### Abstract

The aim of this paper is to find those pairs of generalized quasi-arithmetic means on an open real interval I for which the arithmetic mean is invariant, i.e., to characterize those continuous strictly monotone functions φ, ψ : I → R and Borel probability measures μ, ν on the interval [0, 1] such thatφ^{-1} (underover(∫, 0, 1) φ (t x + (1 - t) y) d μ (t)) + ψ^{-1} (underover(∫, 0, 1) ψ (t x + (1 - t) y) d ν (t)) = x + y (x, y ∈ I) holds. Under at most fourth-order differentiability assumptions and certain nondegeneracy conditions on the measures, the main results of this paper show that there are three classes of the solutions φ, ψ: they are equivalent either to linear, or to exponential or to power functions.

Original language | English |
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Pages (from-to) | 8-23 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 353 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 1 2009 |

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### Keywords

- Gauss-composition
- Generalized quasi-arithmetic mean
- Invariance equation
- Matkowski-Sutô equation

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics