In this article, we define a new class of convexity called generalized $(h-m)$-convexity, which generalizes $h$-convexity and $m$-convexity on fractal set ${\mathbb{R}}^{\alpha }$ $(0<\alpha \le 1)$. Some properties of this new class are discussed. Using local fractional integrals and generalized $(h-m)$-convexity, we generalized Hermite–Hadamard (H–H) and Fejér–Hermite–Hadamard (Fejér–H–H) types inequalities. We also obtained a new result of the Fejér–H–H type for the function whose derivative in absolute value is the generalized $(h-m)$-convexity on fractal sets. As applications, we studied some new inequalities for random variables, numerical integrations and generalized to special means.

在本文中，我们定义了一类新的凸性，称为广义 $（H-米）$-凸性，泛化 $H$-凸性和 $米$分形集上的凸性 ${\mathbb{[R}}^{\alpha }$ $（0<\alpha \le \mathrm{1个}）$。讨论了这个新类的一些属性。使用局部分数积分和广义$（H-米）$-凸性，我们推广了Hermite–Hadamard（H–H）和Fejér–Hermite–Hadamard（Fejér–H–H）类型的不等式。我们还获得了Fejér–H–H型函数的新结果，该函数的绝对值导数是广义的$（H-米）$分形集的凸性。作为应用，我们研究了随机变量，数值积分的新不等式，并推广到了特殊的均值。