{
  "abstract": "In 1986, Shea and Sons obtained the following bound for a meromorphic function $f$ of finite order $ρ$ in the unit disc, under the hypothesis $0<λ(f)\\leq+\\infty$, and for every positive integer $n$: \\[ \\sum_{a\\ne\\infty}δ(a,f) \\leq δ(0,f^{(n)})\\bigl(1+n k(f)\\bigr) +\\frac{2n(ρ+1)}{λ(f)}. \\] Under the condition $0<α(f)\\leq+\\infty$, they also obtained \\[ \\sum_{a\\ne\\infty}δ(a,f) \\leq Δ(0,f^{(n)})\\bigl(1+n k(f)\\bigr) +\\frac{2n}{α(f)}. \\] Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever $0<λ(f)\\leq+\\infty$, one has \\[ \\sum_{a\\ne\\infty}δ(a,f) \\leq δ(0,f^{(n)})\\bigl(1+n k(f)\\bigr) +\\frac{n(ρ+1)}{λ(f)}, \\] and under the condition $0<α(f)\\leq+\\infty$ one has \\[ \\sum_{a\\ne\\infty}δ(a,f) \\leq Δ(0,f^{(n)})\\bigl(1+n k(f)\\bigr) +\\frac{n}{α(f)}. \\] The coefficient $n$ in each of $n(ρ+1)/λ(f)$ and $n/α(f)$ is best possible for every $n$, and the dependence on $ρ$ and $α(f)$ in these terms is also essential. At the end, we extend both estimates to finite subsets $A$ of an $n$-dimensional complex vector space $V\\subset\\mathbb{C}(z)$, with $f^{(n)}$ replaced by the monic differential operator $D_Vf$ of order $n$ whose kernel is $V$. We prove that both extensions are sharp for every $V$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05835",
  "author_total": 1,
  "authors": [
    {
      "name": "Sina Nadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/889h3f75scsrq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05835",
  "fulltext_url": null,
  "id": "889h3f75scsrq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05835",
  "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|Meromorphic and Entire Functions"
    ]
  ],
  "openalex": "W7212002736",
  "pdf_url": "https://arxiv.org/pdf/2609.05835",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/889h3f75scsrq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05835",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05835"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc",
  "title_text": "Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/889h3f75scsrq",
  "venue": "arXiv",
  "versions": []
}