Integral inequalities of Hermite-Hadamard type for functions whose 3rd derivatives are (α,m)-convex

Let f : I ⊆R→R be a convex function on I and a, b∈ I with a< b. The inequality

This inequality is well known in the literature as Hermite-Hadamard’s inequality for convex mappings.

Definition 1.1A function f : I ⊆R= (−∞, +∞) →R is said to be convex if

holds for all x, y ∈ I and t ∈ [0, 1]. Definition 1.2 A function f : [0, b] →R is said to be m-convex if

holds for all x, y ∈ [0, b], t ∈ [0, 1] and m∈ (0, 1]. Definition 1.3 A function f : [0, b] →R is said to be (α,m)-convex if

is valid for all x, y ∈ [0, b] and t ∈ [0, 1], and (α,m) ∈ (0, 1]2.