- Tytuł:
- The converse of the Hölder inequality and its generalizations
- Autorzy:
- Matkowski, Janusz
- Powiązania:
- https://bibliotekanauki.pl/articles/1290537.pdf
- Data publikacji:
- 1994
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
measure space
integrable step functions
conjugate functions
a converse of Hölder inequality
subadditive function
convex function
generalized Hölder-Minkowski inequality - Opis:
- 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 $ʃ_Ω xydμ ≤ ϕ^{-1} (\int_{Ω} ϕ∘x dμ) ψ^{-1} (\int_{Ω} ψ∘x dμ)$ 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.
- Źródło:
-
Studia Mathematica; 1994, 109, 2; 171-182
0039-3223 - Pojawia się w:
- Studia Mathematica
- Dostawca treści:
- Biblioteka Nauki
Artykuł