{
  "abstract": "Let $\\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\\mathbb{D} := \\{z \\in \\mathbb{C} : \\vert{}z\\vert{} < 1\\}$. For $β> 1$, let $\\mathcal{N}(β)$ denote the subclass of $\\mathcal{A}$ satisfying $\\text{Re}\\{1 + z f''(z)/f'(z)\\} < β$ for $z \\in \\mathbb{D}$. The main purpose of this paper is to establish sharp bounds for the pre-Schwarzian norm $\\Vert{}P_f\\Vert{}$ and Schwarzian norm $\\Vert{}S_f\\Vert{}$ for functions $f \\in \\mathcal{N}(β)$, parametrized by $f''(0)$, with special emphasis on the case $f''(0) = 0$. In addition, sharp growth, distortion, and radius results (convexity and concavity) for $\\mathcal{N}(β)$ are obtained. As an application, we determine the sharp pre-Schwarzian norm estimate for harmonic mappings $f = h + \\bar{g}$ whose analytic part $h$ belongs to $\\mathcal{N}(β)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30595",
  "author_total": 2,
  "authors": [
    {
      "name": "Molla Basir Ahamed"
    },
    {
      "name": "Rajesh Hossain"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5jevqkd6txe4q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30595",
  "fulltext_url": null,
  "id": "5jevqkd6txe4q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30595",
  "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": "W7204915888",
  "pdf_url": "https://arxiv.org/pdf/2608.30595",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5jevqkd6txe4q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30595",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30595"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Schwarzian norm estimates for some classes of analytic and harmonic mappings",
  "title_text": "Schwarzian norm estimates for some classes of analytic and harmonic mappings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/5jevqkd6txe4q",
  "venue": "arXiv",
  "versions": []
}