Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm
Volume 60 / 1995
Annales Polonici Mathematici 60 (1995), 221-230
DOI: 10.4064/ap-60-3-221-230
Abstract
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.
