{
  "abstract": "We prove that, for all $x > 0$, the product of modified Bessel functions satisfies \\[ \\frac{1}{2\\sqrt{x^2+ν^2+1/5}}<I_ν(x)K_ν(x)<\\frac{1}{2\\sqrt{x^2+ν^2-1/4}} , \\] where the lower bound holds for $ν>-1$ and the upper bound for $ν\\ge 1/2$, and that the sum of the logarithmic derivatives satisfies \\[ -\\frac{x}{x^2+ν^2-1}<\\frac{I_ν'(x)}{I_ν(x)}+\\frac{K_ν'(x)}{K_ν(x)}<-\\frac{x}{x^2+ν^2+7/20}, \\] where the upper bound holds for $ν>-1$ and the lower bound for $ν\\ge 1$. All four constants are best possible. The four proofs share the same elementary structure: in each case the relevant auxiliary function satisfies a second-order differential inequality, and the inequality follows from the strong maximum principle. Among other applications, we obtain sharp monotonicity properties of the product and improved bounds for the ratios $I_{ν-1}(x)/I_ν(x)$ and $K_{ν+1}(x)/K_ν(x)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07933",
  "author_total": 2,
  "authors": [
    {
      "name": "Javier Segura"
    },
    {
      "name": "Soichiro Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqawvrnan9fq8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07933",
  "fulltext_url": null,
  "id": "bqawvrnan9fq8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07933",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7221026682",
  "pdf_url": "https://arxiv.org/pdf/2610.07933",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqawvrnan9fq8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07933",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07933"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative",
  "title_text": "Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/bqawvrnan9fq8",
  "venue": "arXiv",
  "versions": []
}