Invariance of quasi-arithmetic means with function weights

Let I ⊂ R be a non-trivial interval and let λ , μ , ν : I 2 → ( 0 , 1 ) . We present some results concerning the following functional equation, generalizing the Matkowski–Sutô equation, λ ( x , y ) φ −1 ( μ ( x , y ) φ ( x ) + ( 1 − μ ( x , y ) ) φ ( y ) ) + ( 1 − λ ( x , y ) ) ψ −1 ( ν ( x , y ) ψ ( x ) + ( 1 − ν ( x , y ) ) ψ ( y ) ) = λ ( x , y ) x + ( 1 − λ ( x , y ) ) y , where φ , ψ : I → R are continuous and strictly monotonic unknown functions.