{
  "id": "1210.6478",
  "version": "v1",
  "published": "2012-10-24T10:17:38.000Z",
  "updated": "2012-10-24T10:17:38.000Z",
  "title": "The monotonicity results and sharp inequalities for some power-type means of two arguments",
  "authors": [
    "Zhen-Hang Yang"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "For $a,b>0$ with $a\\neq b$, we define M_{p}=M^{1/p}(a^{p},b^{p})\\text{if}p\\neq 0 \\text{and} M_{0}=\\sqrt{ab}, where $M=A,He,L,I,P,T,N,Z$ and $Y$ stand for the arithmetic mean, Heronian mean, logarithmic mean, identric (exponential) mean, the first Seiffert mean, the second Seiffert mean, Neuman-S\\'{a}ndor mean, power-exponential mean and exponential-geometric mean, respectively. Generally, if $M$ is a mean of $a$ and $b$, then $M_{p}$ is also, and call \"power-type mean\". We prove the power-type means $P_{p}$, $T_{p}$, $N_{p}$, $Z_{p}$ are increasing in $p$ on $\\mathbb{R}$ and establish sharp inequalities among power-type means $A_{p}$, $He_{p}$, $L_{p}$, $I_{p}$, $P_{p}$, $N_{p}$, $Z_{p}$, $Y_{p}$% . From this a very nice chain of inequalities for these means L_{2}<P<N_{1/2}<He<A_{2/3}<I<Z_{1/3}<Y_{1/2} follows. Lastly, a conjecture is proposed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-24T10:17:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E60",
      "26D05",
      "26A48"
    ],
    "keywords": [
      "power-type mean",
      "monotonicity results",
      "second seiffert mean",
      "first seiffert mean",
      "logarithmic mean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.6478Y"
    }
  }
}