In this paper, based on Yang’s fractal theory, the Hermite–Hadamard’s inequalities for generalized $h$-preinvex function are proved. Then, using the local fractional integral identity proposed by Sun [Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals27(5) (2019) 1950071] as auxiliary function, some parameterized local fractional integral inequalities for generalized $h$-preinvex functions are established. For the special cases of the parameters, some generalized Simpson-type, midpoint-type and trapezoidal inequalities are established. Finally, some applications of these inequalities in numerical integration are proposed.

本文基于杨的分形理论，提出了广义的 Hermite-Hadamard 不等式。$H$-preinvex 函数得到证明。然后，利用Sun提出的局部分数积分恒等式[Some localfractionalfractionalignequalitiesforgeneralizedpreinvexfunctionsandappliancestonumericalquadrature, Fractals 27 (5)(2019)1950071]作为辅助函数，一些参数化的广义预倒函数的局部分数积分不等式$H$-建立前逆函数。针对参数的特殊情况，建立了一些广义的Simpson型、中点型和梯形不等式。最后，提出了这些不等式在数值积分中的一些应用。