Sub- and superadditive integral means (Q556990)

Given a continuous strictly monotone function \(f:I\to{\mathbb R}\), the integral mean \(I_f(x_1,\dots,x_n)\) of the variables \(x_1,\dots,x_n\in I\) is defined by \[ I_f(x_1,\dots,x_n):=f^{-1}\bigg(\int_{S_{n-1}} f\big(\mu_1(x_1-x_n)+\cdots+\mu_{n-1}(x_{n-1}-x_n)+x_n\big)d\mu\bigg), \] where \(S_{n-1}\) stands for the \((n-1)\) dimensional simplex \[ S_{n-1}:=\big\{\mu=(\mu_1,\dots,\mu_{n-1})\mid \mu_1,\dots,\mu_{n-1}\geq0,\, \mu_1+\cdots+\mu_{n-1}\leq0\big\}. \] Then \(I_f\) is a symmetric, continuous, \(n\)-variable mean on \(I\). If \(I\) is an open interval which is a semigroup with respect to the usual addition then, assuming the twice continuous differentiability of the functions \(f,g,h:I\to{\mathbb R}\), the main result of the paper obtains three pairwise equivalent sets of necessary and sufficient conditions in order that the inequality \[ I_f(x_1+y_1,\dots,x_n+y_n)\leq I_g(x_1,\dots,x_n)+I_h(y_1,\dots,y_n) \] be valid for a fixed integer \(n\geq2\) and for all \(x_1,\dots,x_n,y_1,\dots,y_n\in I\). Surprisingly, these conditions do not depend on the number \(n\), thus it follows that the above inequalities are equivalent for each fixed \(n\geq2\). Another interesting result states that \(I_f\) is sub- or superadditive on \(I={\mathbb R}\) if and only if \(I_f\) is the arithmetic mean.