Sharp Schoenberg type inequalities and the de Bruin--Sharma problem

TL;DR

This paper confirms two conjectures on Schoenberg type inequalities of order 4, solving the de Bruin--Sharma problem, and introduces a new interpolation framework to derive sharp inequalities for negative orders.

Contribution

It proves two conjectures on Schoenberg inequalities, develops a novel interpolation method, and extends inequalities to negative orders, advancing understanding of Schoenberg type inequalities.

Findings

Confirmed conjectures on order 4 inequalities, solving the de Bruin--Sharma problem.

Developed a new interpolation framework for Schoenberg inequalities.

Derived sharp inequalities for negative orders and discussed dual inequalities.

Abstract

In this paper, we confirm two conjectures proposed by Georgiev, G\'{o}mez-Serrano, Tao, and Wagner~\cite{GGTW25} on Schoenberg type inequalities of order $4$, thereby providing a complete solution to the de Bruin--Sharma problem. We further develop a new interpolation framework to study Schoenberg type inequalities and, in particular, give a new proof of Pereira's result. Motivated by Sendov's conjecture, we then derive sharp Schoenberg type inequalities of orders $-1$ and $-2m$ (with $m \in \mathbb{N}$), as well as non-sharp inequalities valid for all negative orders $p \le -1$. Finally, we discuss a dual counterpart of the Schoenberg type inequalities of order $-2$.