### 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) (x,y ∈ I) and 2f(M_{1} ⊗ M_{2}(x, y)) = f(x) + f(y) (x, y ∈ I) is studied, where M_{1} and M_{2} are two variable strict means on an open real interval I, and M_{1} ⊗ M_{2} 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 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} ⊗ 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.

Original language | English |
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Pages (from-to) | 521-530 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 134 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1 2006 |

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### Keywords

- Functional equation
- Gauss composition
- Mean

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*134*(2), 521-530. https://doi.org/10.1090/S0002-9939-05-08009-3