In this manuscript, we present a collection of inequalities that extend and generalize the important results of the findings from \cite{bas}, with the latter serving as specific cases within our study. Our research takes many forms of fractional differential inequalities, including the comprehensive structure of the Caputo fractional derivative operator with respect to a function $\psi$. Furthermore, we demonstrate the practical applications of $\sigma$-convex functions in the domain of fractional calculus within this framework.