{
  "id": "2102.07842",
  "version": "v1",
  "published": "2021-02-15T20:42:16.000Z",
  "updated": "2021-02-15T20:42:16.000Z",
  "title": "An extension of the Geometric Modulus Principle to holomorphic and harmonic functions",
  "authors": [
    "Matt Hohertz"
  ],
  "comment": "7 pages, 2 figures",
  "categories": [
    "math.CV",
    "math.FA"
  ],
  "abstract": "Kalantari's Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if $p(z) = a_0 + \\sum_{j=k}^n a_j\\left(z-z_0\\right)^j,\\;a_0a_ka_n \\neq 0$, then the complex plane near $z = z_0$ comprises $2k$ sectors of angle $\\frac{2\\pi}{k}$, alternating between arguments of ascent (angles $\\theta$ where $|p(z_0 + te^{i\\theta})| > |p(z_0)|$ for small $t$) and arguments of descent (where the opposite inequality holds). In this paper, we generalize the Geometric Modulus Principle to holomorphic and harmonic functions. As in Kalantari's original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-15T20:42:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30A10",
      "42A99"
    ],
    "keywords": [
      "harmonic functions",
      "holomorphic",
      "kalantaris geometric modulus principle",
      "opposite inequality holds",
      "kalantaris original paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}