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?
[Math] 3 holomorphic functions, sum of absolute values does not have maximum
complex-analysis
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Best Answer
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.