On the Equality Problem of Conjugate Means

Let ${Isubsetmathbb{R}}$ be a nonvoid open interval and let L : I ²→ I be a fixed strict mean. A function M : I ²→ I is said to be an L-conjugate mean on I if there exist ${p,qin,]0,1]}$ and ${varphiin CM(I)}$ such that $$M(x,y):=varphi^{-1}(pvarphi(x)+qvarphi(y)+(1-p-q) varphi(L(x,y)))=:L_varphi^{(p,q)}(x,y),$$ for all ${x,yin I}$ . Here L(x, y) : = A _χ(x, y) ${(x,yin I)}$ is a fixed quasi-arithmetic mean with the fixed generating function ${chiin CM(I)}$ . We examine the following question: which L-conjugate means are weighted quasi-arithmetic means with weight ${rin, ]0,1[}$ at the same time? This question is a functional equation problem: Characterize the functions ${varphi,psiin CM(I)}$ and the parameters ${p,qin,]0,1]}$ , ${rin,]0,1[}$ for which the equation $$L_varphi^{(p,q)}(x,y)=L_psi^{(r,1-r)}(x,y)$$ holds for all ${x,yin I}$ .