# 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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topMatkowski, 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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- [3] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, 1952.
- [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] N. Kuhn, A note on t-convex functions, in: General Inequalities 4, Internat. Ser. Numer. Math. 71, Birkhäuser, Basel, 1984, 269-276.
- [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] 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] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675. Zbl0704.46020
- [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] J. Matkowski, Functional inequality characterizing concave functions in ${(0,\infty )}^{k}$, Aequationes Math. 43 (1992), 219-224. Zbl0756.39017

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