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In the complex plane, set $z=x+iy$. How to prove that the function $xy(x^2-y^2)$ cannot have local maximum or minimum in $|z|<1$? I suspect that the maximum modulus principle plays a role but don't know how to make use of this idea

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    Not exactly a modulus (this is not even nonnegative), but you can see it as the imaginary part of some simple holomorphic function, and as such it is harmonic.2011-12-09
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    @ D.Thomine Yes, I see this, it is the imaginary part of $z^4/4$ or something, but now how is the fact that it is harmonic going to help me?2011-12-09
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    Because harmonic functions also satisfy a maximum principle (http://en.wikipedia.org/wiki/Harmonic_function). By the way, yes, $z^4/4$ works.2011-12-09
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    and I believe this could be done by the usual, pedestrian way (find critical points, etc)2011-12-09

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