### 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 |
---|---|

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 2006 |

### Fingerprint

### 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

**Functional equations involving means and their gauss composition.** / Daróczy, Zoltán; Maksa, Gyula; Páles, Z.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 134, no. 2, pp. 521-530. https://doi.org/10.1090/S0002-9939-05-08009-3

}

TY - JOUR

T1 - Functional equations involving means and their gauss composition

AU - Daróczy, Zoltán

AU - Maksa, Gyula

AU - Páles, Z.

PY - 2006/2

Y1 - 2006/2

N2 - 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.

AB - 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.

KW - Functional equation

KW - Gauss composition

KW - Mean

UR - http://www.scopus.com/inward/record.url?scp=33644556523&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644556523&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-05-08009-3

DO - 10.1090/S0002-9939-05-08009-3

M3 - Article

AN - SCOPUS:33644556523

VL - 134

SP - 521

EP - 530

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -