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 $r\ge 0$, $A{\nabla }_{-r}B+2r(A\nabla B-A♯B)\le A{♯}_{-r}B$ 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 $r\ge 0$, then $\label{reverse_trace} \mathrm{tr}((1+r)A-rB)\leq \mathrm{tr}|A^{1+r}B^{-r} |-r(\sqrt{\mathrm{tr} A} - \sqrt{\mathrm{tr} B})^{2}$.

15 pages, to appear in Rocky Mountain J. Math

References in corpus (1)

• Alternative reverse inequalities for Young's inequality

Cited by in corpus (1)

• Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant