On the balancing property of Matkowski means

Abstract

Let Isubseteq mathbb {R} be a nonempty open subinterval. We say that a two-variable mean M:Itimes Irightarrow mathbb {R} enjoys the balancing property if, for all x,yin I, the equality 1M(M(x,M(x,y)),M(M(x,y),y))=M(x,y)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {M\\big (M(x,M(x,y)),M(M(x,y),y)\\big )=M(x,y)} \\end{aligned}$$\\end{document}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.