, Volume 58, Issue 1-2, pp 69-79
Date: 28 Apr 2010

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 \({x,y\in I}\) .

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