Hermite-Hadamard-Type Fractional Inclusions for Interval-Valued Preinvex Functions

We introduce a new class of interval-valued preinvex functions termed as harmonically h-preinvex interval-valued functions. We establish new inclusion of Hermite–Hadamard for harmonically h-preinvex interval-valued function via interval-valued Riemann–Liouville fractional integrals. Further, we prove fractional Hermite–Hadamard-type inclusions for the product of two harmonically h-preinvex interval-valued functions. In this way, these findings include several well-known results and newly obtained results of the existing literature as special cases. Moreover, applications of the main results are demonstrated by presenting some examples.

A Comprehensive Analysis of Hermite–Hadamard Type Inequalities via Generalized Preinvex Functions

The principal objective of this article is to introduce the idea of a new class of n-polynomial convex functions which we call n-polynomial s-type m-preinvex function. We establish a new variant of the well-known Hermite–Hadamard inequality in the mode of the newly introduced concept. To add more insight into the newly introduced concept, we have discussed some algebraic properties and examples as well. Besides, we discuss a few new exceptional cases for the derived results, which make us realize that the results of this paper are the speculations and expansions of some recently known outcomes. The immeasurable concepts and chasmic tools of this paper may invigorate and revitalize additional research in this mesmerizing and absorbing field.

Estimations of Upper Bounds for n -th Order Differentiable Functions Involving χ -Riemann–Liouville Integrals via γ -Preinvex Functions

A new fractional integral identity is obtained involving n -th order differentiable functions and χ -Riemann–Liouville fractional integrals. Then, some associated estimates of upper bounds involving γ -preinvex functions are obtained. In order to relate some unrelated results, several special cases are discussed.

New Integral Inequalities via Generalized Preinvex Functions

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function and to present inequalities of the Hermite–Hadamard type in the setting of the generalized s-type m–preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite–Hadamard-type inequality using Hölder’s integral inequality, the improved power-mean integral inequality, and the Hölder-İşcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory.

Some New Post-Quantum Integral Inequalities Involving Twice (p,q)-Differentiable ψ-Preinvex Functions and Applications

The main motivation of this article is derive a new post-quantum integral identity using twice (p,q)-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard’s inequality essentially via the class of ψ-preinvex functions. To support our results, we offer some applications to a special means of positive real numbers and twice (p,q)-differentiable functions that are in absolute value bounded as well.

Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

Equilibrium problems and variational inequalities are connected to the symmetry concepts, which play important roles in many fields of sciences. Some new preinvex functions, which are called generalized preinvex functions, with the bifunction ζ(.,.) and an arbitrary function k, are introduced and studied. Under the normed spaces, new parallelograms laws are taken as an application of the generalized preinvex functions. The equilibrium-like problems are represented as the minimum values of generalized preinvex functions under the kζ-invex sets. Some new inertial methods are proposed and researched to solve the higher order directional equilibrium-like problem, Convergence criteria of the our methods is discussed, along with some unresolved issues.

On Some New Inequalities of Hermite–Hadamard Midpoint and Trapezoid Type for Preinvex Functions in (p, q)-Calculus

In this paper, we establish some new Hermite–Hadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for p,q-differentiable preinvex functions in the context of p,q-calculus. We also show that the results established in this paper are generalizations of comparable results in the literature of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.