On homogeneous Lagrange means

Let f be a real differentiable function in an open interval I with one-to-one derivative. We observe that if the Lagrange mean L [ f ] of a generator f is conditionally positively homogeneous, then f must be of the class C∞ and the function g(x) := x f ′ (x)− f (x) , x ∈ I, is also a generator of L [ f ] i.e. that L [g] = L [ f ].We show that this fact and a result on equality of two Lagrange means allow easily to determine all positively homogeneous Lagrange means.