Functional equations involving means and their Gauss composition

Daróczy, Zoltán; Maksa, Gyula; Páles, Zsolt

doi:10.1090/S0002-9939-05-08009-3

Functional equations involving means and their Gauss composition
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by Zoltán Daróczy, Gyula Maksa and Zsolt Páles PDF

Proc. Amer. Math. Soc. 134 (2006), 521-530 Request permission

Abstract:

In this paper the equivalence of the two functional equations \[ f(M_1(x,y))+f(M_2(x,y))=f(x)+f(y) \qquad (x,y\in I) \] and \[ 2f(M_1\otimes M_2(x,y))=f(x)+f(y) \qquad (x,y\in I) \] is studied, where $M_1$ and $M_2$ are two variable strict means on an open real interval $I$, and $M_1\otimes M_2$ denotes their Gauss composition. The equivalence of these equations is shown (without assuming further regularity assumptions on the unknown function $f:I\to \mathbb {R}$) for the cases when $M_1$ and $M_2$ are the arithmetic and geometric means, respectively, and also in the case when $M_1$, $M_2$, and $M_1\otimes M_2$ are quasi-arithmetic means. If $M_1$ and $M_2$ are weighted arithmetic means, then, depending on the algebraic character of the weight, the above equations can be equivalent and also non-equivalent to each other.

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