{
  "id": "1609.00218",
  "version": "v1",
  "published": "2016-09-01T12:57:11.000Z",
  "updated": "2016-09-01T12:57:11.000Z",
  "title": "On Polya' Theorem in Several Complex Variables",
  "authors": [
    "Ozan Günyüz",
    "Vyacheslav Zakharyuta"
  ],
  "comment": "9 pages. arXiv admin note: substantial text overlap with arXiv:1605.09314",
  "journal": "Banach Center Publications 107(2015), 149-157",
  "doi": "10.4064/bc107-0-10",
  "categories": [
    "math.CV"
  ],
  "abstract": "Let $K$ be a compact set in $\\mathbb{C}$, $f$ a function analytic in $\\overline{\\mathbb{C}}\\smallsetminus K$ vanishing at $\\infty $. Let $% f\\left( z\\right) =\\sum_{k=0}^{\\infty }a_{k}\\ z^{-k-1}$ be its Taylor expansion at $\\infty $, and $H_{s}\\left( f\\right) =\\det \\left( a_{k+l}\\right) _{k,l=0}^{s}$ the sequence of Hankel determinants. The classical Polya inequality says that \\[ \\limsup\\limits_{s\\rightarrow \\infty }\\left\\vert H_{s}\\left( f\\right) \\right\\vert ^{1/s^{2}}\\leq d\\left( K\\right) , \\]% where $d\\left( K\\right) $ is the transfinite diameter of $K$. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Polya's inequality, considered by the second author in Math. USSR Sbornik, 25 (1975), 350-364.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-01T12:57:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32A22",
      "32A70",
      "32U35",
      "46E10"
    ],
    "keywords": [
      "complex variables",
      "classical polya inequality says",
      "transfinite diameter",
      "hankel determinants",
      "compact set"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}