{
  "abstract": "Let $\\mathcal{P}$ be the class of polynomials $P$ of degree at least $2$ with $P(0)=0$, $P'(0)=1$, let $S(P)=\\min_{P'(c)=0}|P(c)/c|$ and let $\\mathcal{M}=\\sup_{P\\in\\mathcal{P}}S(P)$ be Smale's mean value constant. Smale proved that $\\mathcal{M}\\le4$, observed that $\\mathbb{M}\\ge 1$, and asked whether $\\mathcal{M}=1$. We show that $\\mathcal{M}$ coincides with $$ \\mathcal{B}=\\sup\\frac1{B(0)}, $$ the supremum running over pairs $(W,B)$ in which $W$ is a normalised univalent map of the standard unit disk $\\mathbb{D}$ and $B$ is a finite Blaschke product with $B(0)>0$ such that $WB$ is locally univalent in $\\mathbb{D}$. Smale's question is therefore a problem in univalent function theory. We also introduce the degree-restricted constants $\\mathcal{B}_d$ for $B$ with $\\mathrm{deg} B \\le d$ and show that $$1.1018586788\\ldots=\\sqrt{\\frac{8081}{6656}} \\le \\mathcal{B}_1 < \\sqrt{\\frac{7}{5}}=1.1832159566\\ldots$$ This implies that $\\mathcal{M}>1.1018$ and we also show that there are polynomials of arbitrarily large degree with $S(P)>1.1018$. The polynomial construction behind these results also settles Problems 1A and 1B of Smale's 1981 paper. For $P\\in\\mathcal P$ and an integer $k\\ge2$ let $S_k(P)=\\min_{P'(c)=0}|P^{(k)}(0)/k!|^{1/(k-1)}|P(c)|$. Smale proved, by Löwner's coefficient estimate for inverses of univalent functions, that $\\sup_{P\\in\\mathcal P}S_k(P)\\le C_k^{1/(k-1)}<4$, where $C_k$ is the $k$-th Catalan number and he asked whether the constant $4$ (Problem 1A), or the constant $2=C_2$ for $k=2$ (Problem 1B), can be reduced. We prove that $\\sup_{P\\in\\mathcal P}S_k(P)=C_k^{1/(k-1)}$ for every $k\\ge2$. Hence the constants $4$ and $2$ cannot be reduced and the answers to both problems are negative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21472",
  "author_total": 2,
  "authors": [
    {
      "name": "Aneesh Jatar"
    },
    {
      "name": "Tuen Wai Ng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1zavrc4add03a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21472",
  "fulltext_url": null,
  "id": "1zavrc4add03a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21472",
  "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",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7222476658",
  "pdf_url": "https://arxiv.org/pdf/2610.21472",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1zavrc4add03a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.21472",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21472"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Smale's mean value constant and admissible univalent Blaschke pairs",
  "title_text": "Smale's mean value constant and admissible univalent Blaschke pairs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/1zavrc4add03a",
  "venue": "arXiv",
  "versions": []
}