### Abstract

Given two continuous functions f, g : I -ℝ R such that g is positive and f/g is strictly monotone, and a probability measure μ on the Borel subsets of [0,1], the two variable mean M_{f,g,μ} : I^{2} → I is defined by M_{f,g,μ}(x,y):=(f/g) ^{-1}(∫_{0}^{1}f(tx+(1-t)y)dμ(t) /∫_{0}^{1}g(tx+(1-t)y)dμ(t))(x,y ∈I). The aim of this paper is to study Minkowski-type inequalities for these means, i.e., to find conditions for the generating functions f_{0},g_{0} : I_{0} → R, f_{1}, g_{1} : I_{1} → R,., f_{n}, g_{n} : In→ R, and for the measure μ such that M_{f0,g0:μ}(x_{1}+.+x_{n},y_{1}+.+y _{n}) ≤ [≥] Mf_{1},g1;μ(x_{1},y _{1})+.+Mf_{n},g0,μ(x_{n},y_{n}) holds for all x_{1},y_{1} ∈ I_{1}, ., x_{n},y _{n} ∈ I_{n} with x_{1} + . + x_{n},y _{1}+.+y_{n} ∈ I_{0}. The particular case when the generating functions are power functions, i.e., when the means are generalized Gini means is also investigated.

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

Pages (from-to) | 743-753 |

Number of pages | 11 |

Journal | Publicationes Mathematicae |

Volume | 78 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2011 |

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### Keywords

- Equality and homogeneity problem
- Generalized cauchy means

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*78*(3-4), 743-753. https://doi.org/10.5486/PMD.2011.5017

**Minkowski-type inequalities for means generated by two functions and a measure.** / Losonczi, László; Páles, Z.

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 78, no. 3-4, pp. 743-753. https://doi.org/10.5486/PMD.2011.5017

}

TY - JOUR

T1 - Minkowski-type inequalities for means generated by two functions and a measure

AU - Losonczi, László

AU - Páles, Z.

PY - 2011

Y1 - 2011

N2 - Given two continuous functions f, g : I -ℝ R such that g is positive and f/g is strictly monotone, and a probability measure μ on the Borel subsets of [0,1], the two variable mean Mf,g,μ : I2 → I is defined by Mf,g,μ(x,y):=(f/g) -1(∫01f(tx+(1-t)y)dμ(t) /∫01g(tx+(1-t)y)dμ(t))(x,y ∈I). The aim of this paper is to study Minkowski-type inequalities for these means, i.e., to find conditions for the generating functions f0,g0 : I0 → R, f1, g1 : I1 → R,., fn, gn : In→ R, and for the measure μ such that Mf0,g0:μ(x1+.+xn,y1+.+y n) ≤ [≥] Mf1,g1;μ(x1,y 1)+.+Mfn,g0,μ(xn,yn) holds for all x1,y1 ∈ I1, ., xn,y n ∈ In with x1 + . + xn,y 1+.+yn ∈ I0. The particular case when the generating functions are power functions, i.e., when the means are generalized Gini means is also investigated.

AB - Given two continuous functions f, g : I -ℝ R such that g is positive and f/g is strictly monotone, and a probability measure μ on the Borel subsets of [0,1], the two variable mean Mf,g,μ : I2 → I is defined by Mf,g,μ(x,y):=(f/g) -1(∫01f(tx+(1-t)y)dμ(t) /∫01g(tx+(1-t)y)dμ(t))(x,y ∈I). The aim of this paper is to study Minkowski-type inequalities for these means, i.e., to find conditions for the generating functions f0,g0 : I0 → R, f1, g1 : I1 → R,., fn, gn : In→ R, and for the measure μ such that Mf0,g0:μ(x1+.+xn,y1+.+y n) ≤ [≥] Mf1,g1;μ(x1,y 1)+.+Mfn,g0,μ(xn,yn) holds for all x1,y1 ∈ I1, ., xn,y n ∈ In with x1 + . + xn,y 1+.+yn ∈ I0. The particular case when the generating functions are power functions, i.e., when the means are generalized Gini means is also investigated.

KW - Equality and homogeneity problem

KW - Generalized cauchy means

UR - http://www.scopus.com/inward/record.url?scp=80053012500&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80053012500&partnerID=8YFLogxK

U2 - 10.5486/PMD.2011.5017

DO - 10.5486/PMD.2011.5017

M3 - Article

VL - 78

SP - 743

EP - 753

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 3-4

ER -