{
  "abstract": "Let $m$ denote the normalized Haar measure on the unit circle $\\mathbb{T}$. For $0<r<\\infty$, define $\\lVert f\\rVert_{r}:=\\left(\\int_{\\mathbb{T}}|f|^{r}\\,\\mathrm{d} m\\right)^{1/r}$, with $\\lVert f\\rVert_{0}$ and $\\lVert f\\rVert_{\\infty}$ interpreted as the geometric mean and the supremum norm, respectively. Set $Q_n(z)=1+z^n$. We prove that, for every nonzero polynomial $p$ of degree $n$ whose zeros all lie on $\\mathbb{T}$, $$ \\frac{\\lVert p\\rVert_s}{\\lVert Q_n\\rVert_s} \\le \\frac{\\lVert p\\rVert_t}{\\lVert Q_n\\rVert_t}, \\qquad 0\\le s\\le t\\le\\infty. $$ This settles Baernstein's quasi-norm monotonicity conjecture. As corollaries, we obtain an $L^r$ extension of Visser's coefficient inequality, the sharp O'Hara--Rodriguez inequality and its higher-power analogues, the Erdős--Szekeres product bound $ \\left\\lVert\\prod_{j=1}^N(1-z^{s_j})\\right\\rVert_\\infty\\ge2\\sqrt N $ for all positive integers $s_1,\\ldots,s_N$ and Agler--McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02009",
  "author_total": 1,
  "authors": [
    {
      "name": "Teng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s715gkfab7a2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02009",
  "fulltext_url": null,
  "id": "0s715gkfab7a2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02009",
  "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.CV"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7215881211",
  "pdf_url": "https://arxiv.org/pdf/2610.02009",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0s715gkfab7a2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.02009",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02009"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero",
  "title_text": "Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/0s715gkfab7a2",
  "venue": "arXiv",
  "versions": []
}