{
  "abstract": "Let $\\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let $L_p(\\mathcal{M})$ denote the associated Haagerup noncommutative $L_p$-space. For $1<p<\\infty$, let $q=p/(p-1)$. For $z\\in L_q(\\mathcal{M})$ and $y\\in L_p(\\mathcal{M})$, denote by $\\langle z,y\\rangle=\\operatorname{Re}\\operatorname{Tr}(z^*y)$ the canonical real duality pairing between $L_q(\\mathcal{M})$ and $L_p(\\mathcal{M})$. Define the normalized duality map by $J_p(x)=\\|x\\|_p^{\\,2-p}u|x|^{p-1}$ for $x=u|x|\\neq0$, and set $J_p(0)=0$. We establish the sharp tangent inequality $$\\|x+y\\|_p^2\\ge\\|x\\|_p^2+2\\langle J_p(x),y\\rangle+(p-1)\\|y\\|_p^2,\\qquad x,y\\in L_p(\\mathcal{M}),\\quad 1<p\\le2.$$ The reverse inequality holds for $2\\le p<\\infty$, and the constant $p-1$ is optimal. Our inequality extends Xu's tangent inequality \\cite{Xu89} from classical $L_p$-spaces to general noncommutative $L_p$-spaces and provides the sharp tangent formulation of the Ricard--Xu convexity inequality \\cite{RX16}. When specialized to Schatten $p$-classes, it also yields the corresponding tangent formulation of the Ball--Carlen--Lieb convexity inequality \\cite{BCL94}. Under a suitable positivity assumption, we further establish a sharp complement to Xu's tangent inequality for Schatten $p$-classes in the range $0<p<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16202",
  "author_total": 2,
  "authors": [
    {
      "name": "Zongben Xu"
    },
    {
      "name": "Teng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4n81tndayy4he.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16202",
  "fulltext_url": null,
  "id": "4n81tndayy4he",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16202",
  "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.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.16202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4n81tndayy4he.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp tangent inequalities in noncommutative $L_p$-spaces",
  "title_text": "Sharp tangent inequalities in noncommutative L_p-spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/4n81tndayy4he",
  "venue": "arXiv",
  "versions": []
}