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1994 | 109 | 2 | 171-182
Tytuł artykułu

The converse of the Hölder inequality and its generalizations

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EN
Abstrakty
EN
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.
Twórcy
  • Department of Mathematics, Technical University, Willowa 2, 43-309 Bielsko-Biała, Poland
Bibliografia
  • [1] R. Cooper, Note on the Cauchy-Hölder inequality, J. London Math. Soc. (2) 26 (1928), 8-9.
  • [2] Z. Daróczy and Z. Páles, Convexity with given infinite weight sequences, Stochastica 11 (1987), 5-12.
  • [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.
  • [8] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675.
  • [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 $(0,∞)^k$, Aequationes Math. 43 (1992), 219-224.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv109i2p171bwm
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