Reverses of the Young inequality 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 $r\geq 0$, $A\nabla_{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp_{-r}B$ 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 $\label{reverse_trace} \mathrm{tr}((1+r)A-rB)\leq \mathrm{tr}|A^{1+r}B^{-r} |-r(\sqrt{\mathrm{tr} A} - \sqrt{\mathrm{tr} B})^{2}$.