On h-convexity

Abstract

We introduce a class of h-convex functions which generalize convex, s-convex, Godunova-Levin functions and P -functions. Namely, the h-convex function is defined as a non-negative function f :where h is a non-negative function, α ∈ (0, 1) and x, y ∈ J . Some properties of h-convex functions are discussed. Also, the Schur-type inequality is given.

The equality case in some recent convexity inequalities

“…A function f : D → R is (T, h)-convex if (1.1) holds for all x, y ∈ D and t ∈ T . It is clear that this generalizes the concepts of convexity (h(t) = t, t ∈ [0, 1], [24], [21]), the Breckner-convexity (h(t) = t s , t ∈]0, 1[, for some s ∈ R, [5], [6]), the Godunova-Levin functions (h(t) = t −1 , t ∈]0, 1[, [10]), the P-functions (h(t) = 1, t ∈ [0, 1], [18]), and the t-convexity (T = {t, 1 − t}, h(t) = t, h(1 − t) = 1 − t, where 0 < t < 1 is a fixed number, Kuhn [14]). For further related results see Burai-Házy [1,2] and Burai-Házy-Juhász [3,4].…”

“…In a recent paper [24] by Varošanec, a common generalization of convex and s-convex functions, Godunova-Levin functions, and P-functions is introduced in the following way: Let I be a nonvoid subinterval of R (the set of all real numbers), h : [0, 1] → R and f : I → R be real-valued functions satisfying the inequality f (tx + (1 − t)y) ≤ h(t)f (x) + h(1 − t)f (y) (1.1) for all x, y ∈ I and t ∈]0, 1[. An even more general notion, the so-called (T, h)-convexity, can be found in Házy [11]: Let X be a real or complex normed space, D ⊂ X be a nonempty convex set, ∅ = T ⊂ [0, 1], and h : T → R be a function.…”

Conformable fractional Hermite-Hadamard inequalities via preinvex functions

“…Noor et al [12] introduced another class of preinvex functions which is called as Godunova-Levin type of s-preinvex functions and noticed that this class also included in the class of h-preinvex functions. For η(y, x) = y − x the class of h-preinvex functions reduces to the class of h-convex functions introduced and studied by Varosanec [19]. Thus it is worth to mention here that the class of h-preinvex is quite unifying one.…”

Integral Inequalities Involving Strongly Convex Functions

“…These extensions and generalizations in the theory of inequalities have made valuable contributions in many areas of mathematics. Some new generalized concepts in this point of view are quasiconvex [1], strongly convex [2], approximately convex [3], logarithmically convex [4], midconvex functions [5], pseudoconvex [6], -convex [7], -convex [8], ℎ-convex [9], delta-convex [10], Schur convex [11][12][13][14][15], and others [16][17][18][19].…”