This paper contains a variety of new integral inequalities for $(s,m)$-convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are $(s,m)$-convex. Inequalities involving the digamma function and special means are deduced as applications.

This paper accomplishes some new Hermite-Hadamard type inequalities for \((\alpha,s,m)\)-convex functions in the context of Caputo-Fabrizio integrals. We establish this new version with the aid of Hölder and power-mean inequalities. Applications to obtained results are given by special means. Furthermore, errors are estimated for the trapezoidal formula and are graphically depicted. Finally, some bounds for the expectation value of the probability density function (which is \((\alpha,s,m)\)-convex) are obtained using the exponential population growth model.