Superquadratic functions in several variables

Abstract

The concept of superquadratic functions in several variables, as a generalization of the same concept in one variable is introduced. Analogous results to results obtained for convex functions in one and several variables are presented. These include refinements of Jensen's inequality and its counterpart, and of SlaterPečarić's inequality.

A Strengthened Form of a General Hardy-Type Inequality Obtained via Superquadraticity and Its Applications

“…For reader convenience, we also recall the definition and some basic properties of superquadratic functions, introduced by Abramovich et al [2] (for more information see also [1,3]). A function Ψ : [0, ∞) → R is called superquadratic provided that for each x ≥ 0 there exists a constant C x ∈ R such that…”

Regularity of weakly subquadratic functions

“…(More precisely, in the paper above, superquadratic functions were considered, but here is an analogous relation between the concepts as above.) Subquadratic (or superquadratic) functions have been investigated by several authors in this sense (cf., e.g., [1,2,[4][5][6][8][9][10][11][12]18]). As a result of the study of the connection between the two different concepts of subquadraticity (cf.…”

Refinement multidimensional dynamic inequalities with general kernels and measures

“…for a nonnegative integrable function g in the space L p (R + ), 1 < p < ∞, where the constant (p/(p -1)) p is the best possible. Rewriting (1) with the function g 1/p instead of g and taking the limit as p → ∞, we get the limiting case of the Hardy inequality, known as the Pólya-Knopp inequality (see [15]), ln g(s) ds dt ≤ e ∞ 0 g(t) dt (2) for positive functions g ∈ L 1 (R + ), where the constant e is the best possible. Recently, Kaijser et al [14] have pointed out that both (1) and (2) are just particular cases of the much more general Hardy-Knopp's inequality ous modifications both in the continuous and discrete settings.…”

“…Rewriting (1) with the function g 1/p instead of g and taking the limit as p → ∞, we get the limiting case of the Hardy inequality, known as the Pólya-Knopp inequality (see [15]), ln g(s) ds dt ≤ e ∞ 0 g(t) dt (2) for positive functions g ∈ L 1 (R + ), where the constant e is the best possible. Recently, Kaijser et al [14] have pointed out that both (1) and (2) are just particular cases of the much more general Hardy-Knopp's inequality ous modifications both in the continuous and discrete settings. They have extensive applications to partial differential and difference equations, harmonic analysis, approximations, number theory, optimization, convex geometry, spectral theory of differential and difference operators, and others (see [24][25][26][27][28]30]).…”