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

Quanyu Tang, Teng Zhang

TL;DR

The paper resolves the de Bruin–Sharma problem by proving the two conjectures for quartic order, pinpointing the exact Ω(n) region as an upper-right convex hull derived from anchor configurations. It introduces a novel complex interpolation framework to transfer sharp endpoint bounds at p=1,2,∞ to a full range of exponents, delivering sharp negative-order inequalities for p=−1 and p=−2m, along with non-sharp bounds for all p≤−1. A major technical feat is the complete solution of the quartic problem via a sharp real inequality for odd n and a sharp two-point-zero configuration analysis, enabling the convex-hull description of Ω(n). The work also develops a dual perspective for order −2 through matrix-integrability concepts, yielding dual Schoenberg-type bounds and equality criteria tied to normality, thereby enriching the zero–critical-point relationship analysis on polynomials.

Abstract

In this paper, we confirm two conjectures proposed by Georgiev, Gómez-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 $-2$.

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

TL;DR

The paper resolves the de Bruin–Sharma problem by proving the two conjectures for quartic order, pinpointing the exact Ω(n) region as an upper-right convex hull derived from anchor configurations. It introduces a novel complex interpolation framework to transfer sharp endpoint bounds at p=1,2,∞ to a full range of exponents, delivering sharp negative-order inequalities for p=−1 and p=−2m, along with non-sharp bounds for all p≤−1. A major technical feat is the complete solution of the quartic problem via a sharp real inequality for odd n and a sharp two-point-zero configuration analysis, enabling the convex-hull description of Ω(n). The work also develops a dual perspective for order −2 through matrix-integrability concepts, yielding dual Schoenberg-type bounds and equality criteria tied to normality, thereby enriching the zero–critical-point relationship analysis on polynomials.

Abstract

In this paper, we confirm two conjectures proposed by Georgiev, Gómez-Serrano, Tao, and Wagner~\cite{GGTW25} on Schoenberg type inequalities of order , 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 and (with ), as well as non-sharp inequalities valid for all negative orders . Finally, we discuss a dual counterpart of the Schoenberg type inequalities of order .

Paper Structure

This paper contains 25 sections, 41 theorems, 462 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The de Bruin--Sharma problem
  3. Schoenberg type inequalities of general orders
  4. Normalized polynomials and related conjectures
  5. Towards Schoenberg type inequalities of negative orders
  6. Solution of the de Bruin--Sharma problem
  7. Notation and tools for the sequel
  8. Matrix analysis
  9. Complex interpolation theory
  10. Revisiting the Schoenberg type inequality of order 1 and Pereira's result
  11. A new proof of the Schoenberg type inequality of order 1
  12. A new proof of Pereira's result
  13. The case where the centroid is at the origin
  14. Schoenberg type inequality of order $-1$
  15. Schoenberg type inequality of order $-2$
  16. ...and 10 more sections

Key Result

Theorem 1.4

Let $n\ge 4$, and introduce the four anchor points in the plane: Let $\operatorname{co}(\cdot)$ denote the convex hull. Define the two closed, convex "upper-right closures" by Then the set $\Omega(n)$ defined in Problem prob:dbs_12345 satisfies

Figures (1)

  • Figure 1: The region $\widetilde{\Omega}_{\mathrm{odd}}(7)$ in the $(\alpha,\beta)$-plane: the feasible set is the region above and to the right of the convex hull of the three points $(\alpha_{\mathrm{odd}}(7),0)$, $(\alpha_{\mathrm{CN}}(7),\beta_{\mathrm{CN}}(7))$, and $(0,\beta_{\mathrm{all}}(7))$.

Theorems & Definitions (102)

  • Conjecture 1.2: GGTW25
  • Conjecture 1.3: GGTW25
  • Theorem 1.4
  • Conjecture 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Conjecture 1.9: Sendov
  • Conjecture 1.10
  • Theorem 1.11: Theorem \ref{['eq:ele-minus-one']}
  • Theorem 1.12: Theorem \ref{['thm:general-2m']}
  • ...and 92 more