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1994-1995 | 60 | 3 | 221-230
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Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm

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Let a and b be fixed real numbers such that 0 < min{a,b} < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that $limsup_{t → 0+} f(t) ≤ 0$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the $L^p$-norm.
Twórcy
  • Department of Mathematics, Technical University, Willowa 2, 43-309 Bielsko-Biała, Poland
autor
  • Rafowa 21, 43-300 Bielsko-Biała, Poland
Bibliografia
  • [1] J. Aczél and J. Dhombres, Functional Equations in Several Variables, Encyclopedia Math. Appl. 31, Cambridge University Press, Cambridge, Sydney, 1989.
  • [2] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, R.I., 1957.
  • [3] 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, 1985.
  • [4] J. Matkowski, On a characterization of $L^p$-norm, Ann. Polon. Math. 50 (1989), 137-144.
  • [5] 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.
  • [6] J. Matkowski, Functional inequality characterizing nonnegative concave functions in (0,∞), ibid. 43 (1992), 219-224.
  • [7] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675.
  • [8] J. Matkowski, $L^p$-like paranorms, in: Selected Topics in Functional Equations and Iteration Theory, Proc. Austrian-Polish Seminar, Graz, 1991, Grazer Math. Ber. 316 (1992), 103-135.
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bwmeta1.element.bwnjournal-article-apmv60z3p221bwm
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