Merikoski, Jorma K. Extending means of two variables to several variables. (English) Zbl 1056.26020 JIPAM, J. Inequal. Pure Appl. Math. 5, No. 3, Paper No. 65, 9 p. (2004). The author presents a method, based on series expansions and symmetric polynomials to extend a mean of two arguments to several variables. For example, he defines the logarithmic mean of \(n\) variables by \[ L(x_1,\dots, x_n)= 1+\sum^\infty_{m=1} {1\over m!} Q_m(u_1,\dots, u_n), \] where \(Q_m(u_1,\dots, u_n)= \left(\begin{smallmatrix} n+m-1\\ m\end{smallmatrix}\right)^{-1} C_m(u_1,\dots, u_n)\) with \(C_m\) denoting the \(m\)th complete symmetric polynomial of \(u_i\geq 0\), and \(1\leq m\leq n\). Here \(x_i= \exp(u_i)\) \((1\leq i\leq n)\). Inequalities, special cases, as well as connections to various means are pointed out. By using divided differences, it is shown that \(L\) coincides with a mean considered by E. Neumann in 1994. This was discovered by S. Mustonen in 1976 (unpublished). Reviewer: József Sándor (Cluj-Napoca) Cited in 3 Documents MSC: 26E60 Means 26D15 Inequalities for sums, series and integrals Keywords:means and their inequalities PDFBibTeX XMLCite \textit{J. K. Merikoski}, JIPAM, J. Inequal. Pure Appl. Math. 5, No. 3, Paper No. 65, 9 p. (2004; Zbl 1056.26020) Full Text: EuDML