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• # Artykuł - szczegóły

## Colloquium Mathematicum

2013 | 133 | 1 | 35-49

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

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.

35-49

wydano
2013

### Twórcy

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