{
  "abstract": "In this paper, we present Lappan's five-point type criteria for $φ$-normality of sense-preserving harmonic mappings and families of harmonic mappings in the unit disc $\\mathbb{D}$. We establish a normality criterion involving the higher-order spherical derivative of order $k$, and $k+2$ distinct test values under suitable conditions. We further reduce the number of test values to $\\left[\\frac{k}{2}\\right]+2$ under an additional differential condition involving the analytic and co-analytic parts. We also prove a generalization in which the spherical derivative is replaced by a polynomial with non-negative real coefficients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19829",
  "author_total": 2,
  "authors": [
    {
      "name": "Gopal Datt"
    },
    {
      "name": "Ritesh Pal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/af18wza1e6wpt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19829",
  "fulltext_url": null,
  "id": "af18wza1e6wpt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19829",
  "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": "W7213611532",
  "pdf_url": "https://arxiv.org/pdf/2609.19829",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/af18wza1e6wpt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19829",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19829"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On $φ$-Normality of Harmonic Mappings and Their Families",
  "title_text": "On φ-Normality of Harmonic Mappings and Their Families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/af18wza1e6wpt",
  "venue": "arXiv",
  "versions": []
}