Tang–Zhang: the sharp order −2 Schoenberg inequality

Q. Tang, T. Zhang, "Sharp Schoenberg type inequalities and the de Bruin–Sharma problem", arXiv:2508.10341v3 (5 Dec 2025), §6, p. 31.

The identity (5.2) leads us to the following conjectural refinements of Theorem 6.1 in the case of order −2.

Conjecture 6.3. Let P(z) = z ∏_{j=1}^n (z − z_j), z_j ∈ C \ {0}, and let w_1, …, w_n denote the zeros of P′(z). Then Σ_{k=1}^n 1/|w_k|² ≤ 3 Σ_{j=1}^n 1/|z_j|² + |Σ_{j=1}^n 1/z_j|². (6.9) Moreover, equality in (6.9) holds whenever z_1, …, z_n all lie on the same line passing through the origin.

In the case n = 2 we can verify Conjecture 6.3 by an explicit computation. (Proposition 6.4.)

Theorem 6.1 there proves the weaker right side (n+3) Σ 1/|z_j|². The exact public definitions and result are:

noncomputable def P {n : ℕ} (z : Fin n → ℂ) : Polynomial ℂ :=
  Polynomial.X * ∏ j, (Polynomial.X - Polynomial.C (z j))

def claim : Prop := ∀ (n : ℕ) (hn : 1 ≤ n) (z : Fin n → ℂ), (∀ j, z j ≠ 0) →
    (((Polynomial.derivative (P z)).roots.map (fun w => ‖w‖⁻¹ ^ 2)).sum ≤
      3 * ∑ j, ‖z j‖⁻¹ ^ 2 + ‖∑ j, (z j)⁻¹‖ ^ 2) ∧
    ((((Polynomial.derivative (P z)).roots.map (fun w => ‖w‖⁻¹ ^ 2)).sum =
      3 * ∑ j, ‖z j‖⁻¹ ^ 2 + ‖∑ j, (z j)⁻¹‖ ^ 2) ↔
        ∀ j, ∃ r : ℝ, z j = (r : ℂ) * z ⟨0, hn⟩)

theorem result : claim

Here n≥1, all z_j are nonzero complex numbers, and roots of the derivative are counted with multiplicity. No simplicity, distinctness, reality or common argument hypothesis is imposed on the nonzero roots. Since P′(0)=∏_j(−z_j)≠0, no critical point is zero. The final equivalence says that all z_j are real multiples of the nonzero first root, which is exactly collinearity on a line through the origin.

The unrestricted conjecture is preregistered in https://github.com/the-omega-institute/trureturing/issues/18487. It is an explicit external conjecture in the first problem tier. The motivation address names its general-degree resolution, including the necessary as well as sufficient equality condition.

The source proves the n=2 case in Proposition 6.4 and gives the weaker Theorem 6.1 bound. Conjecture 6.3 asks for the displayed sharp bound for general n and states the collinear sufficient equality condition. The source locator and the boundary of the literature evidence are in tang2025schoenberg. The formal result proves the full conjecture and strengthens its equality statement to an equivalence.

Write b_j=1/z_j, D=diag(b_j), J for the n by n all-ones matrix, and B=D(I+J)=D+DJ. The perturbation DJ has rank one for n≥1 and nonzero b_j. Reversing the derivative polynomial gives the exact identity

$$ (P')^{\mathrm{rev}}(t)=\left(\prod_j(-z_j)\right)\det(tI-B). $$

Consequently the reciprocal critical points are the eigenvalues of B, with multiplicity. Put s=√(n+1), c=1/(s+1) and

$$ R=I+cJ,\qquad R^{-1}=I-\frac{1}{s(s+1)}J. $$

Then R is Hermitian, R²=I+J, and RBR⁻¹=RDR. Cyclicity of trace yields

$$ |RDR|_F^2 =\operatorname{tr}\bigl(D^*(I+J)D(I+J)\bigr) =3\sum_j|b_j|^2+\left|\sum_j b_j\right|^2. $$

The Schur–Frobenius inequality bounds the sum of squared eigenvalue moduli by this Frobenius norm. Equality holds exactly when RDR is normal. Cancelling R and R⁻¹ reduces normality to

$$ \overline{b_i}b_j=b_i\overline{b_j}\quad\text{for all }i,j. $$

Because the b_j are nonzero, this means each b_j/b_1 is real. Inversion preserves collinearity through the origin, giving precisely the equality criterion in claim.

A finite nonempty family of nonzero complex z_j whose derivative-root reciprocal-square sum exceeds the stated bound would refute the conjecture. Equality for a noncollinear family, or strict inequality for a collinear family, would refute the strengthened equality criterion. All root sums must include algebraic multiplicities; a floating-point root list alone cannot certify any of these alternatives.

D5/S3/Zeros/CriticalPoints/TangZhangOrderMinusTwo.result proves claim. The characteristic-polynomial, reciprocal-root, trace and collinearity identities occur on its proof path. Its matrix estimate is supplied by D5/S3/Zeros/CriticalPoints/SchurFrobeniusInequality.schur_frobenius, which includes the normality equality criterion. Both public theorems have proof_shape: content: the matrix triangularization and equality argument, and the polynomial-to-matrix reduction, supply the substantive proof. The settlement has admission_basis: open-problem-resolution for the preregistered external Conjecture 6.3.

The modules have computational_content.kind=none. The conclusion holds for every admissible finite family and every n≥1, without a degree cutoff.

What the settlement shows

PROVED — reciprocal critical-point spectrum. The private roots_derivative identity in the proof of result identifies the entire multiset of inverse critical points with the characteristic roots of the rank-one perturbation D(I+J). This is an exact multiplicity-preserving identity, rather than an estimate on individual roots.

PROVED — exact symmetrization. The private spectral_inequality, sandwich_reduction and core_trace steps construct R and its inverse and prove that the squared Frobenius norm of RDR is exactly the conjectured right side. Thus Schur's inequality gives the coefficient 3 without any extra degree-dependent loss.

PROVED — equality if and only if collinear. The public result proves both equality directions. The conjecture's “whenever” can therefore be strengthened to “if and only if”, including repeated nonzero roots.

PROVED mathematical consequence — recovery of Theorem 6.1. By the finite Cauchy–Schwarz inequality,

$$ \left|\sum_j b_j\right|^2\le n\sum_j|b_j|^2, $$

so the proved sharp right side is at most (n+3)Σ_j|b_j|². This recovers the source's weaker order −2 bound. This corollary is a mathematical consequence of result and Cauchy–Schwarz; it is not a separately checked conclusion of these Lean modules.

OPEN — higher negative even orders. Whether the same similarity yields sharp bounds for order −2m, including a derivation or refinement of Section 7's order −4 inequality, remains open here. The checked result controls the second powers of reciprocal critical-point moduli; it makes no assertion about higher powers.

The supplied source quotation and locator identify the conjecture in v3; they do not establish the absence of subsequent settlements. No exhaustive absence or worldwide priority is claimed. Theorem 6.1's corollary and the higher-order extension have no additional Lean proof in these modules.