On ψ-convex functions and related inequalities

We introduce the class of $ \psi $-convex functions $ f:[0, \infty)\to \mathbb{R} $, where $ \psi\in C([0, 1]) $ satisfies $ \psi\geq 0 $ and $ \psi(0)\neq \psi(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 $ \psi $-convex functions and a Simpson-type inequality for functions $ f\in C^1([0, \infty)) $ such that $ |f'| $ is $ \psi $-convex. Our obtained results are new and recover several existing results from the literature.