Generalized Fej\'er-Hermite-Hadamard type via generalized $(h-m)$-convexity on fractal sets and applications

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 ${\mathbb{R}}^{\alpha }$ $(0<\alpha \le 1)$. Some properties of this new class are discussed. Using local fractional integrals and generalized $(h-m)$-convexity, we generalized Hermite-Hadamard (H-H) and 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.