# On the Equality Problem of Conjugate Means

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## Abstract

Let $${I\subset\mathbb{R}}$$ be a nonvoid open interval and let L : I 2I be a fixed strict mean. A function M : I 2I 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 $${x,y\in I}$$.

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## Author information

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Correspondence to Zoltán Daróczy.

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK-68040, 81402.

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Daróczy, Z., Dascăl, J. On the Equality Problem of Conjugate Means. Results. Math. 58, 69–79 (2010). https://doi.org/10.1007/s00025-010-0042-4

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• DOI: https://doi.org/10.1007/s00025-010-0042-4