Why means in two arguments are special
The author notes that both the logarithmic mean \(L(x,y)= (x-y)/(\ln x-\ln y)\) and the geometric mean \(G(x,y)= (xy)^{1/2}\) have interesting applications and both are limits, as \(t\to0\), of one-parameter (and also of two-parameter) mean value families, \[ S(x,y)= t(x^{t+1}- y^{t+1})/((t+1)(x^t-y^t)) \quad\text{and}\quad P(x,y)= ((x^t+y^t)/2)^{1/t}, \] respectively, furthermore, both have analogues in \(n\)-variables, and so have their families, but those of \(G\) and \(P\) seem to be far more interesting than those of \(L\) and \(S\). He attributes this to the fact that the \(n\)-variable analogues of \(P\) and \(G\), the power means \(P_n(x_1,\dots,x_n)= ((x^t_1+\cdots+x^t_n)/n)^{1/t}\) and the geometric mean \(G_n(x_1,\dots,x_n)=(x_1x_2\cdots x_n)^{1/n}\) satisfy, in addition to homogeneity, symmetry, internality (lying between min and max), continuity and strict monotonicity, the Kolmogorov [\textit{A. Kolmogoroff}, Rend. Accad. d.L. Roma (6) 12, 388-391 (1930; JFM 56.0198.02)] -Nagumo [\textit{M. Nagumo}, Jap. J. Math. 7, 71-79 (1930; JFM 56.0198.03)] system of functional equations \[ \begin{multlined} M_n(x_1,\dots,x_r,x_{r+1},\dots,x_n)=\\ M_n(M_r(x_1,\dots,x_r),\dots,M_r(x_1,\dots,x_r),x_{r+1},\dots,x_n) (r=2,\dots, n),\end{multlined}\tag{K-N} \] while no \(n\)-variable generalization of \(L\) or \(S\) can satisfy them since these conditions characterize \(P_n\) and \(G_n\); (K-N), however, is meaningless for \(n=2\) (more exactly, a consequence of internality). \{Note: There exist replacements for (K-N) which, with the other above conditions, characterize \(P_n\) and \(G_n\) and which make also sense for \(n=2\) alone, such as bisymmetry which, for \(n=2\) and under symmetry, postulates that \(M_2(M_2(x,y),M_2(u,v))\) be symmetric in its four variables. If symmetry is dropped then the equation \[ M_2(M_2(x,y),M_2(u,v))= M_2(M_2(x,u),M_2(y,v)) \] is satisfied also by weighted geometric and power means; it seems that the author would consider this too to be desirable\}.
scientific article; zbMATH DE number 598843