{
  "id": "2011.01560",
  "version": "v1",
  "published": "2020-11-03T08:33:15.000Z",
  "updated": "2020-11-03T08:33:15.000Z",
  "title": "Irregular finite order solutions of complex LDE's in unit disc",
  "authors": [
    "Igor Chyzhykov",
    "Petro Filevych",
    "Janne Gröhn",
    "Janne Heittokangas",
    "Jouni Rättyä"
  ],
  "comment": "41 pages",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "It is shown that the order and the lower order of growth are equal for all non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coefficient $A$ is analytic in the unit disc and $\\log^+ M(r,A)/\\log(1-r)$ tends to a finite limit as $r\\to 1^-$. A family of concrete examples is constructed, where the order of solutions remain the same while the lower order may vary on a certain interval depending on the irregular growth of the coefficient. These coefficients emerge as the logarithm of their modulus approximates smooth radial subharmonic functions of prescribed irregular growth on a sufficiently large subset of the unit disc. A result describing the phenomenon behind these highly non-trivial examples is also established. En route to results of general nature, a new sharp logarithmic derivative estimate involving the lower order of growth is discovered. In addition to these estimates, arguments used are based, in particular, on the Wiman-Valiron theory adapted for the lower order, and on a good understanding of the right-derivative of the logarithm of the maximum modulus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-03T08:33:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34M10",
      "30D35"
    ],
    "keywords": [
      "irregular finite order solutions",
      "unit disc",
      "lower order",
      "complex ldes",
      "approximates smooth radial subharmonic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}