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Linear functional inequalities-a general theory and new special cases

100%

Pycia M.

Instytut Matematyczny Polskiej Akademii Nauk

This paper solves the functional inequality af(s) + bf(t) ≥ f(αs + βt), s,t > 0, with four positive parameters a,b,α,β arbitrarily fixed. The unknown function f: (0,∞) → ℝ is assumed to satisfy the regularity condition $limsup_{s→ 0+} f(s) ≤ 0$. The paper partitions the space of parameters into regions where the inequality has qualitatively similar classes of solutions, estimates the rate of growth of the solutions, determines their signs, and identifies all the parameters such that the solutions form small nontrivial classes of functions. In addition to the well known cases of convex and subadditive functions, examples of such classes of functions include nonnegative power functions $(0,∞) ∋ t ↦ f(1)t^{p}$ for fixed p ≥ 1, nonpositive power functions $(0,∞) ∋ t↦ f(1)t^{p}$ for fixed p ∈ (0,1], and convex functions satisfying some homogeneity conditions.

Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm

61%

Matkowski J., Pycia M.

Annales Polonici Mathematici

1994-1995

tom 60

nr 3

221-230

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

2 Pycia M.

1 Matkowski J.

1 2006

1 1995

Wyniki wyszukiwania

Linear functional inequalities-a general theory and new special cases

Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm