The invariance of the arithmetic mean with respect to generalized quasi-arithmetic means

Abstract

The aim of this paper is to find those pairs of generalized quasi-arithmetic means on an open real interval I for which the arithmetic mean is invariant, i.e., to characterize those continuous strictly monotone functions ϕ, ψ : I → R and Borel probability measures μ, ν on the interval [0, 1] such thatholds. Under at most fourth-order differentiability assumptions and certain nondegeneracy conditions on the measures, the main results of this paper show that there are three classes of the solutions ϕ, ψ: they are equivalent either to linear, or to exponential or to power functions.

An invariance of geometric mean with respect to generalized quasi-arithmetic means

“…To get our main result, we will need the formulae for the partial derivatives of the mean M ϕ,μ along the diagonal of the Cartesian product I × I . [11].) Let μ be a Borel probability measure.…”

“…Páles considered a common generalization of quasi-arithmetic and Lagrangian means (cf. [10,11]): Given a continuous strictly monotone function ϕ : I → R and a probability measure μ on the Borel subsets of [0, 1], the generalized quasi-arithmetic mean M ϕ,μ : I 2 → I is defined by M(x, y) = M ϕ,μ (x, y) := ϕ −1 1 0 ϕ tx + (1 − t) y dμ(t) , x, y ∈ I.…”

“…J. Matkowski solved the invariance equation involving the arithmetic mean in class of Lagrangian mean-type mappings [13]. In [11], Z. Makó and Zs. Páles investigated the invariance of the arithmetic mean with the generalized quasi-arithmetic means, that is, found the solutions of equation M ϕ,μ (x, y) + M ψ,ν (x, y) = x + y, x, y ∈ I.…”

Invariance of means

“…The plurality of probability Borel measures implies that the class of Makó-Páles means is pretty large and, consequently, the invariance problem studied there has various solutions which strongly depend on some parameters of the used measures, on their moments, in particular. From the numerous results proved in [105] we choose the following two describing the form of pairs (ϕ, ψ) satisfying Eq. (6.1).…”

Invariance of Weighted Bajraktarević Mean with Respect to the Beckenbach-Gini means

“…Matkowski solved the invariance equation involving the arithmetic mean in class of Lagrangian mean-type mappings [14]. In [15], Makó and Páles investigated the invariance of the arithmetic mean with respect to generalized quasi-arithmetic means. The invariance of the geometric mean in class of Lagrangian mean-type mappings has been studied by Głazowska and Matkowski in [16].…”