Extending means of two variables to several variables. (English) Zbl 1056.26020

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 This was discovered by S. Mustonen in 1976 (unpublished).


26E60 Means
26D15 Inequalities for sums, series and integrals
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