bibkey: furuichi2022bounds
authors: S. Furuichi and M. Eghbali Amlashi
year: 2024
title: "On bounds of logarithmic mean and mean inequality chain"
doi: 10.7153/mia-2024-27-27
url: https://arxiv.org/abs/2203.01134
claim: "Remark 3.2(i) of arXiv v2 conjectures that t (cosh t)^(1-r) / sinh t is not positive definite for r < 1."
strata_touched:
  - D5/S3/Analytic/Fourier/PositiveDefiniteImaginaryAxis
  - D5/S3/Quantum/Matrix/FuruichiAmlashiHyperbolicRatio
license: citation-only
triage: anchor

Logarithmic means and hyperbolic positive definiteness

Locator

DOI: https://doi.org/10.7153/mia-2024-27-27

Preprint: https://arxiv.org/abs/2203.01134

S. Furuichi and M. Eghbali Amlashi, On bounds of logarithmic mean and mean inequality chain, Mathematical Inequalities & Applications 27(2) (2024), 379–399. The supplied source identifies Remark 3.2(i) in arXiv version 2 with Remark 2(i) in the journal article.

Source statement

Thus these numerical computations support the conjecture that the function ϕr (t) for r < 1, is not positive definite.

Here ϕr(t) = t (cosh t)^(1-r) / sinh t, continuously extended by ϕr(0)=1. Positive definiteness means that every finite matrix with entries ϕr(x_j-x_i) is positive semidefinite.

Scope

The source reports positive definiteness for r ≥ 1 and supplies numerical obstructions at selected parameters below 1. The imaginary-axis method uses an analytic extension to the disk of radius π/2; its values approach zero on the positive imaginary radius, whereas analytic real positive definite functions have values at least their value at zero there. The analytic obstruction is a separate general result, rather than a claim attributed to this paper.

Source boundary

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