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 Mₙ(A₁,A₂,…,Aₙ) is defined by a symmetrization procedure when the n-tuple (A₁, A₂, …, Aₙ) 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.