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 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.
@article{bwmeta1.element.bwnjournal-article-smv109i2p171bwm,
author = {Janusz Matkowski},
title = {The converse of the H\"older inequality and its generalizations},
journal = {Studia Mathematica},
volume = {108},
year = {1994},
pages = {171-182},
zbl = {0819.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv109i2p171bwm}
}
Matkowski, Janusz. The converse of the Hölder inequality and its generalizations. Studia Mathematica, Tome 108 (1994) pp. 171-182. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv109i2p171bwm/
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