{
  "abstract": "We show that $\\frac1{g(x)}$, with $g(x)=\\frac{x-1}{x\\ln x}$, is a complete Bernstein function and give its Lévy measure explicitly. For $h(x)=\\frac{(x^2-1)\\ln(x+1)}{x^2\\ln x}$, we prove that $h$ is not a Stieltjes function, so $\\frac1h$ is not a complete Bernstein function, but $\\frac1h$ is a Bernstein function. The latter rests on the positivity of a function built from the convolution of the exponential integrals $E_1$ and $E_2$. We also give a related completely monotonic function that is not a Stieltjes function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40718",
  "author_total": 1,
  "authors": [
    {
      "name": "Feng Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/420rbdh8sr849.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40718",
  "fulltext_url": null,
  "id": "420rbdh8sr849",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40718",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40718",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/420rbdh8sr849.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40718",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40718"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Bernstein and Stieltjes properties of some functions involving logarithmic functions",
  "title_text": "Bernstein and Stieltjes properties of some functions involving logarithmic functions",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/420rbdh8sr849",
  "venue": "arXiv",
  "versions": []
}