Let $P$ be a polynomial of degree $n$. For every $r>0$, let $M(r):=\max \{|P(z)| :|z|=r\}$. I want to show that the function $F(r)=\frac{M(r)}{r}$ is monotonically decreasing in $(0, +\infty)$. Second question is: if $F(r_1)=F(r_2)$ for $r_1\neq r_2$, what can be said on polynomial $P$?
For the first question, how can i use maximum modulus principle? for the second, i have no idea....any help?