{
  "id": "2302.08127",
  "version": "v1",
  "published": "2023-02-16T07:34:44.000Z",
  "updated": "2023-02-16T07:34:44.000Z",
  "title": "Matrix Inequalities between $f(A)σf(B)$ and $AσB$",
  "authors": [
    "Manisha Devi",
    "Jaspal Singh Aujla",
    "Mohsen Kian",
    "Mohammad Sal Moslehian"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $A$ and $ B$ be $n\\times n$ positive definite complex matrices, let $\\sigma$ be a matrix mean, and let $f : [0,\\infty)\\to [0,\\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\\prime}(0)(A \\sigma B)\\leq \\frac{f(m)}{m}(A\\sigma B)\\leq f(A)\\sigma f(B)\\leq \\frac{f(M)}{M}(A\\sigma B)\\leq f^{\\prime}(M)(A\\sigma B),$$ where $m$ is the smallest among eigenvalues of $A$ and $B$ and $M$ is the largest among eigenvalues of $A$ and $B$. If $f$ is differentiable concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under some mild conditions. In particular, if $f(x)/x$ is increasing, then $$|||f(A)+f(B)|||\\leq\\frac{f(M)}{M} |||A+B|||\\leq |||f(A+B)|||$$ holds for all $A$ and $B$ with $M\\leq A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski's determinant inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-16T07:34:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A60",
      "15A42"
    ],
    "keywords": [
      "matrix inequalities",
      "minkowskis determinant inequality",
      "positive definite complex matrices",
      "reverse inequalities hold",
      "matrix mean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}