On the balancing property of Matkowski means
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Aequationes Mathematicae
On the balancing property of Matkowski means Tibor Kiss Abstract. 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)
(1)
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. Mathematics Subject Classification. Primary 39B22, Secondary 26E60. Keywords. Balanced means, Balancing property, Aumann’s equation, Matkowski mean, Iteratively quasi-arithmetic mean.
1. Introduction We are going to use the usual notations N, Q, R, and C for the sets of positive integers, rational numbers, real numbers, and complex numbers, respectively. The set of positive real numbers will be denoted by R+ , that is, R+ := {x ∈ R | x > 0}. The identity function and the function which is identically 1 will be denoted by the symbols id and 1, respectively. The research of the author was supported in part by the NKFIH Grant K-134191 and in ´ part by the project no. 2019-2.1.11-TET-2019-00049, which has been implemented with the support provided from the National Research, Development and Innovation Fund of ´ funding scheme. Hungary, financed under the TET
T. Kiss
AEM
Throughout this paper, the subset I ⊆ R will stand for a nonempty open subinterval. For a given function h : I → I and for n ∈ N ∪ {0}, the nth iterate of h will be denoted by h[n] : I → I, where h[0] := id on I, furthermore [n] [n−1] (x) for all x ∈ I, whenever n ∈ N. h (x) := h h A two-place function M : I × I → R will be called a two-variable mean on I or, shortly, a mean on I if min(x, y) ≤ M (x, y) ≤ max(x, y),
(x, y ∈ I).
(2)
If both of the above inequalities are strict whenever x = y, then M is said to be a strict mean. We note that, by their definition, two-variable means are reflexive, that is, we have M (x, x) = x for all x ∈ I. We say that the mean M is strictly monotone if, for all fixed x0 , y0 ∈ I, the functions x → M (x, y0 ),
(x ∈ I)
and
y → M (x0 , y),
(y ∈ I)
are strictly increasing on I. Observe that strictly monotone means are also strict. Finally, the mean M is called symmetric if M (x, y) = M (y, x) holds for all x, y ∈ I. The mean M : I × I → R is said to be a quasi-arithmetic mean if there exists a continuous, strictly monotone function
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