Midpoints and critical points

Abstract

For a degree $5$ real polynomial with roots $x_1\leq \cdots \leq x_5$ and roots $\xi_1\leq \cdots \leq \xi_4$ of its derivative, we set $z_j:=(x_j+x_{j+1})/2$, $1\leq j\leq 4$. We prove that one cannot have at the same time $\min_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \min_{1\leq j\leq 3}(\xi_{j+1}-\xi_j)$ and $\max_{1\leq j\leq 3}(z_{j+1}-z_j)\geq \max_{1\leq j\leq 3}(\xi_{j+1}-\xi_j)$. The result settles a general question about midpoints and critical points of hyperbolic polynomials.