Integral Inequalities of Integer and Fractional Orders for n–Polynomial Harmonically tgs–Convex Functions and Their Applications

The main objective of this article is to introduce the notion of n–polynomial harmonically tgs–convex function and study its algebraic properties. First, we use this notion to present new variants of the Hermite–Hadamard type inequality and related integral inequalities, as well as their fractional analogues. Further, we prove two interesting integral and fractional identities for differentiable mappings, and, using them as auxiliary results, some refinements of Hermite–Hadamard type integral inequalities for both classical and fractional versions are presented. Finally, in order to show the efficiency of our results, some applications for special means and error estimations are obtained as well.