=|f(z)|+|g(z)|+|h(z)|$ has no local maximum in this domain $D$.

Can someone give a sketch of the proof?

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This seems like the maximum modulus theorem; at least that gives that each function independently never reaches a maximum. I think you can use this to show that F also never reaches a maximum. – process91 Nov 11 '11 at 12:38
On second thought, perhaps there are some additional requirements necessary to apply the Maximum Modulus theorem. Someone with more experience in this area should probably answer. – process91 Nov 11 '11 at 13:54
up vote 5 down vote accepted

If $f$ is a non-constant holomorphic function then $|f|$ is strictly subharmonic (away from zeros of $f$, i.e. where $|f|$ is smooth, it is easy to see that $\Delta |f|>0$, since $\Delta\log|f|=0$). This means (by definition) that the mean value of $|f|$ on a circle is strictly greater than the value in the center. $F$ is the sum of 3 strictly subharmonic functions, so is strictly subharmonic, so it can't have local maxima.

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