A Generalization of Hermite–Hadamard–Fejer Type Inequalities for the p -Convex Function via α -Generator

Abstract

In the ${17}^{\mathrm{t}\mathrm{h}}$ century, I. Newton and G. Leibniz found independently each other the basic operations of calculus, i.e., differentiation and integration. And this development broke new ground in mathematics. From 1967 to 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, converting the roles of subtraction and addition into division and multiplication, respectively. And then they generalised this operation. Later, they named this analysis non-Newtonian calculus. This calculus is basically generated by generators. So, in this article, first, we give the definition of the $p$ -convex function due to $\alpha$ -generator. Second, we obtain some new theorems for this function with respect to the $\alpha$ -generator. Third, we get some new theorems using Hermite–Hadamard–Fejer inequality for the $\alpha$ $p$ -convex function. Finally, we show that our obtained results are reduced to the classical case in the special conditions.