Hermite-Hadamard Type Research Articles

Exploration of Hermite–Hadamard-Type Integral Inequalities for Twice Differentiable h-Convex Functions

The significance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set of such inequalities and analyze them using different mathematical identities. This particular class of inequalities is introduced by employing a generalized convexity concept. To validate our work, we create visual graphs and a table of values using specific functions to represent the inequalities. This approach allows us to demonstrate the validity of our findings and further solidify our conclusions. Moreover, we find that some previously published results emerge as special consequences of our main findings. This research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by using generalized fractional operators and expanding the concept of convexity.

On some integral inequalities for $(h,m)$-convex functions in a generalized framework

In this paper, we present some new integral inequalities of Hermite-Hadamard type. To obtain these results, general convex functions of various type are considered such as $(h,m)$-convex functions. The main results extend some previously known inequalities by taking fractional integral operators.

Several Quantum Hermite–Hadamard-Type Integral Inequalities for Convex Functions

The aim of this study was to present several improved quantum Hermite–Hadamard-type integral inequalities for convex functions using a parameter. Thus, a new quantum identity is proven to be used as the main tool in the proof of our results. Consequently, in some special cases several new quantum estimations for q-midpoints and q-trapezoidal-type inequalities are derived with an example. The results obtained could be applied in the optimization of several economic geology problems.

Interval valued Jensen’s inequalities for h-convex functions on time scales

In the paper, we study dynamic h-convexity for interval valued functions. Some generalizations of Jensen’s inequality in interval valued analysis for h-convex functions on time scales are proved in the paper. In seek of applications of generalized Jensen’s inequality, Hermite-Hadamard type inequalities for h-convex functions on time scales are established. Further discrete analogues of newly proved results are also presented in the paper. Some numerical examples are also provided to check the validity of the results.

On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) \left({\tilde{h}}_{1},{\tilde{h}}_{2}) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel

Abstract Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) \left({\tilde{h}}_{1},{\tilde{h}}_{2}) -preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable.

Hermite–Hadamard-Type Inequalities for Coordinated Convex Functions Using Fuzzy Integrals

In this paper, some estimates of third and fourth inequalities in Hermite–Hadamard-type inequalities for coordinated convex functions are proved using the non-additivity of the integrals and Fubini’s theorem for fuzzy integrals. That is, the results are obtained in the fuzzy context and using the Lebesgue measure. Several examples are provided on how to evaluate these estimates in order to illustrate the obtained results.

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.

On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities

Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. On the other hand, it is well known that a twice differentiable function is convex, if and only if it admits a nonnegative second-order derivative. In this paper, we obtain a characterization of a class of twice differentiable functions (including the class of convex functions) satisfying second-order differential inequalities. Some special cases are also discussed.

Some new fractional Hermite-Hadamard type inequalities for functions with co-ordinated extended [formula omitted]-prequasiinvex mixed partial derivatives

This article introduces extended (s,m)-prequasiinvex functions on coordinates, a new form of generalized convex function. Using a previously established identity, we derive new fractional Hermite-Hadamard type integral inequalities for functions whose mixed partial derivatives belong to this new class of functions. The obtained results generalize existing Hermite-Hadamard type inequalities and have numerous applications in mathematics and physics. To demonstrate the utility of our findings, we provide examples of applications to special means, such as arithmetic, harmonic and p-logarithmic means.

Some New Estimates of Fuzzy Integral Inequalities for Harmonically Convex Fuzzy-Number-Valued Mappings via up and down Fuzzy Relation

In this article, the up and down harmonically convex fuzzy-number-valued mapping which is a novel kind of harmonically convex fuzzy-number-valued mapping is introduced. In addition, it is highlighted that the new idea of up and down harmonically convex fuzzy-number-valued mapping (U−O−H convex F−N−V−M), which is a generalization of the previous class, describes a variety of new and classical classes as special cases by employing some mild restrictions. With the help of fuzzy inclusion relation, the new versions of the Hermite–Hadamard-type (HH-type) inequalities for up and down harmonically convex fuzzy-number-valued mappings are established. Then, we introduce a new version of Hermite–Hadamard Fejér-type inequality via fuzzy inclusion relation by using up and down harmonically convex fuzzy-number-valued mapping. Additionally, several instances are given to illustrate our main findings.

Some New Hermite–Hadamard Type Inequalities Pertaining to Generalized Multiplicative Fractional Integrals

There is significant interaction between the class of symmetric functions and other types of functions. The multiplicative convex function class, which is intimately related to the idea of symmetry, is one of them. In this paper, we obtain some new generalized multiplicative fractional Hermite–Hadamard type inequalities for multiplicative convex functions and for their product. Additionally, we derive a number of inequalities for multiplicative convex functions related to generalized multiplicative fractional integrals utilising a novel identity as an auxiliary result. We provide some examples for the appropriate selections of multiplicative convex functions and their graphical representations to verify the authenticity of our main results.

GENERALIZED HERMITE–HADAMARD INCLUSIONS FOR A GENERALIZED FRACTIONAL INTEGRAL

We introduce new generalized fractional integrals for interval-valued functions. Then we prove generalized Hermite–Hadamard type inclusions for interval-valued convex functions using these newly defined generalized fractional integrals. We also show that these results generalize several results obtained in earlier works.

FRACTIONAL HERMITE–HADAMARD INEQUALITY, SIMPSON’S AND OSTROWSKI’S TYPE INEQUALITIES FOR CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF FUNCTIONS

We consider the convexity with respect to a pair of functions and establish a Hermite–Hadamard type inequality for Riemann–Liouville fractional integrals. Moreover, we derive some new Simpson’s and Ostrowski’s type inequalities for differentiable convex mapping with respect to a pair of functions. We also show that the newly established inequalities are the extension of some existing inequalities. Finally, we consider some mathematical examples and graphs to show the validity of the newly established inequalities.

A Generalized Version of Polynomial Convex Functions and Some Interesting Inequalities with Applications

In this article, we introduce a general class of convex functions and proved some of its basic properties. We establish Hermite-Hadamard type inequalities as well as fractional version of Hermite-Hadamard type inequalities by using Riemann-Liouville integral operator. At the end, some application to special means of real numbers are also given. It can be observed from the remarks given in this paper that several exiting results of ligature can be obtained immediacy from our results by taking suitable involved parameters.

MULTIPLICATIVELY PREINVEX P-FUNCTIONS

In this study, using both Hölder’s integral inequality and Hölder-İşcan integral inequality, some inequalities of Hermite Hadamard type obtained for multiplicatively preinvex P-functions are given. Afterwards, the result obtained with Hölder-İşcan inequality has been shown to be better than that obtained with Hölder’s integral inequality.

Hermite-Hadamard Like Inequalities for Exponentially Subadditive Functions via Fractional Integrals

In this work, through the Riemann-Liouville fractional integrals, we give Hermite-Hadamard type inequalities for exponentially sub-additive functions. For the product of exponentially sub-additive functions, we present fractional integral inequalities. It is also shown that the results proved here are the refinements and extensions of several existing results in the field of Hermite-Hadamard like inequalities.

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.

Hermite–Hadamard–Type Fractional–Integral Inequalities for (p,h)–Convex Fuzzy–Interval–Valued Mappings

Hermite–Hadamard–Type Fractional–Integral Inequalities for (p,h)–Convex Fuzzy–Interval–Valued Mappings

On fuzzy fractional integral operators having exponential kernels and related certain inequalities for exponential trigonometric convex fuzzy-number valued mappings

The most important operator in fractional theory that enables the classical theory of integrals to be generalized is the Riemann-Liouville fractional integrals. In this paper, we have introduced new fractional operators in the fuzzy environment known as fuzzy Riemann-Liouville fractional integrals having exponential kernels. All classical fractional integrals that depend upon exponential kernels are exceptional cases of this new one. Moreover, we have defined a new class of convex mappings which is known as exponential trigonometric convex fuzzy-number valued mappings. With the help of this class and the newly proposed fuzzy fractional integral operator, the well-known Hermite-Hadamard type and related inequalities are taken into account in this work. Moreover, some new versions of midpoint Hermite-Hadamard-type inequalities are also established. By applying these definitions, we have amassed some novel and classical exceptional cases that serve as implementations of the key findings. For the purpose of proving the viability of the fuzzy order relations put forth in this research, some nontrivial examples of fuzzy numbered valued convexity are also provided.

On the Integral Representation and the Raşa, Jensen and Hermite–Hadamard Inequalities for Box-Convex Functions

We give the integral representation of Popoviciu’s (Mathematica 8:1–85, 1934) box-(m, n)-convex functions. Based on this integral representation, we obtain a characterization of box-(m, n)-convex orders, which we then use in the proofs of the Raşa type, the Hermite–Hadamard type and the Jensen inequalities for box-(m, n)-convex functions as well as for strongly box-(m, n)-convex functions. We extend the results obtained by Gal and Niculescu (J Math Anal Appl 479:903–925, 2019), and by Gavrea and Gavrea (Mediterr J Math 19:18–33, 2022).