A generalization of convexity via an implicit inequality

We unified several kinds of convexity by introducing the class ${\mathcal{A}}_{\zeta ,w}([0,1]\times I^2)$ of $(\zeta ,w)$-admissible functions $F:[0,1]\times I\times I\to \mathbb{R}$. Namely, we proved that most types of convexity from the literature generate functions $F\in {\mathcal{A}}_{\zeta ,w}([0,1]\times I^2)$ for some $\zeta \in C([0,1])$ and $w\in C^1(I)$ with $w(I)\subset I$ and $w^{\prime }>0$. We also studied some properties of $(\zeta ,w)$-admissible functions and established some integral inequalities that unify various Hermite-Hadamard-type inequalities from the literature.