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 → R and Borel probability measures μ , ν on the interval [ 0 , 1 ] such that φ −1 ( ∫ 0 1 φ ( t x + ( 1 − t ) y ) d μ ( t ) ) + ψ −1 ( ∫ 0 1 ψ ( t x + ( 1 − t ) y ) d ν ( 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 , ψ : they are equivalent either to linear, or to exponential or to power functions.