Conjugate functions, lp-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity

• Autorzy: Matkowski J.

Identyfikatory

DOI

http://dx.doi.org/10.7494/OpMath.2014.34.3.523

• Języki publikacji: EN

Abstrakty

EN

For h : (0,∞) → R, the function h* (t) := th( 1/t ) is called (*)-conjugate to h. This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of (*)-conjugacy are proved. If φ and φ* are bijections of (0,∞) then [formula]. Under some natural rate of growth conditions at 0 and ∞, if φ is increasing, convex, geometrically convex, then [formula] has the same properties. We show that the Young conjugate functions do not have this property. For a measure space (Ω,Σ,μ) denote by S = (Ω,Σ,μ) the space of all μ-integrable simple functions x : Ω → R, Given a bijection φ : (0,∞) → (0,∞) define [formula] by [formula] where Ω(x) is the support of x. Applying some properties of the (*) operation, we prove that if ƒ xy ≤ Pφ(x)Pψ (y) where [formula] and [formula] are conjugate, then φ and ψ are conjugate power functions. The existence of nonpower bijections φ and ψ with conjugate inverse functions [formula] such that Pφ and Pψ are subadditive and subhomogeneous is considered.

Słowa kluczowe

EN

Lp-norm like functional; homogeneity; subhomogeneity; subadditivity; Minkowski inequality; Hölder inequality; converses; generalization of the Minkowski and Hölder inequalities; conjugate functions; complementary functions; Young conjugate functions; convex function; geometrically convex function; Wright convex function; functional equation

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2014
• Tom: Vol. 34, no. 3
• Strony: 523--560
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Matkowski J.

• University of Zielona Góra Faculty of Mathematics, Computer Science and Econometrics ul. Szafrana 4a, 65-516 Zielona Góra, Poland
• Silesian University Institute of Mathematics ul. Bankowa 14, 40-007 Katowice, Poland

Bibliografia

• [1] G. Aumann, Konvexe Funktionen und Induktion bei Ungleichungen zwischen Mittelwerten, S.-B. Math.-Naturw. Abt. Bayer. Akad. Wiss. München, (1933), 405–413.
• [2] Z. Dároczy, Zs. Páles, Convexity with given infinite weight sequences, Stochastica 11 (1987), 5–12.
• [3] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, 2nd ed., 1952.
• [4] E. Hille, R.S. Phillips, Functional Analysis and Semigrops, American Mathematical Society Publications 31, A.M.S., Providence, R.I., 1957.
• [5] W. Jarczyk, J. Matkowski, On Mulholland’s inequality, Proc. Amer. Math. Soc. 130 (2002), 3243–3247.
• [6] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Cauchy’s Equation and Jensen’s Inequality, P.W.N, Uniwersytet Slaski, Warszawa-Kraków-Katowice, 1985.
• [7] J. Matkowski, On a characterization of Lp-norm, Ann. Polon. Math. 50 (1989), 137–144.
• [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 functional inequality characterizing convex functions, conjugacy and a generalization of Hölder’s and Minkowski’s inequalities, Aequationes Math. 40 (1990), 168–180.
• [10] J. Matkowski, A generalization of Hölder’s and Minkowski’s inequalities and conjugate fubctions, Constantin Carathéodory: An International Tribute, Th.M. Rassias (Ed.), 1991 World Scientific Publ. Co., 819-827.
• [11] J. Matkowski, Functional inequality characterizing concave functions in (0,1)k, Aequationes Math. 43 (1992), 219–224.
• [12] J. Matkowski, Lp-like paranorms, in Selected Topics in Functional Equations and Iteration
• Theory, Proceedings of the Austrian-Polish Seminar, Universität Graz, October 24–26, 1991, D. Gronau and L. Reich (Eds.), Grazer Math. Ber. 316 (1992), 103–138.
• [13] J. Matkowski, On a generalization of Mulholland’s inequality, Abh. Math. Sem. Univ. Hamburg 63 (1993), 97–103.
• [14] J. Matkowski, The converse of the Hölder inequality and its generalizations, Studia Math. 109 (1994) 2, 171–182.
• [15] J. Matkowski, On a characterization of Lp-norm and a converse of Minkowski’s inequality, Hiroshima Math. J. 26 (1996), 277–287.
• [16] J. Matkowski, The converse of a generalized Hölder inequality, Publicationes Math. Debrecen 50 (1997), 135–143.
• [17] J. Matkowski, The converse theorem for Minkowski’s inequality, Indag. Math. (N.S.) 15 (2004) 1, 73–84.
• [18] J. Matkowski, The converse theorem for the Minkowski inequality, J. Math. Anal. Appl. 348 (2008), 315–323.
• [19] J. Matkowski, A converse of the Hölder inequality theorem, Math. Inequal. Appl. 12 (2009) 1, 21–32.
• [20] J. Matkowski, Subadditive periodic functions, Opuscula Math. 31 (2011) 1, 75–96.
• [21] J. Matkowski, The Pexider type generalization of the Minkowski inequality, J. Math. Anal. Appl. 393 (2012), 298–310.
• [22] J. Matkowski, Conjugate functions and a short proof of a property of convex functions, (to appear).
• [23] J. Matkowski, Subhomogeneity and subadditivity of the Lp-norm like functionals, J. Math. Anal. Appl. 404 (2013), 172–184.
• [24] J.Matkowski, T. Swiatkowski, Quasi-monotonicity, subadditive bijections of R+ and characterization of Lp-norm, J. Math. Anal. Appl. 154 (1991), 493–506.
• [25] H.P. Mulholland, On generalizations of Minkowski’s inequality in the form of a triangle inequality, Proc. London Math. Soc. 51 (1950), 294–307.
• [26] C.T. Ng, Functions generating Schur-convex sums, General Inequalities 5 (Oberwolfach, 1986), 433–438, Internat. Schriftenreihe Numer. Math., 80, Birkhäuser, Basel, 1987.
• [27] M. Pycia, Linear functional inequalities – a general theory and new special cases, Dissertationes Math. 438 (2006), 62pp.
• [28] M.M. Rao, Z.D. Ren, Theory of Orlicz spaces, Marcel Dekker, Inc., New York-Basel- -Hong Kong, 1991.
• [29] A.W. Roberts, D.L. Varberg, Convex Functions, Academic Press, New York and London, 1973.
• [30] R.A. Rosenbaum, Sub-additive functions, Duke Math. J. 17 (1950), 227–247.