Geodesic
Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm
Annales Polonici Mathematici, Tome 60 (1994) no. 3, pp. 221-230

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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.
DOI : 10.4064/ap-60-3-221-230
Keywords: functional inequality, subadditive functions, homogeneous functions, Banach functionals, convex functions, linear space, cones, measure space, integrable step functions, $L^p$-norm, Minkowski's inequality

Janusz Matkowski 1 ; Marek Pycia 1

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Janusz Matkowski; Marek Pycia. Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm. Annales Polonici Mathematici, Tome 60 (1994) no. 3, pp. 221-230. doi: 10.4064/ap-60-3-221-230

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