{
  "abstract": "We study the radius of absolute monotonicity of rational approximations to the exponential of the form \\[ ϕ(z)=\\frac{P(z)}{(1-z/b)^s}, \\qquad b>0,\\qquad °P\\le s, \\] with approximation orders two and three. For second-order approximation, we prove that the maximal radius in degree $s=5$ is $10$ and determine the unique extremizer. Hence the conjectured value $2s$ by Lóczi and Ketcheson (LMS Journal of Computation and Mathematics, 17 (2014), no.~1, 159--205) remains valid in this case. In contrast, for every integer $s\\ge6$ we construct explicitly a rational function with rational coefficients and exact radius \\[ 2s+\\frac{4(s-3)}{s^2-3s+4}>2s. \\] For third-order approximation, we construct, for every $s\\ge5$, an explicit rational family with exact radius \\[ 2s-1>s-1+\\sqrt{s^2-1}. \\] These constructions disprove the conjectures of Lóczi and Ketcheson concerning rational functions with a single repeated pole.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06941",
  "author_total": 1,
  "authors": [
    {
      "name": "Rachid Ait-Haddou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/878z42fk1zd6a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06941",
  "fulltext_url": null,
  "id": "878z42fk1zd6a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06941",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06941",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/878z42fk1zd6a.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.06941",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06941"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Maximal Absolute Monotonicity of Balanced Rational Functions with a Single Repeated Pole",
  "title_text": "Maximal Absolute Monotonicity of Balanced Rational Functions with a Single Repeated Pole",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/878z42fk1zd6a",
  "venue": "arXiv",
  "versions": []
}