Linear and Multilinear Algebra, ( ISI ), Volume (66), No (6), Year (2017-3) , Pages (1186-1198)

#### Title : ( Some operator inequalities involving operator means and positive linear maps )

Authors: Maryam Khosravi , Mohammad Sal Moslehian , Alemeh Sheikhhosseini ,

Citation: BibTeX | EndNote

#### Abstract

Let $A$ and $B$ be two positive operators with $0 < m \leqslant A, B \leqslant M$ for positive real numbers $M, m, \, \sigma$ be an operator mean and $\sigma^{*}$ be the adjoint mean of $\sigma.$ If $\sigma\leqslant \sigma_1,\sigma_2\leqslant \sigma^*$ and $\Phi$ is a positive unital linear map, then $$\Phi^{p}(A \sigma_{1} B) \leqslant \alpha^{p} \Phi^{p}(A \sigma_{2} B),$$ where $$\alpha= \max \left \lbrace K, 4^{1-\frac{2}{p}}K \right \rbrace,$$ and $K= \frac{(M+m)^2}{4mM}$ is the Kantorovich constant. In addition, for $p\geqslant 4$ $$\Phi^{p}(A \sigma_{1} B) \leqslant \dfrac{1}{16}\left (\dfrac{ K(M^{2}+m^{2})}{mM}\right )^{p} \Phi^{p}(A \sigma_{2} B).$$

#### Keywords

Kantorovich constant; operator mean; operator inequality; positive linear map
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@article{paperid:1063569,
author = {Maryam Khosravi and Sal Moslehian, Mohammad and Alemeh Sheikhhosseini},
title = {Some operator inequalities involving operator means and positive linear maps},
journal = {Linear and Multilinear Algebra},
year = {2017},
volume = {66},
number = {6},
month = {March},
issn = {0308-1087},
pages = {1186--1198},
numpages = {12},
keywords = {Kantorovich constant; operator mean; operator inequality; positive linear map},
}