{
  "id": "1606.08134",
  "version": "v1",
  "published": "2016-06-27T06:22:33.000Z",
  "updated": "2016-06-27T06:22:33.000Z",
  "title": "On Some Subclass of Harmonic Close-to-convex Mappings",
  "authors": [
    "Nirupam Ghosh",
    "A. Vasudevarao"
  ],
  "comment": "18 Pages",
  "categories": [
    "math.CV"
  ],
  "abstract": "Let $\\mathcal{H}$ denote the class of harmonic functions $f$ in $\\mathbb{D}:= \\{z\\in \\mathbb{C}:|z| < 1\\}$ normalized by $f(0) = 0 = f_z(0) -1$. For $\\alpha \\geq 0$, we consider the following class $$\\mathcal{W}^0_{\\mathcal{H}}(\\alpha):= \\{f = h + \\overline{g}\\in\\mathcal{H}: {\\rm Re\\,}(h'(z) + \\alpha z h''(z)) >|g'(z) + \\alpha z g''(z)|, \\quad z\\in \\mathbb{D}\\}. $$ In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class $\\mathcal{W}^0_{\\mathcal{H}}(\\alpha)$. We also prove growth theorem, convolution, convex combination properties for functions in the class $\\mathcal{W}^0_{\\mathcal{H}}(\\alpha)$. Finally, we determine the value of $r$ so that the partial sums of functions in the class $\\mathcal{W}^0_{\\mathcal{H}}(\\alpha)$ are close-to-convex in $|z|<r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-27T06:22:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C45",
      "30C80"
    ],
    "keywords": [
      "harmonic close-to-convex mappings",
      "convex combination properties",
      "coefficient conjecture",
      "harmonic functions",
      "growth theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}