On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom
\textit{A. N. Kolmogorov} [Atti Accad. Naz. Lincei Rend. (6) 12, 388-391 (1930; JFM 56.0198.02)] and \textit{M. Nagumo} [Japan. J. Math. 7, 71-79 (1930; JFM 56.0198.03)] characterized the quasi-arithmetic means \(M_n(x_1,...,x_n)=f^{-1}(f(x_1)+...f(x_n))\) (\(f: I\to\mathbb{R}\) continuous and strictly monotonic; \(I\) a real interval, \(x_j\in I;\: j=1,\dots,n;\: n=2,3,...\)) by \(M_n (n=2,3,...)\) being reflexive (\(M_n(x,...,x)=x\)), symmetric, continuous, strictly increasing in each variable and unchanged when the first \(k\) variables are replaced by their \(M_k\) mean (\(k=2,...,n-1\)). The author offers the remark that the last property and the symmetry can be replaced by the requirement that \(M_n\) remain unchanged when any \(k\) of its variables is replaced by their \(M_k\) mean.
scientific article; zbMATH DE number 1436166