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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?

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If $P(z) = z^n$, then $F(r) = r^{n-1}$ is not decreasing. Did you make a mistake in the definition of $F$? – Greg Martin Nov 5 '12 at 22:47
Possible duplicate of Domination of complex-value polynomial by highest power. – WimC Nov 6 '12 at 17:29
yes, there's an error. Put "increasing" in place of "decreasing" – bateman Nov 6 '12 at 18:29
up vote 0 down vote accepted

my solution: put $P(z)=z^nQ(\frac{1}{z})$. Then Q(z) is entire on $\mathbb C-{0}$. By maximum modulus principle, the function $M^Q(r)=max \{|Q(\frac{1}{z})|:|z|=r\}$ is increasing for $r>0$.

Now, $M(r)=r^nM^Q(r)$ and so $F(r)=r^{n-1}M^Q(r)$ is also increasing. Did i make any mistake?

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