{
  "abstract": "Let $\\mathcal{C}(Φ)$ denote a subclass of class $ \\mathcal{C}$ of convex functions in the unit disc $\\mathbb{D}=\\{z:|z|<1, z\\in \\mathbb{C}\\}$ defined by \\begin{equation*} {Re}\\left \\{1+ \\frac{ zf^{\\prime \\prime}(z)}{f^{\\prime}(z)}\\right \\} \\prec Φ(z), \\end{equation*} where $Φ$ represents the Ma--Minda function in $\\mathbb{D}$. For $f\\in\\mathcal{C}(Φ)$, we find the sharp estimates of $||γ_{2}|-|γ_{1}||$, where $γ_{1}$ and $γ_{2}$ are the first two logarithmic coefficients of $f$. The bound is also obtained for the inverse logarithmic coefficients $Γ_{1}$ and $Γ_{2}$. We then show that, under suitable conditions, the bounds for $||γ_{2}|-|γ_{1}||$ and $||Γ_{2}|-|Γ_{1}||$ are identical, which gives the invariance result between these functionals. We also present bounds for the differences for various classes of univalent functions by choosing particular function $Φ$. Thus our results unify several results studied in the literature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20645",
  "author_total": 3,
  "authors": [
    {
      "name": "Rashid Ali"
    },
    {
      "name": "Mohsan Raza"
    },
    {
      "name": "Umar Raza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/75cvbgkvxrx64.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20645",
  "fulltext_url": null,
  "id": "75cvbgkvxrx64",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20645",
  "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": "W7222690647",
  "pdf_url": "https://arxiv.org/pdf/2610.20645",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/75cvbgkvxrx64.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20645",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20645"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Moduli Difference of Initial Logarithmic and Inverse Logarithmic Coefficients of Ma-Minda Convex Functions",
  "title_text": "On the Moduli Difference of Initial Logarithmic and Inverse Logarithmic Coefficients of Ma-Minda Convex Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/75cvbgkvxrx64",
  "venue": "arXiv",
  "versions": []
}