The converse of the Hölder inequality and its generalizations

Janusz Matkowski¹

¹

Studia Mathematica, Tome 109 (1994) no. 2, pp. 171-182

Résumé

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 μ (A) 1 μ (B) ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if $ʃ_Ω xydμ ≤ ϕ^{-1} (ʃ_{Ω} ϕ∘x dμ) ψ^{-1} (ʃ_{Ω} ψ∘x dμ)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist some broad classes of nonpower bijections ϕ and ψ such that the above inequality holds true. A general inequality which contains integral Hölder and Minkowski inequalities as very special cases is also given.

Détail

DOI : 10.4064/sm-109-2-171-182

Keywords: measure space, integrable step functions, conjugate functions, a converse of Hölder inequality, subadditive function, convex function, generalized Hölder-Minkowski inequality