# The converse of the Hölder inequality and its generalizations

Studia Mathematica (1994)

• Volume: 109, Issue: 2, page 171-182
• ISSN: 0039-3223

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## Abstract

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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 ${ʃ}_{\Omega }xyd\mu \le {\varphi }^{-1}\left({ʃ}_{\Omega }\varphi \circ xd\mu \right){\psi }^{-1}\left({ʃ}_{\Omega }\psi \circ xd\mu \right)$ 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.

## How to cite

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Matkowski, Janusz. "The converse of the Hölder inequality and its generalizations." Studia Mathematica 109.2 (1994): 171-182. <http://eudml.org/doc/216067>.

@article{Matkowski1994,
abstract = {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.},
author = {Matkowski, Janusz},
journal = {Studia Mathematica},
keywords = {measure space; integrable step functions; conjugate functions; a converse of Hölder inequality; subadditive function; convex function; generalized Hölder-Minkowski inequality; integral Hölder and Minkowski inequalities},
language = {eng},
number = {2},
pages = {171-182},
title = {The converse of the Hölder inequality and its generalizations},
url = {http://eudml.org/doc/216067},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Matkowski, Janusz
TI - The converse of the Hölder inequality and its generalizations
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 2
SP - 171
EP - 182
AB - 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.
LA - eng
KW - measure space; integrable step functions; conjugate functions; a converse of Hölder inequality; subadditive function; convex function; generalized Hölder-Minkowski inequality; integral Hölder and Minkowski inequalities
UR - http://eudml.org/doc/216067
ER -

## References

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1. [1] R. Cooper, Note on the Cauchy-Hölder inequality, J. London Math. Soc. (2) 26 (1928), 8-9. Zbl54.0221.01
2. [2] Z. Daróczy and Z. Páles, Convexity with given infinite weight sequences, Stochastica 11 (1987), 5-12.
3. [3] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, 1952.
4. [4] M. Kuczma, An introduction to the theory of functional equations and inequalities, Cauchy's equation and Jensen's inequality, Prace Nauk. Uniw. Śl. 489, Polish Scientific Publishers, Warszawa-Kraków-Katowice, 1985.
5. [5] N. Kuhn, A note on t-convex functions, in: General Inequalities 4, Internat. Ser. Numer. Math. 71, Birkhäuser, Basel, 1984, 269-276.
6. [6] J. Matkowski, Cauchy functional equation on a restricted domain and commuting functions, in: Iteration Theory and Its Functional Equations (Proc. Schloss Hofen, 1984), Lecture Notes in Math. 1163, Springer, Berlin, 1985, 101-106.
7. [7] J. Matkowski, Functional inequality characterizing convex functions, conjugacy and a generalization of Hölder's and Minkowski's inequalities, Aequationes Math. 40 (1990), 168-180. Zbl0715.39013
8. [8] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675. Zbl0704.46020
9. [9] J. Matkowski, A generalization of Holder's and Minkowski's inequalities and conjugate functions, in: Constantin Carathéodory: An International Tribute, Vol. II, World Scientific, Singapore, 1991, 819-827.
10. [10] J. Matkowski, Functional inequality characterizing concave functions in ${\left(0,\infty \right)}^{k}$, Aequationes Math. 43 (1992), 219-224. Zbl0756.39017

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