There is this version of maximum modulus principle:
If $P$ is a non-constant polynomial, then $|P|$ doesn't have a local maximum.
I know that if $P$ is non-constant, then $|P(z)| \stackrel{|z|\to+\infty}{\longrightarrow}+\infty$. But this only tells me that $|P|$ does not have a global maximum. It could have some locals, though. I am not supposed to use the open mapping theorem (I could prove that it implies the principle, though), nor the Gutzmer-Parseval inequality, nor anything related to integration theory.
Any ideas? Thanks.