Online Premeans and Their Computation Complexity

Abstract

We extend some approach to the family of symmetric means (i.e. symmetric functions $$\mathscr {M} :\bigcup _{n=1}^\infty I^n \rightarrow I$$ M : ⋃ n = 1 ∞ I n → I with $$\min \le \mathscr {M}\le \max $$ min ≤ M ≤ max ; I is an interval). Namely, it is known that every symmetric mean can be written in a form $$\mathscr {M}(v_1,\dots ,v_n):=F(f(v_1)+\cdots +f(v_n))$$ M ( v 1 , ⋯ , v n ) : = F ( f ( v 1 ) + ⋯ + f ( v n ) ) , where $$f :I \rightarrow G$$ f : I → G and $$F :G \rightarrow I$$ F : G → I (G is a commutative semigroup). For $$G=\mathbb {R}^k$$ G = R k or $$G=\mathbb {R}^k \times \mathbb {Z}$$ G = R k × Z ($$k \in \mathbb {N}$$ k ∈ N ) and continuous functions f and F we obtain two series of families (depending on k). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize the celebrated families of quasi-arithmetic means ($$G=\mathbb {R}\times \mathbb {Z}$$ G = R × Z ) and Bajraktarević means ($$G=\mathbb {R}^2$$ G = R 2 under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.

Decision making via generalized Bajraktarević means

“…In the next lemma we characterize the subfamily of zero-synergy generalized Bajraktarević means. Its single variable counterpart was proved in [15]. Theorem 7.3.…”

Estimating the Hardy constant of nonconcave Gini means

“…All these results use Kedlaya (or Kedlaya-type) properties [5,6] in their background, which unifies their assumptions. In the most natural setting, we assume that a mean is concave, homogeneous, and repetition invariant (then it is also monotone).These assumptions are relaxed for example using homogenizations techniques [18,19], or by replacing repetition invariance by a weaker axiom [10]. However we have not been able to relax the concavity assumption.…”