bibkey: kosaki2014strong
authors: H. Kosaki
year: 2014
title: "Strong monotonicity for various means"
doi: 10.1016/j.jfa.2014.07.021
url: https://doi.org/10.1016/j.jfa.2014.07.021
claim: "Normalized hyperbolic sine ratios and ratios of Stolarsky means have logarithmic integral and infinite divisibility criteria."
strata_touched:
  - D5/S3/Analytic/Fourier/SinhLogIntegral
  - D5/S3/Quantum/Matrix/StolarskyMean/KernelFactor
license: citation-only
triage: anchor

Hyperbolic sine ratios and strong monotonicity

Locator

DOI: https://doi.org/10.1016/j.jfa.2014.07.021

Publisher locator: https://doi.org/10.1016/j.jfa.2014.07.021

H. Kosaki, Strong monotonicity for various means, Journal of Functional Analysis 267 (2014), 1917–1958.

Mathematical scope

The supplied reading of institutional preprint PS2013-1 identifies Lemma 2(i) and Corollary 3 with logarithmic integral formulas for normalized hyperbolic sine functions and their ratios. Remark 4(ii) records infinite divisibility. Theorem 12 in section 6 treats ratios of two-parameter Stolarsky means.

For p at least one, put E(a,t)=sinh(at)/(at), with value one at zero. The supplied application of Theorem 12 gives infinite divisibility of B(p,t)=E(p,t)^(1/p)/E(p+1,t)^(1/(p+1)). The identity

$${\rho }_p=B_p^{2p}{E}_{p+1}^{-(p-1)/(p(p+1))}{E}_1^{-1/p}$$

then supplies the density-balanced factor used in the Stolarsky curvature decomposition. This does not assert positive definiteness of the whole curvature defect or settle the matrix conjecture by itself. The local Lean proof instead applies the compiled logarithmic integral formula and nonnegative-density criterion directly.

Source boundary

The locators and numbered results are supplied information. The preprint-to-journal numbering correspondence and publisher full text were not independently checked under the offline constraint.