Rocky Mountain Journal of Mathematics, ( ISI ), Volume (46), No (4), Year (2016-10) , Pages (1089-1105)

#### Title : ( Reverses of the Young inequality for matrices and operators )

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#### Abstract

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal{H})$ are positive operators and $r\geq 0$, $A\nabla_{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp_{-r}B$ and prove that equality holds if and only if $A=B$. We also establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A, B$ are positive definite matrices and $r\geq 0$, then $\label{reverse_trace} \mathrm{tr}((1+r)A-rB)\leq \mathrm{tr}\left|A^{1+r}B^{-r} \right|-r\left( \sqrt{\mathrm{tr} A} - \sqrt{\mathrm{tr} B}\right)^{2}$.

#### Keywords

, Young inequality; positive operator, operator mean; unitarily invariant norm; determinant; trace
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@article{paperid:1060671,
title = {Reverses of the Young inequality for matrices and operators},
journal = {Rocky Mountain Journal of Mathematics},
year = {2016},
volume = {46},
number = {4},
month = {October},
issn = {0035-7596},
pages = {1089--1105},
numpages = {16},
keywords = {Young inequality; positive operator; operator mean; unitarily invariant norm; determinant; trace},
}