That for a polynomial of degree $n$, $\frac{M(r)}{r^n}$ is a non-decreasing function of $r$. 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.

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:

