Invariance in a class of Bajraktarević means

Let I⊂RI⊂R be a non-trivial interval, s:I→(0,∞)s:I→(0,∞) be a function, and let φ,ψφ,ψ be real continuous strictly monotonic functions defined on II. We consider the equation A∘(Bsφ,Bsψ)=A, where Bsα denotes the Bajraktarević mean given by Bsα(x,y)=α−1(s(x)s(x)+s(y)α(x)+s(y)s(x)+s(y)α(y)) and AA is the arithmetic mean.