Ostrowski-type inequalities for n-polynomial $\mathscr{P}$-convex function for k-fractional Hilfer–Katugampola derivative

Abstract

In this article, we develop a novel framework to study a new class of convex functions known as n-polynomial $\mathscr{P} $ P -convex functions. The purpose of this article is to establish a new generalization of Ostrowski-type integral inequalities by using a generalized k-fractional Hilfer–Katugampola derivative. We employ this technique by using the Hölder and power-mean integral inequalities. We present analogs of the Ostrowski-type integrals inequalities connected with the n-polynomial $\mathscr{P}$ P -convex function. Some new exceptional cases from the main results are obtained, and some known results are recaptured. In the end, an application to special means is given as well. The article seeks to create an exciting combination of a convex function and special functions in fractional calculus. It is supposed that this investigation will provide new directions in fractional calculus.

Strongly n-polynomial convexity and related inequalities

“…See articles [14,17,19,23,27] and the references therein. We would also like to point out in particular that readers wishing to learn more about the subject of this article and the various types of convexity may refer to the references [3,1,2,4,6,7,9,10,8,11,24,20,21,25,26,29,28,30,31,33].…”

Inequalities for 3-convex functions and applications

“…Recently many generalizations and extensions have been made for the convexity, like: preinvexity [2], GA-convexity [3], strong convexity [4], s-convexity [5], and others. Also some standard inequalities have been defined for different type of convex functions, such as: for PP-convex functions [6], for harmonically convex and harmonically quasi convex functions [7], for interval-valued convex functions [8], for s-convex functions on fractal sets [9],…”

On the refinements of some important inequalities with a finite set of positive numbers

“…This better inequality has attracted a large number of authors who have used Hölder–Iscan integral inequality to provide the best technique in the field of integral inequality linked to Hermit–Hadamard inequality [3–5], Jensen inequality [6, 7], Ostrowski inequality [8], Bullen‐type inequality [9], and Simpson inequality [10], such as refinement, generalization, or extension. H ölder–Iscan integral inequality is overused in various $$$ n $$$‐polynomial convexity [11, 12] and power‐mean integral inequality results [13].…”