Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus.

In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the q ϰ 2 -quantum integral to show midpoint and trapezoidal inequalities for q ϰ 2 -differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via q ϰ 1 -quantum integral is not valid for preinvex functions, and we present its proper form. We use q ϰ 1 -quantum integrals to show midpoint inequalities for q ϰ 1 -differentiable preinvex functions. It is also demonstrated that by considering the limit q → 1 − and η ϰ 2 , ϰ 1 = − η ϰ 1 , ϰ 2 = ϰ 2 − ϰ 1 in the newly derived results, the newly proved findings can be turned into certain known results. [ABSTRACT FROM AUTHOR]