Hadamard Type Inequalities Research Articles

Up and Down h-Pre-Invex Fuzzy-Number Valued Mappings and Some Certain Fuzzy Integral Inequalities

The objective of the current paper is to incorporate the new class and concepts of convexity and Hermite–Hadamard inequality with the fuzzy Riemann integral operators because almost all classical single-valued and interval-valued convex functions are special cases of fuzzy-number valued convex mappings. Therefore, a new class of nonconvex mapping in the fuzzy environment has been defined; up and down h-pre-invex fuzzy-number valued mappings (U.D h-pre-invex F-N∙V∙Ms). With the help of this newly defined class, some new versions of Hermite–Hadamard (HH) type inequalities have been also presented. Moreover, some related inequalities such as HH Fejér- and Pachpatte-type inequalities for U∙D h-pre-invex F-N∙V∙Ms are also introduced. Some exceptional cases have been discussed, which can be seen as applications of the main results. We have provided some nontrivial examples. Finally, we also discuss some future scopes.

Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h1,h2)-Convex Functions Pertaining to Total Order Relation

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯+ω̲2,ω¯−ω̲2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples.

On q-Hermite-Hadamard Inequalities via q − h-Integrals

This paper aims to find Hermite–Hadamard-type inequalities for a generalized notion of integrals called q−h-integrals. Inequalities for q-integrals can be deduced by taking h=0 and are connected with several known results of q-Hermite–Hadamard inequalities. In addition, we analyzed q−h-integrals, q-integrals, and the corresponding inequalities for symmetric functions.

Some Hermite–Hadamard type inequalities for GA‐s‐convex functions in the fourth sense

In the present manuscript, we define the concept of ‐ ‐convex functions in the fourth sense. We obtain the Hermite–Hadamard inequality for the newly introduced class of functions. We also establish some inequalities of the Hermite–Hadamard type for functions whose first derivative in absolute value at certain powers is a ‐ ‐convex function in the fourth sense. In the last, some applications to special means of real numbers are given.

Hermite–Hadamard type inequality for (E, F)-convex functions and geodesic (E, F)-convex functions

The main aim of the present paper is to introduce geodesic (E, F)-convex sets and geodesic (E, F)-functions on a Riemannian manifold. Furthermore, some basic properties of these mappings are investigated. Moreover, the Hadamard-type inequalities for (E, F)-convex functions are proven.

Some New Integral Inequalities Involving Fractional Operator with Applications to Probability Density Functions and Special Means

Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the attention of several mathematicians because of its real-life applications. More importantly, it has turned into a valuable tool for handling elements from perplexing frameworks within different parts of the pure and applied sciences. Integral inequalities, in association with convexity, have a strong relationship with symmetry. The objective of this article is to introduce the notion of operator refined exponential type convexity. Fractional versions of the Hermite–Hadamard type inequality employing generalized R–L fractional operators are established. Additionally, some novel refinements of Hermite–Hadamard type inequalities are also discussed using some established identities. Finally, we present some applications of the probability density function and special means of real numbers.

On Symmetrized Stochastic Harmonically Convexity and Hermite–Hadamard Type Inequalities

Throughout this study, the concept of symmetrized harmonically convex stochastic processes will be discussed in further detail. Some certain characterizations for symmetrized harmonically convex stochastic processes are discussed that use Hermite–Hadamard-type inequalities. A Hyers–Ulam-type stability result for harmonically convex stochastic processes is given as well.

New Hermite–Hadamard Type Inequalities in Connection with Interval-Valued Generalized Harmonically (h1,h2)-Godunova–Levin Functions

As is known, integral inequalities related to convexity have a close relationship with symmetry. In this paper, we introduce a new notion of interval-valued harmonically m,h1,h2-Godunova–Levin functions, and we establish some new Hermite–Hadamard inequalities. Moreover, we show how this new notion of interval-valued convexity has a close relationship with many existing definitions in the literature. As a result, our theory generalizes many published results. Several interesting examples are provided to illustrate our results.

Some Hermite–Hadamard and Hermite–Hadamard–Fejér Type Fractional Inclusions Pertaining to Different Kinds of Generalized Preinvexities

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.

Quantum Integral Inequalities in the Setting of Majorization Theory and Applications

In recent years, the theory of convex mappings has gained much more attention due to its massive utility in different fields of mathematics. It has been characterized by different approaches. In 1929, G. H. Hardy, J. E. Littlewood, and G. Polya established another characterization of convex mappings involving an ordering relationship defined over Rn known as majorization theory. Using this theory many inequalities have been obtained in the literature. In this paper, we study Hermite–Hadamard type inequalities using the Jensen–Mercer inequality in the frame of q˙-calculus and majorized l-tuples. Firstly we derive q˙-Hermite–Hadamard–Jensen–Mercer (H.H.J.M) type inequalities with the help of Mercer’s inequality and its weighted form. To obtain some new generalized (H.H.J.M)-type inequalities, we prove a generalized quantum identity for q˙-differentiable mappings. Next, we obtain some estimation-type results; for this purpose, we consider q˙-identity, fundamental inequalities and the convexity property of mappings. Later on, We offer some applications to special means that demonstrate the importance of our main results. With the help of numerical examples, we also check the validity of our main outcomes. Along with this, we present some graphical analyses of our main results so that readers may easily grasp the results of this paper.

Generalized Preinvex Interval-Valued Functions and Related Hermite–Hadamard Type Inequalities

This study aims to connect the idea of inequalities with Riemann integral operators, which are of interest because of their characteristics and widespread use. We create a new Hermite–Hadamard type integral inequality for an

Hermite–Hadamard-Type Inequalities Involving Harmonically Convex Function via the Atangana–Baleanu Fractional Integral Operator

Fractional integrals and inequalities have recently become quite popular and have been the prime consideration for many studies. The results of many different types of inequalities have been studied by launching innovative analytical techniques and applications. These Hermite–Hadamard inequalities are discovered in this study using Atangana–Baleanu integral operators, which provide both practical and powerful results. In this paper, a symmetric study of integral inequalities of Hermite–Hadamard type is provided based on an identity proved for Atangana–Baleanu integral operators and using functions whose absolute value of the second derivative is harmonic convex. The proven Hermite–Hadamard-type inequalities have been observed to be valid for a choice of any harmonic convex function with the help of examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains.

New Hadamard-type inequality for new class of geodesic convex functions

This paper aims to introduce the concept of (\( E,\mu,\kappa \))-convex function by using special inequality. Hadamard integral inequality for this new class of geodesic convex function in the case of Lebesgue and Sugeno integrals is given.

Hermite–Hadamard and Pachpatte Type Inequalities for Coordinated Preinvex Fuzzy-Interval-Valued Functions Pertaining to a Fuzzy-Interval Double Integral Operator

Many authors have recently examined the relationship between symmetry and generalized convexity. Generalized convexity and symmetry have become a new area of study in the field of inequalities as a result of this close relationship. In this article, we introduce the idea of preinvex fuzzy-interval-valued functions (preinvex F∙I-V∙F) on coordinates in a rectangle drawn on a plane and show that these functions have Hermite–Hadamard-type inclusions. We also develop Hermite–Hadamard-type inclusions for the combination of two coordinated preinvex functions with interval values. The weighted Hermite–Hadamard-type inclusions for products of coordinated convex interval-valued functions discussed in a recent publication by Khan et al. in 2022 served as the inspiration for our conclusions. Our proven results expand and generalize several previous findings made in the body of literature. Additionally, we offer appropriate examples to corroborate our theoretical main findings.

Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings

For left and right λ-preinvex interval-valued functions (left and right λ-preinvex IVFs) in interval-valued Riemann operator settings, we create Hermite–Hadamard (H-H) type inequalities in the current study. Additionally, we create Hermite–Hadamard–Fejér (H-H-Fejér)-type inequalities for preinvex functions of the left and right interval-valued type under some mild conditions. Moreover, some exceptional new and classical cases are also obtained. Some useful examples are also presented to prove the validity of the results.

Some Inequalities of Hermite–Hadamard Type for MT-h-Convex Functions via Classical and Generalized Fractional Integrals

Convexity plays a vital role in pure and applied mathematics specially in optimization theory, but the classical convexity is not enough to fulfil the needs of modern mathematics; hence, it is important to study generalized notion of convexity. Fraction integral operators also become an important tool for solving problems of model physical and engineering processes that are found to be best described by fractional differential equations. The aim of this paper is to study MT-h-convex functions via fractional integral operators. We establish several Hermite–Hadamard-type inequalities for MT-h-convex function via classical and generalized fractional integrals. We also obtain special means related to our results and present some error estimates for the trapezoidal formulas.

Hermite–Hadamard Type Inequalities Involving (k-p) Fractional Operator for Various Types of Convex Functions

We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as (p,h) fractional type inequality and many others, as the (p,h)-convexity is the generalization of the other convex inequalities. As a consequence of the (h,m)-convexity, the fractional inequality of the (s,m)-type is obtained. Many consequences of such fractional inequalities and generalizations are obtained.

NEW ψ - & (k, r)- FRACTIONAL CONFORMABLE INTEGRALS AND INEQUALITIES OF THE HERMITE–HADAMARD TYPE FOR POSITIVE–CONVEX STOCHASTIC PROCESSES

Huang et al in the paper [Some inequalities of the Hermite–Hadamard Type for k-fractional conformable integrals, The Australian Journal of Mathematical Analysis and Applications, 16 (2019), no. 7, pp. 1-9] proved some new Hermite–Hadamard type inequalities for k-fractional conformable integrals for convex functions. In this paper, we extend and generalize the main result of the above-mentioned paper for (k, r)-fractional conformable integrals for positive–convex stochastic process and also point out a mistake (omission) in ([6], Theorem 3.1). In addition, we prove a new Hermite–Hadamard type inequality for ψ-fractional conformable integrals for positive–convex stochastic process.

The Properties of Harmonically cr-h-Convex Function and Its Applications

In this paper, the definition of the harmonically cr-h-convex function is given, and its important properties are discussed. Jensen type inequality, Hermite–Hadamard type inequalities and Fejér type inequalities for harmonically cr-h-convex functions are also established. In addition, some numerical examples are given to verify the accuracy of the results.

Some Generalized Formulas of Hadamard-Type Fractional Integral Inequalities

This paper is aimed at establishing the generalized forms of Riemann-Liouville fractional inequalities of the Hadamard type for a class of functions known as strongly exponentially α , h − m - p -convex functions. These inequalities provide some general formulas from which one can get associated inequalities for various types of exponentially convex and strongly convex functions. Refinements of well-known inequalities are also deducible from the established theorems.