Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

• Autorzy: Janusz Matkowski
• Języki publikacji: EN

Abstrakty

EN

A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f₁,...,f_{k}:I → ℝ$, k ≥ 2, denoted by $A^{[f₁,...,f_{k}]}$, is considered. Some properties of $A^{[f₁,...,f_{k}]}$, including "associativity" assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For a sequence of continuous strictly increasing functions $f_{j}:I → ℝ$, j ∈ ℕ, a mean $A^{[f₁,f₂,...]}: ⋃_{k=1}^{∞} I^{k} → I$ is introduced and it is observed that, except symmetry, it satisfies all conditions of the Kolmogorov-Nagumo theorem. A problem concerning a generalization of this result is formulated.

Kategorie tematyczne

• 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems
• 39B22: Equations for real functions
• 26E60: Means

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 133
• Numer: 1
• Strony: 35-49

Daty

wydano

2013

Twórcy

autor

Janusz Matkowski

• Faculty of Mathematics, Computer Sciences and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

Identyfikatory

DOI

10.4064/cm133-1-3