Hermite-Hadamard Integral Inequalities Research Articles

Hermite-Hadamart Integral Inequality for Differentiable m- Convex Functions

Convexity in connection with integral inequalities is an interesting research domain in recent years. The convexity theory plays a fundamental role in the development of various branches of applied sciences since it includes the theory of convex functions that possesses two important attributes viz. a boundary point is where the maximum value is reached and any local minimum value is a global one. Convexities and inequalities are connected which has a basic character in many branches of pure and applied disciplines. The most important inequality related to convex function is the Hermite-Hadamard integral inequality. The extensions, enhancements and generalizations of this inequality has motivated the researchers in recent years. This paper is an extension of some inequalities connected with difference of the left-hand part as well as the right-hand part from the integral mean in Hermite- Hadamard’s inequality for the case of m- convex functions.

Weighted Fractional Hermite–Hadamard Integral Inequalities for up and down Ԓ-Convex Fuzzy Mappings over Coordinates

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article’s main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as UD-Ԓ-convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down (UD) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for Ԓ, we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard (HH) and Pachpatte types of inequalities using the concepts of coordinated UD-Ԓ-convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples.

Hermite–Hadamard-type inequalities via different convexities with applications

In this paper, we explore a class of Hermite–Hadamard integral inequalities for convex and m-convex functions. The Hölder inequality is used to create this class, which has a wide range of applications in optimization theory. Some trapezoid-type inequalities and midpoint error estimates are investigated. Inequalities for several q-special functions are highlighted. As particular cases, we have included several previous results.

Weighted Hermite-Hadamard integral inequalities for general convex functions.

In this article, starting with an equation for weighted integrals, we obtained several extensions of the well-known Hermite-Hadamard inequality. We used generalized weighted integral operators, which contain the Riemann-Liouville and the $ k $-Riemann-Liouville fractional integral operators. The functions for which the operators were considered satisfy various conditions such as the $ h $-convexity, modified $ h $-convexity and $ s $-convexity.

Some novel refinements of Hermite-Hadamard and Pachpatte type integral inequalities involving a generalized preinvex function pertaining to Caputo-Fabrizio fractional integral operator

<abstract><p>In this article, we aim to introduce and explore a new class of preinvex functions called $ \mathfrak{n} $-polynomial $ m $-preinvex functions, while also presenting algebraic properties to enhance their numerical significance. We investigate novel variations of Pachpatte and Hermite-Hadamard integral inequalities pertaining to the concept of preinvex functions within the framework of the Caputo-Fabrizio fractional integral operator. By utilizing this direction, we establish a novel fractional integral identity that relates to preinvex functions for differentiable mappings of first-order. Furthermore, we derive some novel refinements for Hermite-Hadamard type inequalities for functions whose first-order derivatives are polynomial preinvex in the Caputo-Fabrizio fractional sense. To demonstrate the practical utility of our findings, we present several inequalities using specific real number means. Overall, our investigation sheds light on convex analysis within the context of fractional calculus.</p></abstract>

Q-Analogue of Hermite-Hadamard Type Inequalities for s-Convex Functions in the Breckner Sense

Hermite and Hadamard independently introduced the Herimite-Hadamard inequality for convex functions for the first time. In recent years, a variety of extensions have been made with the use of convex functions by several researchers. In this paper, we have given a variant of the Hermite-Hadamard integral inequality for the s-convex function in the Breckner sense.

New Hermite–Hadamard Integral Inequalities for Geometrically Convex Functions via Generalized Weighted Fractional Operator

The main purpose of this research is to concentrate on the development of new definitions for the weighted geometric fractional integrals of the left-hand side and right-hand side of the function ℵ with regard to an increasing function used as an integral kernel. Moreover, the newly developed class of left-hand side and right-hand side weighted geometric fractional integrals of a function ℵ, by applying an additional increasing function, identifies a variety of novel classes as special cases. This is a development of the previously established fractional integrals by making use of the class of geometrically convex functions. Geometrically convex functions in weighted fractional integrals of a function ℵ in the form of another rising function yield the Hermite–Hadamard inequality type. We also establish a novel midpoint identity and the associated inequalities for a class of weighted fractional integral functions known as geometrically convex with respect to an increasing function and symmetric with respect to the geometric mean of the endpoints of the interval. In order to demonstrate the validity of our research, we present examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains.

GENERALIZED MAPPINGS RELATED TO HERMITE-HADAMARD INEQUALITY

In this paper we introduce two new mapping in connection to Hermite-Hadamard type inequality. Some results concerning these mappings associated to the celebrated Hermite-Hadamard integral inequality for preinvex functions are given.  

Some New Versions of Hermite–Hadamard Integral Inequalities in Fuzzy Fractional Calculus for Generalized Pre-Invex Functions via Fuzzy-Interval-Valued Settings

The purpose of this study is to prove the existence of fractional integral inclusions that are connected to the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for χ-pre-invex fuzzy-interval-valued functions. Some of the related fractional integral inequalities are also proved via Riemann–Liouville fractional integral operator, where integrands are fuzzy-interval-valued functions. To prove the validity of our main results, some of the nontrivial examples are also provided. As specific situations, our findings can provide a variety of new and well-known outcomes which can be viewed as applications of our main results. The results in this paper can be seen as refinements and improvements to previously published findings.

Hermite–Hadamard Type Inequalities Involving k-Fractional Operator for (h¯,m)-Convex Functions

The principal motivation of this paper is to establish a new integral equality related to k-Riemann Liouville fractional operator. Employing this equality, we present several new inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard integral inequality. Additionally, some novel cases of the established results for different kinds of convex functions are derived. This fractional integral sums up Riemann–Liouville and Hermite–Hadamard’s inequality, which have a symmetric property. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. Finally, applications of q-digamma and q-polygamma special functions are presented.

New integral inequalities involving m—convex functions in (p,q)—calculus

In this paper, we establish some new Hermite-Hadamard integral inequalities for m-convex functions via (p,q)-calculus. Also, some known results are recaptured as special cases from our results. At the end, a briefly conclusion is given.

Inequalities on time scales via diamond-Fh integral

In this article, new improvements and generalizations of Hermite-Hadamard integral inequalities for Fh-convex functions on time scales are established. In addition, examples and applications are given to further support the results.

Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the σ˘-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a σ˘-convex function with respect to an increasing function involving the φ-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.

Simpson’s Rule and Hermite–Hadamard Inequality for Non-Convex Functions

In this article we give a variant of the Hermite–Hadamard integral inequality for twice differentiable functions. It represents an improvement of this inequality in the case of convex/concave functions. Sharp two-sided inequalities for Simpson’s rule are also proven along with several extensions.

Some Improvements of the Hermite–Hadamard Integral Inequality

We propose several improvements of the Hermite–Hadamard inequality in the form of linear combination of its end-points and establish best possible constants. Improvements of a second order for the class Φ ( I ) with applications in Analysis and Theory of Means are also given.

The Sugeno fuzzy integral of concave functions

The fuzzy integrals are a kind of fuzzy measures acting on fuzzy sets. They can be viewed as an average membershipvalue of fuzzy sets. The value of the fuzzy integral in a decision making environment where uncertainty is presenthas been well established. Most of the integral inequalities studied in the fuzzy integration context normally considerconditions such as monotonicity or comonotonicity. In this paper, we are trying to extend the fuzzy integrals to theconcept of concavity. It is shown that the Hermite-Hadamard integral inequality for concave functions is not satisfied inthe case of fuzzy integrals. We propose upper and lower bounds on the fuzzy integral of concave functions. We presenta geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.

Refining Hermite-Hadamard integral inequality by two parameters

A new family of refinements of Hermite-Hadamard integral inequality is obtained by using two parameters, which can be used to refine the Heinz inequalities of matrices.

Integral Inequalities via Generalized Geometrically r-Convex Functions

In this paper, we introduce and investigate a new class of generalized convex functions, called generalized geometrically r-convex functions. Some new Hermite-Hadamard integral inequalities via generalized geometrically r-convex functions have been established. Results proved in this paper can be viewed as new significant contributions in this area of research.

Generalized r-Convex Functions and Integral Inequalities

In this paper, we introduce and investigate a new class of generalized convex functions, known as generalized $r$-convex function. Some new Hermite-Hadamard integral inequalities via generalized $r$-convex functions have been established. Results proved in this paper can be viewed as significant new contributions in this area of research.

MORE RESULTS ON HERMITE-HADAMARD TYPE INEQUALITY THROUGH (<i>α, m</i>)-PREINVEXITY

We establish various inequalities for n-times differentiable mappings that are connected with illustrious Hermite-Hadamard integral inequality for mapping whose absolute values of derivatives are (<i>α, m</i>)-preinvex function. The new integral inequalities are then applied to some special means.