The authors offer as ``main result'' necessary and sufficient conditions on \((a_0,b_0,a_1,b_1,a_2,b_2)\in\mathbb R^6\) for \(S_{a_0,b_0}(x+y)\leq S_{a_1,b_1}(x)+S_{a_2,b_2}(y)\) for all \((x,y)\in ]0,\infty[^4\) and the consequence that the inequality is `best' if \((a_0,b_0)=(a_1,b_1)=(a_2,b_2).\) Here \(S_{a,b}((u,v))\) is defined by \((\frac{u^a +v^a} {u^b+v^b})^{1/(b-a)}\) if \(a\neq b\) and by its limit as \(b\to a\) if \(a=b\). As (another) ``main result'' necessary and sufficient conditions for equality of such mean values are offered.