{
  "abstract": "Let $d_i(m)$ denote the coefficient of $x^i$ in the Boros--Moll polynomial $P_m(x)$. We prove that the sequence $\\{d_{i-1}(m)d_{i+1}(m)/d_i(m)^2\\}_{i=1}^{m-1}$ is strictly log-concave for every $m\\ge4$, settling a conjecture of Chen and Gu. For every $m\\ge 3$, we show that the consecutive ratios $d_i(m)/d_{i-1}(m)$ form a strictly log-convex sequence for $1\\le i\\le\\lfloor m/2\\rfloor+1$ and a strictly log-concave sequence for $\\lfloor m/2\\rfloor\\le i\\le m$, as conjectured by Zhao. For each fixed $i\\ge 1$, we establish $(-1)^{r-1}Δ_m^r\\log d_i(m)>0$ for $r\\ge 2$ and $m\\ge i$, where $Δ_m$ denotes the forward difference with respect to $m$. For $i=0$, we obtain the opposite inequalities. We thereby prove a conjecture of Jiang and Wang and two further conjectures of Zhao. For every integer $i\\ge 0$, we prove strict infinite log-monotonicity of the sequence $\\{d_i(m-1)d_i(m+1)/(\\binom{m}{i+1}d_i(m)^2)\\}_{m\\ge i+1}$, strengthening another conjecture of Zhao. We further show that every positive real power of the binomially normalized ratio sequence is a Hausdorff moment sequence with a strictly totally positive Hankel matrix. We also obtain asymptotic formulas for forward differences of arbitrary fixed order, resolving a conjecture of Jiang and Wang. For integers $i\\ge 1$, we prove that the sequence $\\{2^n d_i(i)/d_i(i+n)\\}_{n\\ge 0}$ is an infinitely divisible Hausdorff moment sequence and that the Hankel matrix $[1/d_i(i+j+k)^τ]_{j,k\\ge 0}$ is strictly totally positive for every $τ>0$. For the integral interpolation, we show that the Hausdorff moment property of the normalized reciprocals holds precisely when the positive real coefficient index is at least $1/4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25381",
  "author_total": 3,
  "authors": [
    {
      "name": "Cindy C. Y. Gu"
    },
    {
      "name": "Yafeng Liu"
    },
    {
      "name": "Matthew H. Y. Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5j7pdvgmpx1jy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25381",
  "fulltext_url": null,
  "id": "5j7pdvgmpx1jy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25381",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7223018031",
  "pdf_url": "https://arxiv.org/pdf/2610.25381",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5j7pdvgmpx1jy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25381",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25381"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Inequalities for Higher-Order Logarithmic Differences of Boros--Moll Coefficients",
  "title_text": "Inequalities for Higher-Order Logarithmic Differences of Boros--Moll Coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/5j7pdvgmpx1jy",
  "venue": "arXiv",
  "versions": []
}