# Generalized Homogeneity of Means

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- ISRN Mathematical Analysis
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## Abstract

We extend the classical notions of translativity and homogeneity of means to 𝐹 -homogeneity, that is, invariance with respect to an operation 𝐹 ∶ 𝐼 × 𝐼 → 𝐼 . We find the shape of 𝐹 for the arithmetic weighted mean and then the general form of 𝐹 for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by 𝐹 -homogeneity with respect to a single operation 𝐹 , one needs to take two of such operations in order to characterize a mean.

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