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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Functional equations involving means and their Gauss composition


Authors: Zoltán Daróczy, Gyula Maksa and Zsolt Páles
Journal: Proc. Amer. Math. Soc. 134 (2006), 521-530
MSC (2000): Primary 39B22, 39B12; Secondary 26A51, 26B25
Published electronically: July 18, 2005
MathSciNet review: 2176021
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Abstract: In this paper the equivalence of the two functional equations

\begin{displaymath}f(M_1(x,y))+f(M_2(x,y))=f(x)+f(y) \qquad(x,y\in I) \end{displaymath}

and

\begin{displaymath}2f(M_1\otimes M_2(x,y))=f(x)+f(y) \qquad(x,y\in I) \end{displaymath}

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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Additional Information

Zoltán Daróczy
Affiliation: Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
Email: daroczy@math.klte.hu

Gyula Maksa
Affiliation: Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
Email: maksa@math.klte.hu

Zsolt Páles
Affiliation: Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
Email: pales@math.klte.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08009-3
PII: S 0002-9939(05)08009-3
Keywords: Mean, Gauss composition, functional equation
Received by editor(s): April 29, 2003
Received by editor(s) in revised form: September 29, 2004
Published electronically: July 18, 2005
Additional Notes: This research was supported by the Hungarian Scientific Research Fund (OTKA) Grants T-043080 and T-038072.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2005 American Mathematical Society