Fractional Hermite-Hadamard Inequalities Research Articles

Some New Improvements for Fractional Hermite–Hadamard Inequalities by Jensen–Mercer Inequalities

This article’s objective is to introduce a new double inequality based on the Jensen–Mercer (JM) inequality, known as the Hermite–Hadamard–Mercer inequality. We use the (JM) inequality to build a number of generalized trapezoid‐type inequalities. Moreover, in addition to the (JM) inequality, we also use the Hölder inequality and the power mean inequality. Finally, a few examples are given to highlight the main points of our outcomes.

I.V-CR-γ-Convex Functions and Their Application in Fractional Hermite–Hadamard Inequalities

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR-γ-convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen’s inequalities using CR-γ-convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard’s, Hermite-Hadamard-Fejer, and Pachpatte’s type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations.

Some New Hermite-Hadamard Type Inequalities Pertaining to Fractional Integrals with an Exponential Kernel for Subadditive Functions

The class of symmetric function interacts extensively with other types of functions. One of these is the class of convex functions, which is closely related to the theory of symmetry. In this paper, we obtain some new fractional Hermite–Hadamard inequalities with an exponential kernel for subadditive functions and for their product, and some known results are recaptured. Moreover, using a new identity as an auxiliary result, we deduce several inequalities for subadditive functions pertaining to the new fractional integrals involving an exponential kernel. To validate the accuracy of our results, we offer some examples for suitable choices of subadditive functions and their graphical representations.

On the novel Hermite-Hadamard inequalities for composite inverse functions

The goal of this research is to discover some identities in the general form of the sum of left and right-sided weighted fractional integrals of a function concerning to another function. Using composite convex functions, several fractional Hermite-Hadamard inequalities are derived. The veracity of the inequalities established is demonstrated by drawing graphs of such relationships. Furthermore, our findings generalize a number of previously published outcomes. These findings will aid in the study of fractional differential equations and fractional boundary value problems with unique solutions.

Certain fractional inequalities via the Caputo Fabrizio operator

The Caputo Fabrizio fractional integral operator is one of the key concepts in fractional calculus. It is involved in many concrete and practical issues. In the present study, we have discussed some novel ideas to fractional Hermite-Hadamard inequalities within a Caputo Fabrizio fractional integral framework. The fractional integral under investigation is used to establish some new fractional Hermite-Hadamard inequalities. The findings of this study can be seen as a generalization and extension of numerous earlier inequalities via convex function. In addition, we demonstrate a few applications of our findings to special means of real numbers.

Some General Fractional Integral Inequalities Involving LR–Bi-Convex Fuzzy Interval-Valued Functions

The main objective of this paper is to introduce a new class of convexity called left-right–bi-convex fuzzy interval-valued functions. We study this class from the perspective of fractional Hermite–Hadamard inequalities, involving a new fractional integral called the left-right–AB fractional integral. We discuss several special cases that demonstrate that our results are quite unifying. We provide non-trivial numerical examples regarding special means for positive real numbers in order to check the validity of our outcomes.

Fractional Hermite-Hadamard type inequalities for subadditive functions

In this paper, we establish different variants of fractional Hermite-Hadamard inequalities for subadditive functions via Riemann-Liouville fractional integrals. Moreover, we offer some fractional integral inequalities for the product of two subadditive functions via Riemann-Liouville fractional integrals. It is also shown that the inequalities offered in this research are the generalization of the already given inequalities for convex functions and subadditive functions.

Generalized fractional Hermite-Hadamard inequalities

Generalized fractional Hermite-Hadamard inequalities

On the Hermite\u2013Hadamard type inequality for \u03c8-Riemann\u2013Liouville fractional integrals via convex functions

In this paper, we establish a new Hermite–Hadamard inequality involving left-sided and right-sided ψ-Riemann–Liouville fractional integrals via convex functions. We also show two basic ψ-Riemann–Liouville fractional integral identities including the first order derivative of a given convex function, and these will be used to derive estimates for some fractional Hermite–Hadamard inequalities. Finally, we give some applications to special means of real numbers.

Fractional Hermite-Hadamard inequalities containing generalized Mittag-Leffler function.

The objective of this paper is to establish some new refinements of fractional Hermite-Hadamard inequalities via a harmonically convex function with a kernel containing the generalized Mittag-Leffler function.

New Riemann-Liouville fractional Hermite-Hadamard inequalities via two kinds of convex functions

In this paper, we establish another important integral identity for once differentiable function involving Riemann-Liouville fractional integrals, which will be used to derive some new Riemann-Liouville fractional Hermite-Hadamard inequalities via r-convex function and geometric-arithmetically s-convex function respectively.

On some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions and their applications

In this paper, we establish two fractional integral equalities involving once and twice differential functions. Then, we apply such equalities to give some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions. Some applications to special means of positive real numbers are also given.

Fractional Hermite-Hadamard inequalities for differentiable s-Godunova-Levin functions

In this paper, we consider the class of s-Godunova-Levin functions. We derive a new fractional integral identity for differentiable function. Using this new identity, we establish some new fractional Hermite-Hadamard type inequalities for the class of differentiable s-Godunova-Levin functions.

Generalized fractional Hermite-Hadamard inequalities for twice differentiable s-convex functions

Some Hermite-Hadamard type inequalities via Riemann-Liouville fractional integral for twice differentiable functions having some s-convexity of second kind properties are established. A class of s-affine of second kind functions is identified such as these inequalities are sharp.

On the fractional Hermite-Hadamard type inequalities for (α,m)-logarithmically convex functions

The purpose of this paper is to establish some refinements of Riemann-Liouville fractional Hermite-Hadamard inequalities for (?,m)-logarithmically convex functions. By using our fractional integrals identities, we present some interesting and new left type fractional Hermite-Hadamard inequalities for once and twice differentiable (?,m)-logarithmically convex functions via powerful series.

Fractional Hermite-Hadamard Inequalities for some New Classes of Godunova-Levin Functions

In this paper, some new classes of Godunova-Levin functions are introduced and investigated. Several new fractional Hermite-Hadamard inequalities are derived for these new classes of Godunova-Levin functions. The ideas and techniques of this paper may stimulate and inspire further research.

Riemann-Liouville fractional Hermite-Hadamard inequalities. Part II: for twice differentiable geometric-arithmetically s-convex functions

Abstract Motivated by the definition of geometric-arithmetically s-convex functions in (Shuang et al. in Analysis 33:197-208, 2013) and second-order fractional integral identities in (Zhang and Wang in J. Inequal. Appl. 2013:220, 2013; Wang et al. in Appl. Anal. 2012, doi:10.1080/00036811.2012.727986), we establish some interesting Riemann-Liouville fractional Hermite-Hadamard inequalities for twice differentiable geometric-arithmetically s-convex functions via beta function and incomplete beta function. MSC:26A33, 26A51, 26D15.

Riemann-Liouville fractional Hermite-Hadamard inequalities. Part I: for once differentiable geometric-arithmetically s-convex functions

Abstract By using the definition of geometric-arithmetically s-convex functions in (Analysis 33:197-208, 2013) and first-order fractional integral identities in (Math. Comput. Model. 57:2403-2407, 2013; J. Appl. Math. Stat. Inform. 8:21-28, 2012; Comput. Math. Appl. 63:1147-1154, 2012), we present some interesting Riemann-Liouville fractional Hermite-Hadamard inequalities for differentiable geometric-arithmetically s-convex functions. Both beta function and incomplete beta function are used in the desired estimations. MSC:26A33, 26A51, 26D15.

Fractional Hermite-Hadamard inequalities for ( α , m ) -logarithmically convex functions

By means of two fundamental fractional integral identities, we derive two classes of new Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals for once and twice differentiable (α,m)-logarithmically convex functions, respectively. The main novelty of this paper is that we use powerful series to describe our estimations.MSC:26A33, 26A51, 26D15.