## 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

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

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

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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