Reverse Young-type inequalities for matrices and operators

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 $r\geq 0$, $A\nabla _{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp _{-r}B$. We also prove that equality holds if and only if $A=B$. In addition, we establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A$ and $B$ are positive definite matrices and $r\geq 0$, then $\label {reverse_trace} \mbox {tr\,}((1+r)A-rB)\leq \mbox {tr}|A^{1+r}B^{-r} |-r( \sqrt {\mbox {tr\,} A} - \sqrt {\mbox {tr} B})^{2}$.