Some reverse mean inequalities for operators and matrices

Abstract

Abstract In this paper, we present some new reverse arithmetic–geometric mean inequalities for operators and matrices due to Lin (Stud. Math. 215:187–194, 2013). Among other inequalities, we prove that if A,B∈B(H) $A, B\in B(\mathcal{H})$ are accretive and 0<mI≤ℜ(A),ℜ(B)≤MI $0< {mI}\le \Re (A), \Re (B)\le {MI}$, then, for every positive unital linear map Φ, Φ2(ℜ(A+B2))≤(K(h))2Φ2(ℜ(A♯B)), $$\begin{aligned} \varPhi ^{2} \biggl(\Re \biggl(\frac{A+B}{2} \biggr) \biggr)\le \bigl(K(h) \bigr)^{2}\varPhi ^{2} \bigl(\Re (A\sharp B) \bigr), \end{aligned}$$ where K(h)=(h+1)24h $K(h)=\frac{(h+1)^{2}}{4h}$ and h=Mm $h=\frac{M}{m}$. Moreover, some reverse harmonic–geometric mean inequalities are also presented.

Keywords

• Positive linear maps
• Arithmetic–geometric–harmonic mean
• Sector matrix
• Inequality