Hermite–Hadamard, Fejér and Pachpatte-Type Integral Inequalities for Center-Radius Order Interval-Valued Preinvex Functions

Abstract

The objective of this manuscript is to establish a link between the concept of inequalities and Center-Radius order functions, which are intriguing due to their properties and widespread use. We introduce the notion of the CR (Center-Radius)-order interval-valued preinvex function with the help of a total order relation between two intervals. Furthermore, we discuss some properties of this new class of preinvexity and show that the new concept unifies several known concepts in the literature and also gives rise to some new definitions. By applying these new definitions, we have amassed many classical and novel special cases that serve as applications of the key findings of the manuscript. The computations of cr-order intervals depend upon the following concept B=⟨Bc,Br⟩=⟨B¯+B̲2,B¯−B̲2⟩. Then, for the first time, inequalities such as Hermite–Hadamard, Pachpatte, and Fejér type are established for CR-order in association with the concept of interval-valued preinvexity. Some numerical examples are given to validate the main results. The results confirm that this new concept is very useful in connection with various inequalities. A fractional version of the Hermite–Hadamard inequality is also established to show how the presented results can be connected to fractional calculus in future developments. Our presented results will motivate further research on inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems.

Keywords

• total-order relation
• \({\mathcal{CR}}\)-preinvexity
• center-radius (\({\mathcal{CR}}\))-order
• interval-valued functions
• Hermite–Hadamard inequality
• Fejér inequality