{
  "id": "2210.14023",
  "version": "v1",
  "published": "2022-10-25T13:47:41.000Z",
  "updated": "2022-10-25T13:47:41.000Z",
  "title": "On norm inequalities related to the geometric mean",
  "authors": [
    "Shaima'a Freewan",
    "Mostafa Hayajneh"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $A_i$ and $B_i$ be positive definite matrices for all $i=1,\\cdots,m.$ It is shown that $$\\left|\\left|\\sum_{i=1}^m(A_i^2\\sharp B_i^2)^r\\right|\\right|_1\\leq\\left|\\left|\\left(\\left(\\sum_{i=1}^mA_i\\right)^{\\frac{pr}{_2}}\\left(\\sum_{i=1}^mB_i\\right)^{pr}\\left(\\sum_{i=1}^mA_i\\right)^{\\frac{rp}{_2}}\\right)^{\\frac{1}{p}}\\right|\\right|_1,$$for all $p>0$ and for all $r\\geq1.$ We conjecture this inequality is also true for all unitarily invariant norms. We give an affirmative answer to the case of $m=2,$ $p\\geq1$, $r\\geq1$ and for all unitarily invariant norms. In other words, it is shown that $$\\left|\\left|\\left|\\left(A^{^2}\\sharp B^{^2}\\right)^{r}+\\left(C^{^2}\\sharp D^{^2}\\right)^{r}\\right|\\right|\\right|\\leq \\left|\\left|\\left|\\left(\\left(A+C\\right)^{^\\frac{rp}{_2}}\\left(B+D\\right)^{{rp}}\\left(A+C\\right)^{^\\frac{rp}{_2}}\\right)^{\\frac{1}{_p}}\\right|\\right|\\right|,$$for all unitarly invariant norms, for all $p\\geq1$ and for all $r\\geq1$, where $A,B,C,D$ are positive definite matrices. This gives an affirmative answer to the conjecture posed by Dinh, Ahsani and Tam in the case of $m=2$. The preceding inequalities directly lead to a recent result of Audenaert \\cite{ANIFP}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-25T13:47:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric mean",
      "norm inequalities",
      "inequality",
      "positive definite matrices",
      "unitarily invariant norms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}