In this paper, the author introduced the concept of generalized harmonically convex function on fractal sets R α (0 < α 1) of real line numbers and established generalized Hermite-Hadamard's inequalities for generalized harmonically convex function. Then, by creating a local fractional integral identity, obtained some Hermite-Hadamard type inequalities of these classes of functions.

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In this paper, we establish a local fractional integral identity with a parameter λ on Yang's fractal sets. Using this identity, by generalized power mean inequality and generalized Hölder inequality, two Hermite‐Hadamard type local fractional integral inequalities for generalized harmonically convex functions are established. By giving some special values to the parameter, some inequalities with specific form can be obtained. Some applications to generalized special means are given.

The concept of generalized h-preinvex function on real linear fractal sets $R^{\beta }$ R β ($0 < \beta \le 1$ 0 < β ≤ 1 ) is introduced, which extends generalized preinvex, generalized s-preinvex, generalized Godunova–Levin preinvex, and generalized P-preinvex functions. In addition, some Hermite–Hadamard type inequalities for these classes of functions involving local fractional integrals are established. Lastly, the upper bounds for generalized expectation, generalized rth moment, and generalized variance of a continuous random variable are given to illustrate the applications of the obtained results.

By using thefractal theory and the methods of weight function, a Hilbert-type fractal integral inequality and its equivalent form are given. Their constant factors are proved being the best possible, and their applications are discussed briefly.