Some reverse mean inequalities for operators and matrices

• Main Authors: Chaojun Yang, Yaxin Gao, Fangyan Lu
• Format: Article
• Language: English
• Published: SpringerOpen 2019-04-01
• Series: Journal of Inequalities and Applications
• Subjects: Positive linear maps; Arithmetic–geometric–harmonic mean; Sector matrix; Inequality
• Online Access: http://link.springer.com/article/10.1186/s13660-019-2070-2

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.