On the equality of quasi-arithmetic and Lagrangian means

The paper deals with the equality problem of quasi-arithmetic and Lagrangian means which is to determine all pairs of continuous strictly monotone functions φ,ψ:I→Rφ,ψ:I→R such that, for all x,y∈Ix,y∈I,φ−1(φ(x)+φ(y)2)=ψ−1(1y−x∫xyψ(t)dt) holds. The main result of the paper shows that there are 6 different cases of equality.