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

• Autorzy: J. Matkowski , T. Świątkowski
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

subadditive function; homeomorphisms of $ℝ_+$; Mulholland's inequality; convex function; iteration; measure space; the converse of Minkowski's inequality

Kategorie tematyczne

• 26D15: Inequalities for sums, series and integrals
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 26A18: Iteration
• 26A51: Convexity, generalizations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1993
• Tom: 143
• Numer: 1
• Strony: 75-85

Daty

wydano

1993

otrzymano

1992-09-21

Twórcy

autor

J. Matkowski

• Department of Mathematics, Technical University, Willowa 2, 43-309 Bielsko-Biała, Poland

autor

T. Świątkowski

• 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.