On Seiffert-like means

We investigate the representation of homogeneous, symmetric means in the form M(x,y)=\frac{x-y}{2f((x-y)/(x+y))}. This allows for a new approach to comparing means. As an example, we provide optimal estimate of the form (1-\mu)min(x,y)+ \mu max(x,y)<= M(x,y)<= (1-\nu)min(x,y)+ \nu max(x,y) and M((x+y)/2-\mu(x-y)/2,(x+y)/2+\mu(x-y)/2)<= N(x,y)<= M((x+y)/2-\nu(x-y)/2,(x+y)/2+\nu(x-y)/2) for some known means