Functional equations involving means and their gauss composition

Zoltán Daróczy, Gyula Maksa, Zsolt Páles

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

In this paper the equivalence of the two functional equations f(M 1(x,y)) + f(M2(x,y)) = f(x) + f(y) (x,y ∈ I) and 2f(M1 ⊗ M2(x, y)) = f(x) + f(y) (x, y ∈ I) is studied, where M1 and M2 are two variable strict means on an open real interval I, and M1 ⊗ M2 denotes their Gauss composition. The equivalence of these equations is shown (without assuming further regularity assumptions on the unknown function f : I → R) for the cases when M1 and M2 are the arithmetic and geometric means, respectively, and also in the case when M1, M2, and M1 ⊗ M2 are quasi-arithmetic means. If M 1 and M2 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.

Original languageEnglish
Pages (from-to)521-530
Number of pages10
JournalProceedings of the American Mathematical Society
Volume134
Issue number2
DOIs
Publication statusPublished - Feb 1 2006

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Keywords

  • Functional equation
  • Gauss composition
  • Mean

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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