Warianty tytułu
Abstrakty
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.
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.
Słowa kluczowe
Tematy
Kategoryzacja MSC:
- 26D05: Inequalities for trigonometric functions and polynomials
- 26B25: Convexity, generalizations
- 26D07: Inequalities involving other types of functions
- 26A09: Elementary functions
- 26A51: Convexity, generalizations
- 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
- 39B62: Functional inequalities, including subadditivity, convexity, etc.
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
438
Liczba stron
62
Liczba rozdzia³ów
Opis fizyczny
Daty
wydano
2006
Twórcy
autor
- Department of Economics, Massachusetts Institute of Technology, E52-391, 50 Memorial Dr., Cambridge, MA 02142, U.S.A.
- Leon Koźmiński Academy of Entrepreneurship and Management, Jagiellońska 59, 03-301 Warszawa, Poland
Bibliografia
Języki publikacji
EN |
Uwagi
Identyfikator YADDA
bwmeta1.element.bwnjournal-rm-doi-10_4064-dm438-0-1
Identyfikatory
DOI
10.4064/dm438-0-1
Kolekcja
DML-PL
