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