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 → R such that, for all x , y ∈ I , φ − 1 ( φ ( x ) + φ ( y ) 2 ) = ψ − 1 ( 1 y − x ∫ x y ψ ( t ) d t ) holds. The main result of the paper shows that there are 6 different cases of equality.