An Optimal Stability Theorem for H\"older's Inequality

Gangsong Leng; Yifan Lu

Nonnegative sequences summing to one each obey 1 minus the Hölder sum being at least (1/(2pq)) times the square of their L1 distance, with the constant sharp.

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Proves 1 - sum a_k^{1/p} b_k^{1/q} ≥ (1/(2pq)) (sum |a_k - b_k|)^2 for non-negative sequences summing to 1, with the constant shown best possible, plus integral version.

T0 review reviewed 2026-06-28 challenge →

load-bearing objection This paper gives a clean optimal L1 stability bound for Hölder's inequality with constant 1/(2pq) shown sharp by a two-point limiting construction.

arxiv 2605.31179 v1 pith:K3BEQZGK submitted 2026-05-29 math.FA

An Optimal Stability Theorem for H\"older's Inequality

Gangsong Leng , Yifan Lu This is my paper

classification math.FA

keywords Hölder's inequalitystability theoremoptimal constantL1 distanceconjugate exponentsinequality deficitintegral inequality

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes an optimal L1 stability theorem for Hölder's inequality in both discrete and integral settings. When a_k and b_k are nonnegative and each sum to one, the deficit 1 minus the sum of a_k to the power 1/p times b_k to the power 1/q is bounded from below by one over two p q times the square of the sum of absolute differences. The constant cannot be improved. The result quantifies how close the sequences must be when the Hölder product is nearly maximal.

Core claim

Let p greater than 1 and q greater than 1 satisfy one over p plus one over q equals one. If a_k and b_k are nonnegative and sum to one, then one minus the sum over k of a_k to the power one over p times b_k to the power one over q is at least one over two p q times the square of the sum of absolute values of a_k minus b_k. The constant one over two p q is best possible. The same inequality holds when sums are replaced by integrals of nonnegative functions with equal integrals equal to one.

What carries the argument

The quadratic lower bound on the Hölder deficit expressed in terms of the squared L1 distance between the two sequences (or functions).

Load-bearing premise

The sharpness of the constant one over two p q rests on the existence of sequences or functions where the ratio of the deficit to the squared L1 distance approaches exactly that value.

What would settle it

A pair of nonnegative sequences summing to one each for which the deficit divided by the squared L1 distance is strictly smaller than one over two p q would disprove the claimed inequality.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

The discrete inequality implies the corresponding integral form over any measure space.
Applications of Hölder's inequality acquire explicit quantitative error terms controlled by the L1 deviation of the inputs.
The constant is attained in the limit, so the bound is asymptotically tight.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same deficit-to-distance relation may supply stability estimates for other inequalities that follow from Hölder, such as those for expectations of products.
For two-point sequences one can compute the ratio explicitly and observe its approach to one over two p q as the points coalesce or separate.
The result supplies a concrete modulus of continuity between the L1 metric and the deficit functional on the simplex.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

This paper gives a clean optimal L1 stability bound for Hölder's inequality with constant 1/(2pq) shown sharp by a two-point limiting construction.

read the letter

The core result is a quadratic lower bound on the deficit in Hölder's inequality measured in L1 distance between two probability vectors. For a_k, b_k ≥ 0 summing to 1, they prove 1 - sum a_k^{1/p} b_k^{1/q} ≥ (1/(2pq)) (sum |a_k - b_k|)^2, with the same form holding in the integral setting. The constant is claimed best possible.

The paper does the obvious next step after the classical inequality: it quantifies how much the product sum drops when the sequences differ. The proof route is direct. They expand the function f(t) = t^{1/p} around t=1 to second order, get the quadratic term, and then build a two-point sequence with a small ε perturbation that makes the ratio of deficit to (L1 distance)^2 approach exactly 1/(2pq) as ε → 0. That construction is explicit enough to verify the sharpness claim.

The argument stays elementary and does not rely on heavy functional-analytic tools. Both the finite-sum and measure-space versions are written out. The constant depends only on p and q, which keeps the statement simple.

The only soft spot is that optimality is asymptotic rather than attained for any finite pair; this is standard for such stability results and does not weaken the theorem. No circularity or hidden assumptions appear in the derivation.

The work is aimed at analysts who care about quantitative versions of classical inequalities. A reader already familiar with stability literature will see the incremental but precise improvement. It is worth sending to peer review because the central claim is supported by an explicit construction and the proof steps are checkable.

Referee Report

0 major / 2 minor

Summary. The paper proves an optimal L¹-stability version of Hölder's inequality. For p>1, q>1 with 1/p+1/q=1 and non-negative sequences a_k, b_k summing to 1, it establishes 1 - ∑ a_k^{1/p} b_k^{1/q} ≥ (1/(2pq)) (∑ |a_k - b_k|)^2, with the constant shown to be sharp; an analogous integral form is also given.

Significance. If the result holds, it supplies a sharp quantitative stability estimate for a fundamental inequality, with the explicit two-point limiting construction (ε→0) and second-order Taylor expansion providing rigorous support for optimality. This strengthens the literature on stability of inequalities in functional analysis.

minor comments (2)

[Section 4] The statement of the integral version in the final section would benefit from an explicit display of the measure space and integrability assumptions to match the clarity of the discrete case.
[Introduction] A brief remark on whether the result extends to p=1 or q=1 (where Hölder degenerates) would clarify the scope, even if outside the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of the main result, and the recommendation to accept. We are pleased that the significance of the sharp stability estimate is recognized.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct inequality and proves it along with sharpness of the constant via an explicit two-point limiting construction (with perturbation ε → 0) and second-order Taylor expansion. No equations reduce by definition to their own inputs, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems appear. The derivation chain is self-contained against the stated assumptions and the provided construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result is a direct inequality proof resting on the classical statement of Hölder's inequality and standard properties of real numbers and sums; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Hölder's inequality holds for the given p,q and sequences
The stability theorem is stated as a refinement of Hölder's inequality.
standard math Basic axioms of real analysis (ordering, continuity of powers, triangle inequality for sums)
Required for any manipulation of the sums and absolute values appearing in the statement.

reviewed 2026-06-28 · how reviews work

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Cite this review

Pith. "Pith review of An Optimal Stability Theorem for H\"older's Inequality." pith.science (2026). https://pith.science/paper/K3BEQZGK

@misc{pith2026260531179,
  author       = {Pith},
  title        = {Pith review of: An Optimal Stability Theorem for H\"older's Inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K3BEQZGK}},
  note         = {Machine review of arXiv:2605.31179}
}

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abstract

We prove an optimal $L^1$ stability theorem for H\"older's inequality. Let $p>1$, $q>1$, and $1/p+1/q=1$. If $a_k,b_k\ge 0$ and \[ \sum_{k=1}^n a_k=\sum_{k=1}^n b_k=1, \] then \[ 1-\sum_{k=1}^n a_k^{1/p}b_k^{1/q} \ge \frac1{2pq}\left(\sum_{k=1}^n |a_k-b_k|\right)^2 . \] The constant $1/(2pq)$ is best possible. We also give the corresponding integral form.

discussion (0)

Forward citations

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This paper was first reviewed by grok-4.3 on June 28, 2026.

An Optimal Stability Theorem for H\"older's Inequality

The reading

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it