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1993 | 143 | 1 | 75-85
Tytuł artykułu

Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities

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Let ϕ be an arbitrary bijection of $ℝ_+$. We prove that if the two-place function $ϕ^{-1}[ϕ (s)+ϕ (t)]$ is subadditive in $ℝ^2_+$ then $ϕ $ must be a convex homeomorphism of $ℝ_+$. This is a partial converse of Mulholland's inequality. Some new properties of subadditive bijections of $ℝ_+$ are also given. We apply the above results to obtain several converses of Minkowski's inequality.
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autor
  • Department of Mathematics, Technical University, Willowa 2, 43-309 Bielsko-Biała, Poland
  • Institute of Mathematics, Technical University, Al. Politechniki 11, 90-924 Łódź, Poland
Bibliografia
  • [1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York 1966.
  • [2] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publishers and Silesian University, Warszawa-Kraków-Katowice 1985.
  • [3] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675.
  • [4] J. Matkowski and T. Świątkowski, Quasi-monotonicity, subadditive bijections of $ℝ_+$, and characterization of $L^p$-norm, J. Math. Anal. Appl. 154 (1991), 493-506.
  • [5] J. Matkowski and T. Świątkowski, On subadditive functions, Proc. Amer. Math. Soc., to appear.
  • [6] H. P. Mulholland, On generalizations of Minkowski's inequality in the form of a triangle inequality, Proc. London Math. Soc. 51 (1950), 294-307.
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bwmeta1.element.bwnjournal-article-fmv143i1p75bwm
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