Hermite–Hadamard Type Inequalities Involving (k-p) Fractional Operator for Various Types of Convex Functions

Abstract

We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as (p,h) fractional type inequality and many others, as the (p,h)-convexity is the generalization of the other convex inequalities. As a consequence of the (h,m)-convexity, the fractional inequality of the (s,m)-type is obtained. Many consequences of such fractional inequalities and generalizations are obtained.

Hermite Hadamard Type Inequalities Involving (k-p) Fractional Operator with (α, h − m) − p convexity

“…The following Theorem generalizes the Theorem 1 from the recently published paper [40] about k − p fractional inequalities.…”

“…Corollary 3. Setting α, l, m = 1 in the previously derived inequality, we obtain Theorem 4 from the paper [40] , namely we obtain…”

“…Different types of fractional integrals and derivatives were defined throughout the years, we refer the interested reader to the following books [18,28,44] for more information on the matter. The motivation for this paper comes from the recently published paper by Stojiljković et al [40] where the authors established some Theorems regarding k − p fractional inequalities. In this paper, we generalize the obtained inequalities.…”

“…The left and right sided (k − p) Riemann-Liouville fractional integral operators are given by Recently published paper by Stojiljković. et al [40] proved some inequalities regarding k−p fractional operators. We state two of them for the completeness because the Theorems in this paper generalize the results in the recently published paper.…”

Generalized m -preinvexity on fractal set and related local fractional integral inequalities with applications

“…The most special inequality among all the fundamental inequalities is the Hermite-Hadamard inequality, initiated by Hadamard, see [10]. Many crucial results, which are generalizations, refinements, and improvements of the classical Hermite-Hadamard inequality are associated with various generalizations of convex function, we refer to [20,[23][24][25]41] and references there in.…”