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Different types of functions have been used in many various areas of mathematics until today, and today they are used to obtain new equality and inequalities in fractional analysis, which is one of the important areas of mathematics. In this study, we obtained integral inequalities for preinvex functions with the help of Atangana Baleanu fractional integral operator. [ABSTRACT FROM AUTHOR]

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher's interest even more. [ABSTRACT FROM AUTHOR]

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m , t g s) -type s-preinvex function, Godunova–Levin (s , m) -preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings. [ABSTRACT FROM AUTHOR]

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It's undeniably true that fractional calculus has been the focus point for numerous researchers in recent couple of years. The writing of the Caputo-Fabrizio fractional operator has been on many demonstrating and real-life issues. The main objective of our article is to improve integral inequalities of Hermite-Hadamard and Pachpatte type incorporating the concept of preinvexity with the Caputo-Fabrizio fractional integral operator. To further enhance the recently presented notion, we establish a new fractional equality for differentiable preinvex functions. Then employing this as an auxiliary result, some refinements of the Hermite-Hadamard type inequality are presented. Also, some applications to special means of our main findings are presented.

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher’s interest even more.

Fuzzy variational problems have received significant attention over the past decade due to a number of successful applications in fields such as optimal control theory and image segmentation. Current literature on fuzzy variational problems focuses on the necessary optimality conditions for finding the extrema, which have been studied under several differentiability conditions. In this study, we establish the sufficient conditions for the existence of minimizers for fuzzy variational problems under a weaker notion of convexity, namely preinvexity and Buckley–Feuring differentiability. We further discuss their application in a cost minimization problem. [ABSTRACT FROM AUTHOR]

Fractional derivative and integral operators are often employed to present new generalizations of mathematical inequalities. The introduction of new fractional operators has prompted another direction in different branches of mathematics and applied sciences. First, we investigate and prove new fractional equality. Considering this equality as the auxiliary result, we attain some estimations of a Hermite–Hadamard type inequality involving s-preinvex, s-Godunova–Levin preinvex, and prequasi invex functions. In addition, we investigate a fractional order Hadamard–Fejér inequality and some of its refinements pertaining to h-preinvexity via a non-conformable fractional integral operator. Finally, we present a Pachpatte type inequality for the product of two preinvex functions. The findings as well as the special cases presented in this research are new and applications of our main results. [ABSTRACT FROM AUTHOR]

As a generalization of convex functions and derivatives, in this paper, the authors study the concept of a symmetric derivative for preinvex functions. Using symmetrical differentiation, they discuss an important characterization for preinvex functions and define symmetrically pseudo-invex and symmetrically quasi-invex functions. They also generalize the first derivative theorem for symmetrically differentiable functions and establish some relationships between symmetrically pseudo-invex and symmetrically quasi-invex functions. They also discuss the Fritz John type optimality conditions for preinvex, symmetrically pseudo-invex and symmetrically quasi-invex functions using symmetrical differentiability. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce ( h 1 , h 2 ) $(h_{1},h_{2})$ -preinvex interval-valued function and establish the Hermite–Hadamard inequality for preinvex interval-valued functions by using interval-valued Riemann–Liouville fractional integrals. We obtain Hermite–Hadamard-type inequalities for the product of two interval-valued functions. Further, some examples are given to confirm our theoretical results.

Abstract In this research, we derive two generalized integral identities involving the q ϰ 2 $q^{\varkappa _{2}}$ -quantum integrals and quantum numbers, the results are then used to establish some new quantum boundaries for quantum Simpson’s and quantum Newton’s inequalities for q-differentiable preinvex functions. Moreover, we obtain some new and known Simpson’s and Newton’s type inequalities by considering the limit q → 1 − $q\rightarrow 1^{-}$ in the key results of this paper.

In this paper, we introduce a new class of generalized convex functions called (h, r)-preinvex functions, and establish some new Hermite-Hadamard type inequalities under (h, r)-preinvexity. [ABSTRACT FROM AUTHOR]

In this study, we establish a new weighted integral identity and used it to establish some new weighted Ostrowski‐type inequalities for functions whose modulus of the first derivatives are preinvex, which the obtained results represent generalizations as well as refinements for some known results. Finally, we explain our results by giving some applications dealing with the random variables and the special means. [ABSTRACT FROM AUTHOR]

The generalized fractional integral has been one of the most useful operators for modelling non-local behaviors by fractional differential equations. It is considered, for several integral inequalities by introducing the concept of exponentially (s, m)-preinvexity. These variants derived via an extended Mittag-Leffler function based on boundedness, continuity and Hermite-Hadamard type inequalities. The consequences associated with fractional integral operators are more general and also present the results for convexity theory. Moreover, we point out that the variants are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally [ABSTRACT FROM AUTHOR]

In this paper, we define and introduce some new concepts of the higher order strongly preinvex functions and higher order strongly monotone operators involving an arbitrary bifunction. Some new relationships among various concepts of higher order strongly preinvex functions have been established. We have shown that the optimality conditions for the preinvex functions can be characterized by class of higher order strongly variational-like inequalities, which appears to be new ones. As a novel applications of the higher order strongly preinvex functions, we have obtained the parallelogram-like laws for the uniformly Banach spaces. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results. [ABSTRACT FROM AUTHOR]

In this paper, some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions are established. Also, new inequalities involving multiplicative integrals are obtained for product and quotient of preinvex and multiplicatively preinvex functions.

In this paper, some new classes of the strongly exponentially generalized preinvex functions involving an auxiliary non-negative function and a bifunction are introduced. New relationships among various concepts of strongly exponentially generalized preinvex functions are established. It is shown that the optimality conditions of differentiable strongly exponentially generalized preinvex functions can be characterized by exponentially variational-like inequalities. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

In this paper, we consider some new classes of log-preinvex functions. Several properties of the log-preinvex functions are studied. We also discuss their relations with convex functions. Several interesting results characterizing the log-convex functions are obtained. Optimality conditions of differentiable strongly log-preinvex are characterized by a class of variational-like inequalities. Results obtained in this paper can be viewed as significant improvement of previously known results. [ABSTRACT FROM AUTHOR]

In this paper, we define and introduce some new concepts of the relative strongly preinvex functions and relative strongly monotone operators with respect to the auxiliary non-negative function and bifunction. We establish some new relationships among various concepts of relative strongly preinvex functions. As special cases, one can obtain various new and known consequences of our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results. [ABSTRACT FROM AUTHOR]

In this paper we present the analogue of Hermite-Hadamard inequality for preinvex functions via Caputo fractional derivatives. Moreover, we establish a new integral identity and derive some new trapezium type inequalities for preinvex functions via Caputo fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we introduce (h 1 , h 2) -preinvex interval-valued function and establish the Hermite–Hadamard inequality for preinvex interval-valued functions by using interval-valued Riemann–Liouville fractional integrals. We obtain Hermite–Hadamard-type inequalities for the product of two interval-valued functions. Further, some examples are given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this research, we derive two generalized integral identities involving the q ϰ 2 -quantum integrals and quantum numbers, the results are then used to establish some new quantum boundaries for quantum Simpson's and quantum Newton's inequalities for q-differentiable preinvex functions. Moreover, we obtain some new and known Simpson's and Newton's type inequalities by considering the limit q → 1 − in the key results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Hermite-Hadamard's inequality for higher order differentiable preinvex functions using Riemann-Liouville integrals. First, we give a new generalized deffnition of preinvex function and then extend the inequalities for n-times differentiable (s, m)-preinvex functions. Some previous results as special cases and applications to special means are also given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive a new quantum analogue of an integral identity by using (p; q)-calculus. As a consequence of this identity, some new estimates for Ostrowski type inequality for (p, q)- differentiable ŋ-convex and ŋ-quasi-convex functions are obtained. Moreover, some new estimates for Hermite-Hadamard type inequality for (p; q)- differentiable ŋ-convex and ŋ-quas-iconvex functions are given as well. [ABSTRACT FROM AUTHOR]

The main objective of this article is to establish integral identity relating the left side of Hermite-Hadamard type inequality. By using this identity, we establish some new Hermite-Hadamard type integral inequalities for functions whose mixed partial derivatives are co-ordinated preinvex. These consequences generalize numerous outcomes established in previous studies for these classes of functions. [ABSTRACT FROM AUTHOR]

This paper aims to establish k -fractional integral inequalities with multiple parameters involving the first differentiable mappings. Applications of our results to random variables are also given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of harmonic preinvex functions, which are called harmonic h-preinvex functions. Several new Ostrowski-type inequal- ities for harmonic h-preinvex functions via Riemann-Liouville fractional integrals are established. Some special cases are also discussed, which appears to be new ones. The results obtained in this paper continue to hold for these cases. Interested readers are encouraged to find the applications of the harmonic h-preinvex functions in pure and applied sciences. This is an interesting topic for future research. [ABSTRACT FROM AUTHOR]

An inequality of Ostrowski's type for preinvex functions is introduced. Applications to some special means are considered. [ABSTRACT FROM AUTHOR]

Abstract: In this work continuous-time programming problems of vector optimization are considered. Firstly, a nonconvex generalized Gordan’s transposition theorem is obtained. Then, the relationship with the associated weighting scalar problem is studied and saddle point optimality results are established. A scalar dual problem is introduced and duality theorems are given. No differentiability assumption is imposed. [Copyright &y& Elsevier]

In Refs. [J. Math. Anal. Appl. 258:287-308, 2001; J. Math. Anal. Appl. 256:229-241, 2001], Yang and Li presented a characterization of preinvex functions and semistrictly preinvex functions under a certain set of conditions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions; we also give a characterization of strictly preinvex functions under mild conditions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new class of generalized convex function is introduced, which is called the strongly α-preinvex function. We study some properties of strongly α-preinvex function. In particular, we establish the equivalence among the strongly α-preinvex functions, strongly α-invex functions and strongly αη-monotonicity under some suitable conditions. As special cases, one can obtain several new and previously known results for α-preinvex (invex) functions. [Copyright &y& Elsevier]

Some new fractional Ostrowski's inequalities for functions whose first derivatives are nonconvex are established. [ABSTRACT FROM AUTHOR]