Results in Mathematics

, Volume 58, Issue 1–2, pp 69–79 | Cite as

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

Mathematics Subject Classification (2000)

39B22 

Keywords

Mean functional equation quasi-arithmetic mean conjugate mean 

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© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of DebrecenDebrecen, Pf. 12Hungary

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