{
  "abstract": "The class $M_α$ of $α$-convex functions, introduced by Mocanu in 1969, interpolates between starlike ($α=0$) and convex ($α=1$) mappings and has traditionally been studied only for $α\\geq0$. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes $M_α$ for every real $α\\neq0$ through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of $β$ for which $M_α\\subset C_β$ when $α\\geq1$; we also show that the reciprocal inclusion $C_β\\subset M_α$ fails for every $α>1$. Building on this, we obtain a sharp Fekete--Szegő inequality, valid for every $α\\notin\\{0,-1,-1/2\\}$, and sharp bounds for the first two logarithmic coefficients of $M_α$, which recover the classical values for starlike and convex functions as special cases. We also determine the order of $M_α$, establish a piecewise sharp bound for its Schwarzian norm together with an explicit quasiconformal extension criterion via the Ahlfors--Weill theorem, and compute the Schwarzian norm of the $α$-Koebe function explicitly for $α=1/n$, $n\\in\\mathbb N$, verified for $n\\leq9$ and numerically supported for $n<50$, leading to a conjecture for the general case. Finally, we propose a conjectural sharp bound for the second Hankel determinant of $M_α$, consistent with the known values $1$ and $1/8$ for the starlike and convex subclasses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.21574",
  "author_total": 4,
  "authors": [
    {
      "name": "Víctor Bravo"
    },
    {
      "name": "Pablo Carrasco"
    },
    {
      "name": "Rodrigo Hernández"
    },
    {
      "name": "Osvaldo Venegas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cpq2w8hczdqbd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.21574",
  "fulltext_url": null,
  "id": "cpq2w8hczdqbd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.21574",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7165661084",
  "pdf_url": "https://arxiv.org/pdf/2606.21574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cpq2w8hczdqbd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2606.21574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.21574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp coefficient, logarithmic and Schwarzian estimates for $α$-convex functions on the real line",
  "title_text": "Sharp coefficient, logarithmic and Schwarzian estimates for α-convex functions on the real line",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/cpq2w8hczdqbd",
  "venue": "arXiv",
  "versions": []
}