2020/10/04 by Kiss, Tibor
#26E60 #39B22 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.01536
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 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.