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

Quanyu Tang; Teng Zhang

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

Quanyu Tang, Teng Zhang

arXiv:2508.10341v3Aug 14, 2025math.CVmath.FA

TL;DR

The paper studies the de Bruin–Sharma problem and the lack of general sharp Schoenberg type inequalities beyond selected positive orders. The paper combines earlier quartic estimates with new sharp zero-configuration inequalities and complex interpolation applied to ℓp spaces and Schatten p-classes. The paper confirms both Conjecture 1.2 and Conjecture 1.3 and provides a new proof of Pereira’s result (1.6), with a partial refinement when the centroid of the zeros is at the origin.

Problem
The paper studies the de Bruin–Sharma problem and the lack of general sharp Schoenberg type inequalities beyond selected positive orders.
Source: §1.1 · pp. 2–3
Method
The paper combines earlier quartic estimates with new sharp zero-configuration inequalities and complex interpolation applied to ℓp spaces and Schatten p-classes.
Source: §1.1 · p. 3, §4 · p. 18
Results
The paper confirms both Conjecture 1.2 and Conjecture 1.3 and provides a new proof of Pereira’s result (1.6), with a partial refinement when the centroid of the zeros is at the origin.
Source: §1.1 · p. 3, §4 · p. 18
Takeaways & Limitations
The results provide a complete solution to the de Bruin–Sharma problem and extend Schoenberg type inequalities to selected negative orders and a dual order −2 setting.
Source: §1.1 · p. 3
The sharp order −1 factor cannot serve as an endpoint operator norm for standard Banach-couple interpolation, and equality configurations for order −4 are not fully characterized.
Source: §7 · p. 33, §9 · p. 43

Abstract

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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 $-2$.

1. Introduction

The paper solves the de Bruin–Sharma problem by confirming two order-4 Schoenberg conjectures, develops interpolation methods for general orders, and establishes sharp and non-sharp results for negative orders.

1.1. The de Bruin–Sharma problem.

Both conjectures of Georgiev, Gómez-Serrano, Tao, and Wagner [12] are confirmed, completing the de Bruin–Sharma problem through sharp inequalities for the relevant zero configurations.The proof combines earlier quartic estimates with new sharp inequalities.Source: §1.1 · p. 3
Theorem 1.4 describes the feasible parameter region using convex-hull upper-right closures, with separate formulations for odd and even degrees.For n = 7, Figure 1 gives the region above and to the right of the convex hull of three anchor points.Source: §1.1 · p. 3, Figure 1 · p. 4

1.2. Schoenberg type inequalities of general orders.

Sharp higher-order inequalities remain scarce, especially for non-even orders, because matrix-trace expressions are difficult to express through polynomial zeros.Existing general-order bounds can remain non-sharp even at p = 4.Source: §1.2 · pp. 3–4
Complex interpolation supplies a new proof of Pereira’s result and a partial refinement under the zero-centroid condition by connecting ℓp spaces with Schatten p-classes.The framework interpolates between the endpoint cases p = 1, p = 2, and p = ∞.Source: §1.2 · p. 4

1.3. Normalized polynomials and related conjectures.

The paper motivates stronger normalized-polynomial inequalities from Sendov’s conjecture, while noting that the proposed strengthening fails for every 0 < λ < 1.The strengthening reduces to Sendov’s conjecture as λ → ∞.Source: §1.3 · p. 5

1.4. Towards Schoenberg type inequalities of negative orders.

The paper derives a sharp Schoenberg type inequality of order −1 and states a result for general negative even orders −2m, applying for every integer m ≥ 1.Source: §1.4 · pp. 6–7
Complex interpolation yields generally non-sharp inequalities across the full range p ≤ −1, complementing the sharp results at selected negative orders.The negative-order analysis treats the ranges (−∞, −2] and [−2, −1] separately.Source: §1.4 · pp. 6–7
A dual counterpart of the order −2 Schoenberg inequality is obtained for general complex polynomials under nonzero-root and distinct-critical-point assumptions.The stated equality condition requires each r_k(w_k − G)^2 to be non-negative.Source: §1.4 · pp. 7–8

2. Solution of the de Bruin–Sharma problem

This section confirms Conjectures 1.2 and 1.3 through sharp inequalities for complex zero configurations and uses them to solve the de Bruin–Sharma problem, including the odd-n case of Theorem 1.4. It evaluates the relevant polynomial quantities and applies analogous contradictions to exclude points outside eOmega_even(n) from Omega(n).

Conjecture 1.2

For odd n > 4, a sharp zero-configuration inequality combined with the Kushel–Tyaglov inequality (1.3) proves Conjecture 1.2.The sharp bound is obtained from a real inequality whose equality configuration has two distinct values with multiplicities (n − 1)/2 and (n + 1)/2.Source: §2 · pp. 8, 10, 11

Conjecture 1.3

For every n ≥4, a sharp zero-configuration inequality combined with the Cheung–Ng inequality (1.2) proves Conjecture 1.3.The underlying real inequality is established using an explicit sum-of-squares identity, with equality at (a, …, a, −(n −1)a).Source: §2 · pp. 8, 11, 12, 13

Complete solution

The proof of the odd-n branch of Theorem 1.4 uses the two conjectures, convexity, and test polynomials to establish Ω(n) = eΩodd(n).The test configurations include zeros with one exceptional value and configurations built from two repeated values.Source: §2 · pp. 13–14
The even-n branch begins by combining Cauchy–Schwarz with Conjecture 1.3 to obtain Ω(n) ⊃eΩeven(n), followed by test polynomials with balanced and exceptional zero configurations.The supplied passage presents the setup and test configurations for this branch but does not include its concluding result.Source: §2 · p. 14

2. Solution of the de Bruin–Sharma problem

The computed values for p3 are substituted into inequality (1.4) as part of the exclusion argument.The passage records |z_j|^4 = n, |z_j|^2 = n^2, and the corresponding sum involving |ξ_k|^4.Source: §2 · p. 15
The analogous polynomial p2 also contradicts (2.6), so no (α, β) outside eΩ_even(n) belongs to Ω(n), yielding the claimed characterization.The passage concludes that Ω(n) equals the specified set, although the displayed equality is truncated in the supplied text.Source: §2 · p. 15

3. Notation and tools for the sequel

This section establishes matrix-analysis notation and tools connecting polynomial zeros and critical points to eigenvalue and singular-value methods. It then introduces compatible Banach couples and complex interpolation for transferring endpoint bounds to intermediate exponents.

3.1. Matrix analysis.

The paper defines ℓp(V) using the ℓp norm and Sp using the Schatten p-norm, providing the spaces used in later interpolation arguments.Source: §3.1 · p. 15
The characteristic polynomial of a matrix built from the zeros has the same zeros as z p′(z), linking matrix spectra to polynomial derivatives.Source: §3.1 · pp. 15–16
Polynomial critical points are represented as eigenvalues of a matrix constructed from the polynomial’s zeros.The matrix has the same eigenvalue set as the polynomial’s critical points.Source: §3.1 · p. 16
Matrix inequalities compare eigenvalue and singular-value quantities, with equality in the stated trace inequality exactly when the matrix is normal.The section also introduces singular-value comparisons for diagonal matrices multiplied by arbitrary matrices.Source: §3.1 · p. 16

3.2. Complex interpolation theory.

Complex interpolation is formulated for compatible Banach couples and bounded endpoint operators, producing bounded operators between the interpolated spaces.The section also recalls the corresponding interpolation results for noncommutative Schatten classes and Lp spaces.Source: §3.2 · p. 17

4. Revisiting the Schoenberg type inequality of order 1 and Pereira’s result

The section develops new proofs of the order-1 Schoenberg inequality and Pereira’s result, using matrix inequalities and complex interpolation. Under a zero-centroid condition, it obtains stronger bounds than Pereira’s result for p ≥ 2 and weaker non-sharp bounds for 1 ≤ p < 2.

4.2. A new proof of Pereira’s result.

Complex interpolation combines endpoint estimates to provide a new proof of Pereira’s result for general exponents p ≥ 1.The argument interpolates between ℓp and Schatten p-class estimates and uses eigenvalue–singular value comparison.Source: §4 · p. 18, §4.2 · pp. 20–21

4.1. A new proof of the Schoenberg type inequality of order 1.

Theorem 4.1 strengthens the previously known order-1 Schoenberg inequality, from which inequality (1.5) follows.The proof uses singular values of a projected diagonal matrix, majorization, and Schur-convexity.Source: §4.1 · pp. 18, 20

4.3. The case where the centroid is at the origin.

Under the centroid-at-the-origin assumption, Theorem 4.4 gives a bound stronger than Pereira’s result for p ≥ 2.The proof interpolates endpoint estimates on the centroid-zero subspace.Source: §4.2 · p. 21, §4.3 · pp. 22, 23, 25
The interpolation framework has a limitation: interpolated spaces coincide with ℓp only as vector spaces, with generally unequal equivalent norms when n ≥ 3.This norm-equivalence issue qualifies the interpolation identification used in the centroid-zero argument.Source: §4.3 · p. 22
For 1 ≤ p < 2, the corresponding centroid-zero estimates are non-sharp and are presented as a weaker partial result toward Conjecture 1.6.The section also proves the relevant constant is best possible in an auxiliary estimate, with equality at z = e1.Source: §4.3 · pp. 22, 23, 25

5. Schoenberg type inequality of order −1

This section establishes identities relating zeros and critical points and applies them to a Schoenberg-type inequality of order −1. It also records a direct consequence for the case a = 1 of Conjecture 1.9.

5. Schoenberg type inequality of order −1

The section’s normalized-polynomial framework compares sums of negative powers of critical-point and zero moduli.The zeros of the derivative are denoted by w1, …, wn, and identities relating zeros and critical points are established.Source: §5 · p. 26
The case a = 1 of Conjecture 1.9 follows directly from identity (5.1) when all zeros satisfy |zj + 1| ≤ 1.The conclusion is obtained by applying the identity to each zero under that modulus condition.Source: §5 · p. 26

2. Multiplying by 2 and summing

The section derives a negative-order Schoenberg inequality by comparing elementary symmetric functions of inverse critical-point and inverse zero moduli. It identifies equality for zeros lying on a common ray and uses this family to establish sharpness.

2. Multiplying by 2 and summing

The central inequality is ek(|w|^-1) ≤ (k + 1)ek(|z|^-1) for k = 1, …, n.It is obtained by applying an earlier lemma to the polynomial and combining the resulting estimate with the section’s identities.Source: §2 · p. 26
Equality holds for every k when all zeros lie on the same ray emanating from the origin.The equality analysis rotates and rescales the configuration so the zeros and critical points are positive real numbers.Source: §2 · pp. 26–27
The resulting order −1 Schoenberg inequality is sharp because equality occurs for a nontrivial family of zero configurations.The authors do not attempt to characterize all equality cases.Source: §2 · pp. 27–28

6. Schoenberg type inequality of order −2

The section proves a sharp order −2 Schoenberg inequality by representing critical points through a matrix and applying Schur’s inequality. Equality holds exactly when all zeros coincide.

Equality characterization

The matrix B is normal if and only if z1 = · · · = zn, so equality in the order −2 inequality occurs exactly for coincident zeros.The proof compares BB∗ and B∗B and shows that normality forces all reciprocal zero parameters to coincide.Source: §6 · pp. 28–30

Matrix reduction

The matrix representation of the critical points reduces the inequality to a comparison involving eigenvalues and singular values.The zeros of P′ are identified with the eigenvalues of a matrix constructed from the zeros of P, then its inverse is analyzed.Source: §6 · pp. 28–29

Sharp inequality

Schur’s inequality yields the desired order −2 bound, with equality precisely when the associated matrix B is normal.The trace computation of BB∗ supplies the explicit bound, and equality in Schur’s inequality characterizes the equality case.Source: §6 · p. 29

Sharpness and refinement

The resulting bound is sharp, while a comparison with an elementary estimate shows that the latter is weaker.The section also introduces conjectural refinements of the order −2 inequality and verifies the proposed refinement when n = 2.Source: §6 · pp. 30–31

7. Schoenberg type inequality of order −4

The section derives a sharp order −4 Schoenberg inequality by applying Schur’s inequality to B². Equality includes coincident-zero configurations but is not characterized solely by them.

Sharp inequality

Applying Schur’s inequality to B² produces the order −4 inequality and identifies equality with normality of B².The proof expands traces of BB∗ and B∗B into diagonal and rank-one terms before matching the stated bound.Source: §7 · pp. 32–33

Equality cases

Coincident zeros guarantee equality because B becomes a scalar multiple of the Hermitian matrix I + J.This makes B normal, hence B² normal, which is the equality condition from Schur’s inequality.Source: §7 · p. 33
Unlike the order −2 case, equality in the order −4 inequality can occur when B² is normal even though B is not normal.For n = 2 and z1 = −z2 ≠ 0, B² is a scalar multiple of the identity, yielding equality.Source: §7 · p. 33

8. Schoenberg type inequality of order −2m

The section extends the matrix-trace method to every even negative order −2m, producing sharp inequalities and explicit equality cases. It then derives cleaner modulus-only inequalities for orders −4 and −6.

General even orders

For every integer m ≥ 1, the method gives a sharp upper bound for the sum of |wk|^−2m, with equality exactly when all zeros coincide.The proof expands tr((F + αK)^m), bounds lower-order terms by the highest-order term, and verifies sharpness on coincident-zero polynomials.Source: §8 · pp. 34, 35, 37

Proof strategy

The proof’s modulus-only reduction replaces mixed phase-dependent quantities with expressions involving the highest-order sum and n.The direct formulas in terms of zeros become complicated, motivating the weaker but cleaner bound.Source: §8 · pp. 34, 36

General even orders

The resulting modulus-only inequality settles the even-order family and, in particular, the case p = 2m of Conjecture 1.13.Its equality condition is z1 = · · · = zn.Source: §8 · pp. 35, 37

Orders −4 and −6

At order −4, the specialization m = 2 yields a modulus-based inequality with equality if and only if z1 = · · · = zn.The bound is obtained from the general theorem but is weaker than the earlier order −4 inequality because of a Horn-inequality estimate.Source: §8 · pp. 37–38
At order −6, the specialization m = 3 gives an analogous modulus-based inequality, again with equality exactly for coincident zeros.The section records this as the next concrete application of the general even-order theorem.Source: §8 · p. 38

9. An interpolation approach to negative orders

The section develops complex-interpolation bounds for negative Schoenberg orders by applying interpolation simultaneously to ℓp spaces and Schatten classes. The resulting bounds cover broad exponent ranges but are generally non-sharp away from selected endpoints.

Orders p ≥ 2

Complex interpolation between p = 2 and p = ∞ yields inequalities for every p ≥ 2.The endpoint identifications use equal-norm interpolation for ℓ∞ and ℓ2, and for S∞ and S2.Source: §9 · pp. 39–40

Sharpness

The interpolation family is generally non-optimal: at p = 4, the sharp fourth-order inequality is strictly stronger than the interpolated bound.The paper explicitly compares the constants and concludes that non-optimality should persist for finite p > 2.Source: §9 · p. 40
The interpolation endpoints are sharp at p = 2 and in the limit p → ∞.At p = 2 the constant matches the optimal order −2 result, while the limit recovers the sharp minimum-modulus bound.Source: §9 · p. 40

Orders 1 ≤ p ≤ 2

Interpolating between p = 1 and p = 2 gives bounds for 1 ≤ p ≤ 2 using the exact operator norm √(n + 3).The p = 1 endpoint follows from rank-one images of standard basis vectors, while p = 2 supplies the matching Schatten bound.Source: §9 · pp. 41–42

Interpolation limitation

The sharp factor 2 from the polynomially constrained setting cannot serve as an interpolation endpoint because the admissible set is nonlinear and is not a Banach space.Consequently, the interpolation proof must use the coarser genuine operator bound on all of ℓ1(Cn).Source: §9 · p. 43

10. Dual Schoenberg type inequalities

The section develops a dual framework for Schoenberg inequalities through matrix integrability and full integrals of polynomials. It derives a dual inequality of order −2, including explicit equality conditions and a specialization to distinct critical points.

10. Dual Schoenberg type inequalities

The section frames the dual problem as seeking lower bounds for sums over reciprocal squared critical-point moduli in terms of the polynomial zeros.The dual formulation treats P(z) as a full integral of P′(z).Source: §10 · p. 43

10.1. Matrix integrability and full integrability of polynomials.

Matrix B is integrable exactly when its characteristic polynomial has a full integral, yielding a dual Schoenberg inequality as an application.A full integral P of q satisfies P′ = q and contains every multiple zero of q as a zero; existence is guaranteed when q has at most one multiple zero.Source: §10.1 · p. 44

10.2. Dual Schoenberg type inequality of order −2.

Theorem 10.1 converts an integral matrix construction into an upper bound for the sum involving 1/|z_j|^2 in terms of the diagonal entries w_k.The proof uses the eigenvalues of A^-1, a Frobenius-norm computation, and the relation d_k = 1/w_k.Source: §10.2 · pp. 44–45
Equality in the matrix inequality holds if and only if the integral matrix A is normal.Normality is equivalent to normality of A^-1 and is reduced through the block structure to explicit conditions on the integrator.Source: §10.2 · pp. 44, 46
For a full integral whose derivative has simple zeros a_i and multiple zeros b_j, the specialized inequality holds exactly when r_j(a_j − G)^2 is non-negative for every simple zero.The multiplicities satisfy α_j ≥ 2, and the equality analysis uses the choice of integrator minimizing the Frobenius norm.Source: §10.2 · pp. 45–46
When all critical points of P are distinct, P is a full integral of P′, giving the dual Schoenberg inequality of order −2 for distinct w_k.In this case, equality is characterized by r_k(w_k − G)^2 being non-negative for every k.Source: §10.2 · p. 46

Appendix A. Proof of the sum-of-squares identity (2.3)

The appendices supply algebraic proofs of the paper’s sum-of-squares and identity formulas, then verify Conjecture 1.6 for p = 6. The p = 6 proof combines Gram-matrix positivity with moment inequalities and identifies the equality configuration.

Appendix A. Proof of the sum-of-squares identity (2.3)

The sum-of-squares identity (2.3) follows by symmetrizing the relevant sums, applying identities (A1)–(A5), and collecting terms until the S1S3 terms cancel.The final expression matches the target identity after division by (m + 1)m(m − 2).Source: Appendix A · pp. 49–50

Appendix B. Proof of the identities (5.1) and (5.2)

Appendix B derives identities (5.1) and (5.2) by expressing derivatives of P through Q and its logarithmic derivative at the zeros of P and P′.Substitution of the resulting relations yields (5.2).Source: Appendix B · pp. 51–52

Appendix C. Verification of Conjecture 1.6 in the case p = 6

Conjecture 1.6 is confirmed for p = 6 by rewriting the target inequality in terms of moments m2, m4, and m6 and proving the resulting quantity is nonnegative.The proof assumes m2 > 0 after handling the zero case separately.Source: Appendix C · pp. 52–54
The p = 6 argument uses a positive-semidefinite Gram matrix built from 1, Z, and Z|Z|^2, together with Jensen’s inequality for the convex function x^3.The moments satisfy m2 = EX, m4 = EX^2, and m6 = EX^3.Source: Appendix C · pp. 53–54
Equality in the resulting corollary holds if and only if all zeros z_j lie on a straight line in the complex plane.The corollary combines the preceding theorems for zeros z_j and critical points w_j.Source: Appendix C · pp. 52, 54

Appendix D. Proof of Theorem 9.4

Appendix D proves Theorem 9.4 using sharp even-exponent estimates and complex interpolation. It also establishes sharpness by evaluating the associated rank-one construction on vectors with equal coordinate magnitudes.

Appendix D. Proof of Theorem 9.4

For every integer m ≥ 1, the endpoint estimate is sharp, with equality attained for vectors whose coordinates have equal magnitudes.Taking the supremum yields the operator norm formula involving C_m(n)^(1/(2m)).Source: Appendix D · pp. 54–56
Sharpness is verified by choosing |u_1| = ··· = |u_n| = r, making F a scalar matrix and K a rank-one matrix with an explicitly determined spectrum.The resulting matrix has one distinguished eigenvalue and n − 1 eigenvalues equal to r^2.Source: Appendix D · p. 55
Theorem 9.4 extends the sharp even-exponent bounds to every p ≥ 2 by complex interpolation between the endpoint spaces S_2m and S_(2m+2).The compatible interpolation scales identify the interpolated spaces with ℓ_p(C^n) and S_p.Source: Appendix D · p. 56