Optimal bounds of classical and non-classical means in terms of Q means

We show optimal bounds of the form Qα<M<Qβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_\alpha<M<Q_\beta $$\end{document}, where Qα(x,y)=A(x,y)A2(x,y)(1-α)A2(x,y)+αG2(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q_\alpha (x,y)={\mathsf {A}}(x,y)\frac{{\mathsf {A}}^2(x,y)}{(1-\alpha ){\mathsf {A}}^2(x,y)+\alpha {\mathsf {G}}^2(x,y)} \end{aligned}$$\end{document}and M belongs to a broad class of classical homogeneous, symmetric means of two variables.