An Estimation of Different Kinds of Integral Inequalities for a Generalized Class of Godunova–Levin Convex and Preinvex Functions via Pseudo and Standard Order Relations - Zareen A. Khan; Waqar Afzal (Journal of Function Spaces)
The connection between generalized convexity and analytic operators is deeply rooted in functional analysis and operator theory. To put the ideas of preinvexity and convexity even closer together, we might state that preinvex functions are extensions of convex functions. Integral inequalities are developed using different types of order relations, each of which behaves differently. There have been recent developments in literature presenting the idea of h-Godunova–Levin convex and preinvex functions and their application to various inequalities. This article introduces the more generalized class of preinvexity referred to as h1,h2-Godunova–Levin preinvex functions for the first time in the context of pseudo-order relations. The results are also developed with the help of h1,h2-Godunova–Levin convex functions. By applying these concepts, Hermite–Hadamard, Fejér, Ostrowski, Simpson, Hölder’s inequalities were developed. Using these two classes, some of the results are developed using pseudointerval order relation and some of the results are developed using standard order relation when our interval is degenerated. In addition, we address the Milne inequality problem, which had not been adjusted for interval-valued functions using inclusion order in order to demonstrate the unique characteristics of this order relation. A number of nontrivial examples support the development of these results.