{
  "abstract": "Let $Ω\\subset\\mathbb{R}^n$ be a bounded domain with $n\\geq2$. The well-known Payne-Pólya-Weinberger conjecture for consecutive eigenvalue ratios asserts that $$ \\frac{λ_{m+1}(Ω)}{λ_m(Ω)}\\leq\\left(\\frac{j_{n / 2,1}}{j_{n / 2-1,1}}\\right)^2,\\quad m=1,2,3,\\cdots. $$ Here, $j_{p,k}$ denotes the $k$-th positive zero of the Bessel function $J_p$, and $λ_j(Ω)$ are the Dirichlet eigenvalues of $Ω$, counted with multiplicity and ordered as $$ 0<λ_1(Ω)<λ_2(Ω)\\leqλ_3(Ω)\\leq\\cdots. $$ The cases $m=1,2,3$ were established by Ashbaugh and Benguria in their seminal works. In this paper, we give an affirmative answer to the Payne-Pólya-Weinberger conjecture for $m=4$ by combining a quadratic-form identity with a Brouwer degree argument. We also extend the corresponding spectral inequality to arbitrary bounded open sets in $\\mathbb{R}^n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31088",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhen Song"
    },
    {
      "name": "Ruixiao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3tza4ay7e6vkd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31088",
  "fulltext_url": null,
  "id": "3tza4ay7e6vkd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31088",
  "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.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223283445",
  "pdf_url": "https://arxiv.org/pdf/2610.31088",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3tza4ay7e6vkd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31088",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31088"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian",
  "title_text": "A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/3tza4ay7e6vkd",
  "venue": "arXiv",
  "versions": []
}