  In this paper we study the problem of extending means to means of higher
order. We show how higher order means can be inductively defined and
established in general metric spaces, in particular, in convex metric spaces.
As a particular application, we consider the positive operators on a Hilbert
space under the Thompson metric and show that the operator logarithmic mean
admits extensions of all higher orders, thus providing a positive solution to a
problem of Petz and Temesi.
