0

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.

Ivo Terek
  • 77,665
  • You can use mean value property of holomorphic function. http://math.stackexchange.com/questions/72885/mean-value-property-for-holomorphic-functions – Math.StackExchange Apr 17 '15 at 22:35
  • 1
    You can do it purely algebraically, if you're not going to use any of the powerful results of basic complex analysis. By a change of variables, assume your local maximum is at $z=0$, and show that you can choose $z$ with $|z|=\delta$ so that $(z^n+a_{n-1}z^{n-1}+\dots+a_1z)/a_0>0$. – Ted Shifrin Apr 18 '15 at 01:04
  • Ok, I'll try. Thanks! :)

    (I think you meant $|\cdot|$ instead of $(\cdot)$ there)

     – Ivo Terek Apr 18 '15 at 01:13

0 Answers0