On $\psi$-convex functions and related inequalities

We introduce the class of $\psi$-convex functions $f:[0,\mathrm{\infty })\to \mathbb{R}$, where $\psi \in C([0,1])$ satisfies $\psi \ge 0$ and $\psi (0)\ne \psi (1)$. 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 $f\in C^1([0,\mathrm{\infty }))$ such that $|f^{\prime }|$ is $\psi$-convex. Our obtained results are new and recover several existing results from the literature.