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

László Losonczi, Zsolt Páles

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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 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.

Original languageEnglish
Pages (from-to)743-753
Number of pages11
JournalPublicationes Mathematicae
Volume78
Issue number3-4
DOIs
Publication statusPublished - Sep 26 2011

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Keywords

  • Equality and homogeneity problem
  • Generalized cauchy means

ASJC Scopus subject areas

  • Mathematics(all)

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