{
  "abstract": "In this paper, we study noncommutative analogues of the classical Cauchy and Lagrange mean value theorems in the setting of free noncommutative function theory. For noncommutative functions on a uniformly open noncommutative set, we prove a Cauchy-type mean value inequality involving the right noncommutative difference operator $Δ_R$ and the G-derivative, and we establish a mean value theorem for noncommutative curves. Furthermore, we characterize uniform comparison (Lipschitz-type) inequalities at all matrix levels and prove a factorization theorem for noncommutative functions. The proofs combine the Hahn--Banach theorem, the fundamental theorem of calculus along a line segment, and the direct-sum compatibility of noncommutative functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34032",
  "author_total": 3,
  "authors": [
    {
      "name": "Yali Du"
    },
    {
      "name": "Jun Jie Miao"
    },
    {
      "name": "Tianjiao Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3bpy97dc7vxg3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34032",
  "fulltext_url": null,
  "id": "3bpy97dc7vxg3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34032",
  "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"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7223948817",
  "pdf_url": "https://arxiv.org/pdf/2610.34032",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3bpy97dc7vxg3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34032",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34032"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Noncommutative Mean Value Theorems for Free Noncommutative Functions",
  "title_text": "Noncommutative Mean Value Theorems for Free Noncommutative Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/3bpy97dc7vxg3",
  "venue": "arXiv",
  "versions": []
}