I am trying to prove or disprove the following inequality.

$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$

where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)=\{ x^*Ax:||x||_2=1\}$ is the numerical range of $A$ and $||\cdot||_2$ is the spectral norm.

I know that for $P(x)=x^n$ (power inequality) or $P$ is linear or for $A$ normal(as $W(A)$ contains all the eigenvalue of $A$) or all coefficient of $P$ is real and of the same sign(using triangle inequality) , the inequality would be true.

I have used computer program for randomly generated matrix or the nilpotent matrix with $2\times2,3\times3,10\times10$ , and the inequality turns out to be true.

I have tried to prove it but I cannot succeed. I attempt to show $w(P(A))\leq \max_{\alpha\in W(A)} |P(\alpha)|$ where $w(\cdot)$ is the numerical radius since $||M||_2\leq 2 w(M),\forall M$. But it turns out not to be true always.

Have you encountered this inequality somewhere or any idea about the approaches you would try to prove this? Or you could give some conditions so that the inequality is true? Or maybe relax the range of $\alpha$ (not just $W(A)$) so that we have a similar inequality?

Any help would be appreciated!!

notgo hunting in the literature $\endgroup$ – Yemon Choi Feb 3 '15 at 2:26open problemknown as Crouzeix conjecture. $\endgroup$ – Federico Poloni Feb 12 '15 at 11:29