Some new inequalities for generalized h‐convex functions involving local fractional integral operators with Mittag‐Leffler kernel

Abstract

In this paper, we firstly construct two local fractional integral operators with Mittag‐Leffler kernel on Yang's fractal sets. Then, two local fractional integral identities with the first‐ and second‐order derivatives are derived. With these as auxiliary tools, we establish some new Hermite‐Hadamard–type local fractional integral inequalities involving the local fractional integral operators with Mittag‐Leffler kernel for generalized h‐convex functions. In addition, we obtain some special inequalities when the parameter β and function h take special values. Finally, two examples are given to illustrate the application of the results.

Integral Inequalities for Generalized Approximately h-convex Functions on Fractal Sets via Generalized Local Fractional Integrals

“…[2,8,10,14,18]. Wenbing Sun established various types of Integral inequalities for different generalizations of convex functions in fractal theory [20][21][22][23][24][25]. According to our knowledge the study of approximately h−convex functions has not been carried out in fractal domain.…”

Error bounds on the Hermite-Hadamard- and Bullen-type inequalities for $2\alpha$-local fractional derivative

“…Moreover, it is necessary to note the generalization of local fractional integrals. For example, with the help of local fractional integrals involving Mittag-Leffler kernels, Sun [38] researched two fractal identities and related fractal Hermite-Hadamard-type inequalities for generalized h-convexity. By virtue of the same integrals, Vivas-Cortez et al [41] extended Hermite-Hadamard-type inequalities for generalized ( h1 , h2 )-preinvexity.…”

Some Hermite‐Hadamard's type local fractional integral inequalities for generalized γ‐preinvex function with applications

“…See previous studies. [7][8][9][10][11][12] On the other hand, many researchers extended definitions of convexity on fractal sets to analyze Hermite-Hadamard type inequalities; see previous works [13][14][15][16][17][18][19][20] for further readings. Particularly, some Bullen type integral inequalities for differentiable convex functions are provided includes Caputo derivative and integral and Atangana-Baleanu integral operator; see literature [21][22][23] for more details.…”