Limit properties of quasi-arithmetic means

Abstract

Limit properties of the class { M g λ } λ∈(0,∞) of all quasi-arithmetic means generated by λ-powers of a given generator g are studied. Special types of generators of quasi-arithmetic means that uniquely correspond to the additive generators of continuous Archimedean t-norms or t-conorms are considered. It is shown that for λ → ∞ , the situation is similar to that for t-norms and t-conorms [6]. For λ → 0 + , the limit operators are quasi-geometric means. Finally, the limit properties of the class { M g α } α∈(0,∞) of all quasi-arithmetic means generated by functions g α , g α ( x)= g( x α ) are investigated.