The invariance of the arithmetic mean with respect to generalized quasi-arithmetic means

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.