Schoenberg type inequalities
Quanyu Tang
TL;DR
This work advances the geometry of polynomial zeros and their critical points by establishing a sharp Schoenberg-type inequality of order $6$ under the centroid-zero condition and introducing the first known order-$1$ (odd-order) inequality, addressing open questions of Kushel and Tyaglov. The order-$6$ result employs a matrix-analytic approach using the D-companion framework and trace estimates, producing a bound that depends on power sums of the zeros and attains equality precisely when the zeros are collinear. The order-$1$ inequality follows from coefficient relations between zeros and critical points and a weak log-majorization argument, yielding a tight bound on the sum of the moduli of the critical points under the centroid condition. The paper further applies Schoenberg's inequality to a Sendov-type scenario, deriving conditions under which a critical point lies near a prescribed center, and presents improved high-order bounds for all $r\ge 2$, refining prior results and highlighting potential avenues for higher-order odd cases. Overall, the results deepen the understanding of zero–critical-point relations and connect to Sendov's conjecture, with potential implications for higher-order inequalities and convex/majorization techniques.
Abstract
In the geometry of polynomials, Schoenberg's conjecture, now a theorem, is a quadratic inequality between the zeros and critical points of a polynomial whose zeros have their centroid at the origin. We call its generalizations to other orders Schoenberg type inequalities. While inequalities of order four have been previously established, little is known about other orders. In this paper, we present a Schoenberg type inequality of order six, as well as a novel inequality of order one, representing the first known result in the odd-order case. These results partially answer two open problems posed by Kushel and Tyaglov. We also make a connection to Sendov's conjecture.