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

Abstract We show optimal bounds of the form $$Q_\alpha<M<Q_\beta $$ Q α < M β , where $$\begin{aligned} Q_\alpha (x,y)={\mathsf {A}}(x,y)\frac{{\mathsf {A}}^2(x,y)}{(1-\alpha ){\mathsf {A}}^2(x,y)+\alpha {\mathsf {G}}^2(x,y)} \end{aligned}$$ ( x , y ) = A 2 1 - + G and M belongs to a broad class classical homogeneous, symmetric means two variables.