Article ID Journal Published Year Pages File Type
4620433 Journal of Mathematical Analysis and Applications 2009 16 Pages PDF
Abstract

The aim of this paper is to find those pairs of generalized quasi-arithmetic means on an open real interval I   for which the arithmetic mean is invariant, i.e., to characterize those continuous strictly monotone functions φ,ψ:I→Rφ,ψ:I→R and Borel probability measures μ,νμ,ν on the interval [0,1][0,1] such thatφ−1(∫01φ(tx+(1−t)y)dμ(t))+ψ−1(∫01ψ(tx+(1−t)y)dν(t))=x+y(x,y∈I) holds. Under at most fourth-order differentiability assumptions and certain nondegeneracy conditions on the measures, the main results of this paper show that there are three classes of the solutions φ,ψφ,ψ: they are equivalent either to linear, or to exponential or to power functions.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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