{
  "abstract": "For the Jacobi family with parameters $(0,β)$, $β\\geq-1/3$, we prove a continuous comparison theorem for the associated weighted Jacobi--Radau functions. When two consecutive functions are parametrised by the same Prüfer phase, the function of higher degree attains that phase closer to $θ=0$ and has strictly larger amplitude. In particular, the moduli of all corresponding relative extrema increase strictly with the degree. The lower bound $-1/3$ is sharp for this continuous statement: when $-1<β<-1/3$, the amplitude inequality is reversed at sufficiently small positive phases. This local reversal does not determine the optimal range for the discrete extremal inequalities. For $-1/3\\leqβ\\leq0$, the positive Jacobi product formula identifies the endpoint values of the Lebesgue functions with the global Lebesgue constants, so $(Λ_n^{(0,β)})_{n\\geq0}$ is strictly increasing. At $β=0$ the weighted Radau functions reduce to $P_m^{(0,-1)}$. We thereby recover the theorem of Wong and Zhang and obtain, through an exact total-variation formula, a short proof of the Qu--Wong theorem on Legendre Lebesgue constants. The representation and the termwise comparison together realise the alternative-expression approach proposed by Qu and Wong, without asymptotic expansions, error bounds, or finite numerical verification. The proof is based on the equal-phase Prüfer architecture developed in the first author's earlier preprint arXiv:2608.01404; it applies that architecture to the problem posed by Qu and Wong.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13030",
  "author_total": 2,
  "authors": [
    {
      "name": "K. Castillo"
    },
    {
      "name": "P. -C. Hang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fj8w8rk1sjnzh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13030",
  "fulltext_url": null,
  "id": "fj8w8rk1sjnzh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13030",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7213083459",
  "pdf_url": "https://arxiv.org/pdf/2609.13030",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fj8w8rk1sjnzh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.13030",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13030"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants",
  "title_text": "Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/fj8w8rk1sjnzh",
  "venue": "arXiv",
  "versions": []
}