In the paper, we introduce the generalized convex function on fractal sets (0 1)real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen's inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.

Let l ∈ N and ⃗ A = (A1, . . . , A l ) and ⃗ f = (f1, . . . , f l ) be 2 finite collections of functions, where every function Ai has derivatives of order mi and f1, . .Suppfi. The generalized higher commutator generated by the multilinear fractional integral is then given bythe authors establish the boundedness of I ⃗ A α,m on the product Lebesgue space, Triebel-Lizorkin space, and Lipschitz space.