TITLE: Increasing polynomial with real coefficients
QUESTION [5 upvotes]: Let $P$ be polynomial with real coefficients such that $P(0)=0$ and $P$ is increasing for $x>0$.
I want to maximize $|P(z)|$ on a closed (complex) disk $D(x,r)$ with center $x$ and radius $r$, where $x>0$ and $0<r<x$. More precisely, I want to know if the maximum is necessarily attained on $x+r$ or not. 
If we also assume that the coefficients of $P$ are positive, then I can show that this holds. But is it in general true?

REPLY [2 votes]: Counterexample: Define $p(z) = 1 + (z-1) -(z-1)^3 + (z-1)^5.$ Then $p(0)=0,$ and $p'(z) = 1 -3(z-1)^2 + 5(z-1)^4.$ Because $1-3u+5u^2 > 0$ for real $u,$ we have $p'(z) > 0$ for all real $z.$ Thus $p$ is strictly increasing on $\mathbb R.$
Consider the closed discs $D(1,r), 0<r<1.$ As $r\to 1^-,$ we have
$$p(1+r) = 1+r -r^3 + r^5 \to 2,$$
while
$$|p(1+ri)| = |1 +i(r+r^3 + r^5)| \to |1+3i| = \sqrt {10}.$$
Since $\sqrt {10} > 2,$ we must have $|p(1+ri)|>p(1+r)$ for $r$ close to, but less than, $1.$