Weighted arithmetic-geometric operator mean inequalities.

Abstract

In this paper, we refine and generalize some weighted arithmetic-geometric operator mean inequalities due to Lin (Stud. Math. 215:187-194, 2013) and Zhang (Banach J. Math. Anal. 9:166-172, 2015) as follows: Let A and B be positive operators. If $0<m\le A\le m^{\text{'}}<M^{\text{'}}\le B\le M$ or $0<m\le B\le m^{\text{'}}<M^{\text{'}}\le A\le M$ , then for a positive unital linear map Φ, $\begin{array}{c}{\Phi }^2\left(A{\nabla }_{\alpha }B\right)\le {\left[\frac{K\left(h\right)}{S\left(h^{\text{'}r}\right)}\right]}^2{\Phi }^2\left(A{♯}_{\alpha }B\right),\\ {\Phi }^2\left(A{\nabla }_{\alpha }B\right)\le {\left[\frac{K\left(h\right)}{S\left(h^{\text{'}r}\right)}\right]}^2{\left[\Phi \right(A\left){♯}_{\alpha }\Phi \right(B\left)\right]}^2,\\ {\Phi }^{2p}\left(A{\nabla }_{\alpha }B\right)\le \frac{1}{16}{\left[\frac{K^2\left(h\right){(M^2+m^2)}^2}{S^2\left(h^{\text{'}r}\right)M^2{m}^2}\right]}^p{\Phi }^{2p}\left(A{♯}_{\alpha }B\right),\\ {\Phi }^{2p}\left(A{\nabla }_{\alpha }B\right)\le \frac{1}{16}{\left[\frac{K^2\left(h\right){(M^2+m^2)}^2}{S^2\left(h^{\text{'}r}\right)M^2{m}^2}\right]}^p{\left[\Phi \right(A\left){♯}_{\alpha }\Phi \right(B\left)\right]}^{2p},\end{array}$ where $\alpha \in [0,1]$ , $K\left(h\right)=\frac{{(h+1)}^2}{4h}$ , $S\left(h^{\text{'}}\right)=\frac{h^{\text{'}\frac{1}{h^{\text{'}}-1}}}{elog{h}^{\text{'}\frac{1}{h^{\text{'}}-1}}}$ , $h=\frac{M}{m}$ , $h^{\text{'}}=\frac{M^{\text{'}}}{m^{\text{'}}}$ , $r=min\{\alpha ,1-\alpha \}$ and $p\ge 2$ .