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

Volume 133 / 2013

Janusz Matkowski Colloquium Mathematicum 133 (2013), 35-49 MSC: Primary 26A24, 26E60, 39B22. DOI: 10.4064/cm133-1-3


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.


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

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