Results in Mathematics

, Volume 58, Issue 1–2, pp 69–79

# On the Equality Problem of Conjugate Means

Article

## Abstract

Let $${I\subset\mathbb{R}}$$ be a nonvoid open interval and let L : I2I be a fixed strict mean. A function M : I2I 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) : = Aχ(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 $${x,y\in I}$$.

39B22

### Keywords

Mean functional equation quasi-arithmetic mean conjugate mean

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