On ψ-convex functions and related inequalities

We introduce the class of ψ-convex functions f: [0, ∞) → ℝ, where ψ ∈ C([0, 1]) satisfies ψ ≥ 0 and ψ(0) ≠ ψ(1). This class includes several types of convex functions introduced in previous works. We first study some properties of such functions. Next, we establish a double Hermite-Hadamard-type inequality involving ψ-convex functions and a Simpson-type inequality for functions f ∈ C¹([0, ∞)) such that | f^′ | is ψ-convex. Our obtained results are new and recover several existing results from the literature.