On the balancing property of Matkowski means

Let $$I\subseteq \mathbb {R}$$ I ⊆ R be a nonempty open subinterval. We say that a two-variable mean $$M:I\times I\rightarrow \mathbb {R}$$ M : I × I → R enjoys the balancing property if, for all $$x,y\in I$$ x , y ∈ I , the equality $$\begin{aligned} {M\big (M(x,M(x,y)),M(M(x,y),y)\big )=M(x,y)} \end{aligned}$$ 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.