{
  "id": "1803.07657",
  "version": "v1",
  "published": "2018-03-20T21:07:05.000Z",
  "updated": "2018-03-20T21:07:05.000Z",
  "title": "Bounds for modified Struve functions of the first kind and their ratios",
  "authors": [
    "Robert E. Gaunt"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We obtain a simple two-sided inequality for the ratio $\\mathbf{L}_\\nu(x)/\\mathbf{L}_{\\nu-1}(x)$ in terms of the ratio $I_\\nu(x)/I_{\\nu-1}(x)$, where $\\mathbf{L}_\\nu(x)$ is the modified Struve function of the first kind and $I_\\nu(x)$ is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for $I_\\nu(x)/I_{\\nu-1}(x)$ to immediately deduce bounds for $\\mathbf{L}_\\nu(x)/\\mathbf{L}_{\\nu-1}(x)$. We note some consequences and obtain further bounds for $\\mathbf{L}_\\nu(x)/\\mathbf{L}_{\\nu-1}(x)$ by adapting techniques used to bound the ratio $I_\\nu(x)/I_{\\nu-1}(x)$. We apply these results to obtain new bounds for the condition numbers $x\\mathbf{L}_\\nu'(x)/\\mathbf{L}_\\nu(x)$, the ratio $\\mathbf{L}_\\nu(x)/\\mathbf{L}_\\nu(y)$ and the modified Struve function $\\mathbf{L}_\\nu(x)$ itself. Amongst other results, we obtain two-sided inequalities for $x\\mathbf{L}_\\nu'(x)/\\mathbf{L}_\\nu(x)$ and $\\mathbf{L}_\\nu(x)/\\mathbf{L}_\\nu(y)$ that are given in terms of $xI_\\nu'(x)/I_\\nu(x)$ and $I_\\nu(x)/I_\\nu(y)$, respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-20T21:07:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C20",
      "26D07",
      "33C10"
    ],
    "keywords": [
      "modified struve function",
      "first kind",
      "modified bessel function",
      "simple two-sided inequality",
      "immediately deduce bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}