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

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