### Abstract

Given a continuous strictly monotone function φ : I → ℝ and a probability measure μ on the Borel subsets of [0,1], the two variable mean M_{φ,μ} : I^{2} → I is defined by M _{φ,μ}(x, y) := φ^{-1}(∫_{0}^{1} φ(tx + (1 - t)y)dμ(t)) (x, y ε I. This class of means includes quasi-arithmetic as well as Lagrangian means. The aim of this paper is to study their equality problem, i.e., to characterize those pairs (φ,μ) and (ψ, ν) such that M_{φ,μ}(x, y) = M_{ψ,ν}(x, y) (x,t ε I holds. Under at most fourth-order differentiability assumptions for the unknown functions φ and ψ, a complete description of the solution set of the above functional equation is obtained.

Original language | English |
---|---|

Pages (from-to) | 407-440 |

Number of pages | 34 |

Journal | Publicationes Mathematicae |

Volume | 72 |

Issue number | 3-4 |

Publication status | Published - 2008 |

### Fingerprint

### Keywords

- Equality problem
- Generalized quasi-arithmetic means

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*72*(3-4), 407-440.

**On the equality of generalized quasi-arithmetic means.** / Makó, Zita; Páles, Z.

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 72, no. 3-4, pp. 407-440.

}

TY - JOUR

T1 - On the equality of generalized quasi-arithmetic means

AU - Makó, Zita

AU - Páles, Z.

PY - 2008

Y1 - 2008

N2 - Given a continuous strictly monotone function φ : I → ℝ and a probability measure μ on the Borel subsets of [0,1], the two variable mean Mφ,μ : I2 → I is defined by M φ,μ(x, y) := φ-1(∫01 φ(tx + (1 - t)y)dμ(t)) (x, y ε I. This class of means includes quasi-arithmetic as well as Lagrangian means. The aim of this paper is to study their equality problem, i.e., to characterize those pairs (φ,μ) and (ψ, ν) such that Mφ,μ(x, y) = Mψ,ν(x, y) (x,t ε I holds. Under at most fourth-order differentiability assumptions for the unknown functions φ and ψ, a complete description of the solution set of the above functional equation is obtained.

AB - Given a continuous strictly monotone function φ : I → ℝ and a probability measure μ on the Borel subsets of [0,1], the two variable mean Mφ,μ : I2 → I is defined by M φ,μ(x, y) := φ-1(∫01 φ(tx + (1 - t)y)dμ(t)) (x, y ε I. This class of means includes quasi-arithmetic as well as Lagrangian means. The aim of this paper is to study their equality problem, i.e., to characterize those pairs (φ,μ) and (ψ, ν) such that Mφ,μ(x, y) = Mψ,ν(x, y) (x,t ε I holds. Under at most fourth-order differentiability assumptions for the unknown functions φ and ψ, a complete description of the solution set of the above functional equation is obtained.

KW - Equality problem

KW - Generalized quasi-arithmetic means

UR - http://www.scopus.com/inward/record.url?scp=45149120981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45149120981&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:45149120981

VL - 72

SP - 407

EP - 440

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 3-4

ER -