Abstract
We prove that whenever the selfmapping (M1,…,Mp):Ip→Ip, (p∈N and Mi-s are p-variable means on the interval I) is invariant with respect to some continuous and strictly monotone mean K:Ip→I then for every nonempty subset S⊆{1,…,p} there exists a uniquely determined mean KS:Ip→I such that the mean-type mapping (N1,…,Np):Ip→Ip is K-invariant, where Ni:=KS for i∈S and Ni:=Mi otherwise. Moreover min(Mi:i∈S)≤KS≤max(Mi:i∈S). Later we use this result to: (1) construct a broad family of K-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.
Highlights
We prove that whenever the selfmapping ( M1, . . . , M p ) : I p → I p, (p ∈ N and Mi -s are p-variable means on the interval I) is invariant with respect to some continuous and strictly monotone mean K : I p → I for every nonempty subset S ⊆ {1, . . . , p} there exists a uniquely determined mean KS : I p → I such that the mean-type mapping ( N1, . . . , Np ) : I p → I p is K-invariant, where Ni := KS for i ∈ S and Ni := Mi otherwise
Assume that K : I p → I is a continuous and monotone mean which is invariant with respect to the mean type mapping M := ( M1, . . . , M p )
Let K : I p → I be a continuous and monotone mean which is invariant with respect to the mean type mapping M := ( M1, . . . , M p )
Summary
In the whole paper I ⊂ R stands for an interval, p ∈ N, p > 1 is fixed, and N p := {1, . . . , p}. A function M : I p → I is called a mean on I if min x1 , ..., x p ≤ M x1 , ..., x p ≤ max x1 , ..., x p , x1 , ..., x p ∈ I , or, briefly, if min x ≤ M (x) ≤ max x,. For a mean-type mapping M : I p → I p we denote a projection onto the i-th coordinate by [M]i : I p → I. In this case we obviously have [M]i = Mi for all i ∈ N p.
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