{
  "id": "2406.08932",
  "version": "v1",
  "published": "2024-06-13T09:02:08.000Z",
  "updated": "2024-06-13T09:02:08.000Z",
  "title": "Inequalities for 1/(1-cos(x)) and its derivatives",
  "authors": [
    "Horst Alzer",
    "Henrik L. Pedersen"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove that the function $g(x)= 1 / \\bigl( 1 - \\cos(x) \\bigr)$ is completely monotonic on $(0,\\pi]$ and absolutely monotonic on $[\\pi, 2\\pi)$, and we determine the best possible bounds $\\lambda_n$ and $\\mu_n$ such that the inequalities $$ \\lambda_n \\leq g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y) \\quad (n \\geq 0 \\,\\,\\, \\mbox{even}) $$ and $$ \\mu_n \\leq g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y) \\quad (n \\geq 1 \\,\\,\\, \\mbox{odd}) $$ hold for all $x,y\\in (0,\\pi)$ with $x+y\\leq \\pi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-13T09:02:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A48",
      "26D05",
      "11B68"
    ],
    "keywords": [
      "inequalities",
      "derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}