Where the $M(r)=\operatorname{Max}_{\mid z\mid=r}f(z)$, where $f(z)=p_n(x)$, a polynomial of degree $n$.

My first attempt: maybe this is related to the Cauchy's inequality of estimating derivatives. Maybe consider the integral $\displaystyle \int \frac{f(z)}{z^{n+1}}\mathrm{d}z$?

Another attempt: The inequality $\displaystyle \frac{M(r)}{r^{n}}\leq\frac{M(R)}{R^n}$ remotely assembles the Hadamard three circle theorem.

  • $\begingroup$ Hi, please do not edit a question to invalidate existing answers. I've rolled back to the version answered by mrf below. For the interested reader: the OP asked a follow-up math.stackexchange.com/questions/909322 $\endgroup$ – Willie Wong Aug 26 '14 at 11:02
  • $\begingroup$ @williewong thank you $\endgroup$ – Wilson of Gordon Aug 26 '14 at 13:14

This is not true. Take for example $f(z) = 1+z^2$. Then $M(0) = 1$ and $M(r) \approx r^2$ for $r$ large. Hence $M(r)/r^2$ blows up at $r=0$, and tends to $1$ as $r\to\infty$.

In fact, for this example, the function $M(r)/r^2$ is actually decreasing:

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  • $\begingroup$ No wonder I can't prove it, it isn't even true! $\endgroup$ – Wilson of Gordon Aug 25 '14 at 8:16
  • $\begingroup$ Then when is it an increasing function? It seems like I remembered the inequality in a totally opposite way $\endgroup$ – Wilson of Gordon Aug 25 '14 at 8:25

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