Means of Positive Numbers and Matrices

It is shown that any two-variable-mean of positive matrices (studied by Kubo and Ando) can be extended to more variables. The $n$-variable-mean $M_n(A_1,A_2,\ldots,A_n)$ is defined by a symmetrization procedure when the $n$-tuple $(A_1, A_2, \ldots, A_n)$ is ordered, is monotone in each variable, and satisfies the transformer inequality. This approach is motivated by the paper of Ando, Li, and Mathias on geometric means [Linear Algebra Appl., 385 (2004), pp. 305-334]. Special attention is paid to the logarithmic mean. It is conjectured that for matrix means the symmetrization procedure converges for all triplets.