\(h\)-strongly \(E\)-convex functions

. Starting from strongly E -convex functions introduced by E. A. Youness, and T. Emam, from h -convex functions introduced by S. Varoˇsanec and from the more general concept of h -convex functions introduced by A. H´azy we deﬁne and study h -strongly E -convex functions. We study some properties of them.

In the following lines we recall the definition of h-convex functions introduced in [7] by S. Varošanec. We consider I and J intervals in R, (0, 1) ⊆ J and the real non-negative functions h : J → R, f : I → R, h = 0. Definition 7. [7]The function f : I → R is called h-convex on I or is said to belong to the class SX (h, I) if for every pair of points x, y ∈ I and every λ ∈ (0, 1), the following inequality is satisfied: In [1] Bombardelli and Varošanec omitted the assumption that f and h are non-negative. We recall now the definitions of h-convex functions introduced in [

PROPERTIES OF h-STRONGLY E-CONVEX FUNCTIONS
Starting from strongly E-convex functions and from h-convex functions in the sense of Házy we define and study h-strongly E-convex functions.
In the following lines we consider a map E : R n → R n and a strongly Econvex set A ⊂ R n . We also consider the functions h : if for every pair of points x, y ∈ A, α ∈ [0, 1] and λ ∈ [0, 1], the following inequality is satisfied: Proof. We put λ = 1 in (1) and we obtain (2).
Hence the function F is h-strongly E-convex on A.
We consider a strongly E-convex set A ⊂ R n , a function f : R n → R, and a function ϕ : R → R linear and nondecreasing.
Theorem 12. If the function f : R n → R is h-strongly E-convex on A then the composite function ϕ • f is h-strongly E-convex on A.
Theorem 13. If the function f : R n → R is non-negative and differentiable h-strongly E-convex on a strongly E-convex set A and h is a non-negative function with the property h (λ) ≤ λ for every λ ∈ [0, 1] then for every x, y ∈ A.
Proof. Since f is h-strongly E-convex on A, and hence By taking α → 0, we get Dividing by λ > 0 and taking λ → 0, we obtain for each x, y ∈ A.
The following theorem provides a characterization of h-strongly E-convex functions with respect to the E-monotonicity of the gradient of map, similar with that obtain from E-convex functions, by Soleimani-Damaneh in [3].
Theorem 15. If the function f : R n → R is non-negative and differentiable h-strongly E-convex on a strongly E-convex set A and h is a non-negative function with the property h (λ) ≤ λ for every λ ∈ [0, 1] then (4) (∇f (E (x)) − ∇f (E (y))) (E (x) − E (y)) ≥ 0 for every x, y ∈ A.
Proof. Since f is h-strongly E-convex on A, from theorem (13) we have , for every x, y ∈ A. Adding these two inequalities we obtain (∇f (E (x)) − ∇f (E (y))) (E (x) − E (y)) ≥ 0 for every x, y ∈ A.
If E (M ) ⊆ M and the function h is positively then the set M is strongly E-convex.