On the balancing property of Matkowski means.

Let I ⊆ R be a nonempty open subinterval. We say that a two-variable mean M : I × I → R enjoys the balancing property if, for all x , y ∈ I , the equality 1 M (M (x , M (x , y)) , M (M (x , y) , y)) = M (x , y) holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that M is analytic, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes regular quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are continuously differentiable. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of Matkowski means. [ABSTRACT FROM AUTHOR]