On the Equality Problem of Conjugate Means

Abstract

Let ${I\subset\mathbb{R}}$ 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 ${p,q\in\,]0,1]}$ and ${\varphi\in CM(I)}$ such that

$$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q) \varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$

for all ${x,y\in I}$. Here L(x, y) : = A _χ(x, y) ${(x,y\in I)}$ is a fixed quasi-arithmetic mean with the fixed generating function ${\chi\in CM(I)}$. We examine the following question: which L-conjugate means are weighted quasi-arithmetic means with weight ${r\in\, ]0,1[}$ at the same time? This question is a functional equation problem: Characterize the functions ${\varphi,\psi\in CM(I)}$ and the parameters ${p,q\in\,]0,1]}$, ${r\in\,]0,1[}$ for which the equation

$$L_\varphi^{(p,q)}(x,y)=L_\psi^{(r,1-r)}(x,y)$$

holds for all ${x,y\in I}$.