Linear Algebra and its Applications, ( ISI ), Volume (437), No (3), Year (2012-5) , Pages (1016-1024)

#### Title : ( A general double inequality related to operator means and positive linear maps )

Authors: Rupinderjit Kaur , Mandeep Singh , Jaspal Singh Aujla , Mohammad Sal Moslehian ,

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#### Abstract

Let $A,B\\\\in \\\\mathbb{B}(\\\\mathscr{H})$ be such that $0<b_{1}I \\\\leq A \\\\leq a_{1}I$ and $0<b_{2}I \\\\leq B \\\\leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\\\\;\\\\; i=1,2$ and $\\\\Phi:\\\\mathbb{B}(\\\\mathscr{H})\\\\rightarrow\\\\mathbb{B}(\\\\mathscr{K})$ be a positive linear map. We show that for any operator mean $\\\\sigma$ with the representing function $f$, the double inequality $$\\\\omega^{1-\\\\alpha}(\\\\Phi(A)\\\\#_{\\\\alpha}\\\\Phi(B))\\\\le (\\\\omega\\\\Phi(A))\\\\nabla_{\\\\alpha}\\\\Phi(B)\\\\leq \\\\frac{\\\\alpha}{\\\\mu}\\\\Phi(A\\\\sigma B)$$ holds, where $\\\\mu=\\\\frac{a_{1}b_{1}(f(b_{2}a_{1}^{-1})-f(a_{2}b_{1}^{-1}))}{b_{1}b_{2}-a_{1}a_{2}},~$ $\\\\nu=\\\\frac{a_{1}a_{2}f(b_{2}a_{1}^{-1})-b_{1}b_{2}f(a_{2}b_{1}^{-1})}{a_{1}a_{2}-b_{1}b_{2}},~$ $\\\\omega=\\\\frac{\\\\alpha \\\\nu}{(1-\\\\alpha)\\\\mu}$ and $\\\\#_{\\\\alpha}$ ($\\\\nabla_{\\\\alpha}$, resp.) is the weighted geometric (arithmetic, resp.) mean for $\\\\alpha \\\\in (0,1)$. As applications we present several generalized operator inequalities including Diaz--Metcalf and reverse Ando type inequalities. We also give some related inequalities involving Hadamard product and operator means

#### Keywords

, Positive operator, operator mean, positive linear map, Cauchy--Schwarz inequality, Ando inequality, Diaz--Metcalf type inequality, reverse inequality.
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@article{paperid:1027721,
author = {Rupinderjit Kaur and Mandeep Singh and Jaspal Singh Aujla and Sal Moslehian, Mohammad},
title = {A general double inequality related to operator means and positive linear maps},
journal = {Linear Algebra and its Applications},
year = {2012},
volume = {437},
number = {3},
month = {May},
issn = {0024-3795},
pages = {1016--1024},
numpages = {8},
keywords = {Positive operator; operator mean; positive linear map; Cauchy--Schwarz inequality; Ando inequality; Diaz--Metcalf type inequality; reverse inequality.},
}

%0 Journal Article
%T A general double inequality related to operator means and positive linear maps
%A Rupinderjit Kaur
%A Mandeep Singh
%A Jaspal Singh Aujla