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Abstract Under the new concept of s- ( α , m ) $(\alpha,m)$ -convex functions, we obtain some new Hermite–Hadamard inequalities with an s- ( α , m ) $(\alpha,m)$ -convex function. We use these inequalities to estimate Riemann–Liouville fractional integrals with second-order differentiable convex functions to enrich the Hermite–Hadamard-type inequalities. We give some applications to special means.