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

Janusz Matkowski¹

¹ Faculty of Mathematics, Computer Sciences and Econometrics University of Zielona Góra Szafrana 4a 65-516 Zielona Góra, Poland

Colloquium Mathematicum, Tome 133 (2013) no. 1, pp. 35-49

Résumé

A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f_{1},\ldots ,f_{k}:I\rightarrow \mathbb {R}$, $k\geq 2,$ denoted by $A^{[ f_{1},\ldots ,f_{k}] },$ is considered. Some properties of $A^{[ f_{1},\ldots ,f_{k}] }$, 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 $f_{j}:I\rightarrow \mathbb {R}$, $j\in \mathbb {N}$, a mean $A^{[f_{1},f_{2},\ldots ]}: \bigcup _{k=1}^{\infty }I^{k}\rightarrow I$ 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.

Détail

DOI : 10.4064/cm133-1-3

Keywords: generalization weighted quasi arithmetic mean generated continuous increasing decreasing functions ldots rightarrow mathbb geq denoted ldots considered properties ldots including associativity assumed kolmogorov nagumo theorem shown convex affine functions involving type means considered invariance quasi arithmetic mean respect special mean type mapping built generalized means applied solving functional equation sequence continuous strictly increasing functions rightarrow mathbb mathbb mean ldots bigcup infty rightarrow introduced observed except symmetry satisfies conditions kolmogorov nagumo theorem problem concerning generalization result formulated