On the balancing property of Matkowski means

• Published in: Aequationes mathematicae Vol. 95; no. 1; pp. 75 - 89
• Main Author: Kiss, Tibor
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2021; Springer Nature B.V
• Subjects: Analysis; Arithmetic; Balancing; Combinatorics; Mathematical analysis; Mathematics; Mathematics and Statistics; Balanced means; Balancing property; Primary 39B22; Secondary 26E60; Matkowski mean; Aumann’s equation; Iteratively quasi-arithmetic mean
• ISSN: 0001-9054; 1420-8903
• DOI: 10.1007/s00010-020-00758-7

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 .