Extending a two-variable mean to a multi-variable mean

We consider the problem of extending any two-variable mean M(·, ·) to a multi-variable mean, using no other tool than M(·, ·) itself. Pálfia proposed an iterative procedure that consists in evaluating successively two-variable means according to a cyclic pattern. We propose here a variant of his procedure to improve the convergence speed. Our approach consists in re-ordering the iterates after each iteration in order to speed up the transfer of information between successive iterates.