Archiv der Mathematik, ( ISI ), Volume (122), No (6), Year (2024-4) , Pages (659-669)

Title : ( Operator mean inequalities and Kwong functions )

Authors: Nahid Gharakhanlu , Mohammad Sal Moslehian , Hamed Najafi ,

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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 $m$ represents the smallest eigenvalues of $A$ and $B$ and $M$ represents the largest eigenvalues of $A$ and $B$. If $f$ is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain 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.

Keywords

, Concave function, convex function, matrix mean, determinant inequality, unitarily invariant norm
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@article{paperid:1098418,
author = {ناهید قراخانلو and Sal Moslehian, Mohammad and Najafi, Hamed},
title = {Operator mean inequalities and Kwong functions},
journal = {Archiv der Mathematik},
year = {2024},
volume = {122},
number = {6},
month = {April},
issn = {0003-889X},
pages = {659--669},
numpages = {10},
keywords = {Concave function; convex function; matrix mean; determinant inequality; unitarily invariant norm},
}

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%0 Journal Article
%T Operator mean inequalities and Kwong functions
%A ناهید قراخانلو
%A Sal Moslehian, Mohammad
%A Najafi, Hamed
%J Archiv der Mathematik
%@ 0003-889X
%D 2024

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