Reverses of Young and Heinz inequalities for positive linear operators

Let A, B be invertible positive operators on a Hilbert space H. We present some improved reverses of Young type inequalities, in particular, (1−ν)2ν(A∇B)+(1−ν)2(1−ν)H2ν(A,B)≥2(1−ν)(A♯B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (1-\nu)^{2\nu}(A\nabla B)+(1-\nu)^{2(1-\nu)}H_{2\nu}(A,B) \geq2(1-\nu ) (A\sharp B) $$\end{document} and (1−ν)2νH2ν(A,B)+(1−ν)2(1−ν)(A∇B)≥2(1−ν)(A♯B),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (1-\nu)^{2\nu}H_{2\nu}(A,B)+(1-\nu)^{2(1-\nu)}(A\nabla B) \geq2(1-\nu ) (A\sharp B), $$\end{document} where 0≤υ≤12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\leq\upsilon\leq\frac{1}{2}$\end{document}.