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Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

This paper's main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics. [ABSTRACT FROM AUTHOR]

This paper proves an equality for the case of differentiable convex functions including the conformable fractional integrals. By using this equality, we establish several parameterized inequalities with the help of the conformable fractional integrals. Several inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, we present previously achieved results and new results by using special cases of the obtained theorems. [ABSTRACT FROM AUTHOR]

A. M. Mercer prove a new version of well-known Jensen inequality which is called Jensen-Mercer inequality [16]. By using Jensen-Mercer inequality, Kian andMoslehian establish a new variant of Hermite-Hadamard inequality which is called Hermite-Jensen-Mercer inequality [15]. In this paper, we extend the Hermite-Jensen-Mercer inequality for quantum integrals by utilizing Jensen-Mercer inequality. Then we investigate the connections between our results and those in earlier works. Moreover we give some examples to illustrate the main results in this paper. This is the first paper focusing on Hermite-Jensen-Mercer inequalities for quantum integrals. [ABSTRACT FROM AUTHOR]

This paper concentrates on addressing fractional inequalities for exponential type convex functions. By means of exponential-type convexity, we firstly establish Hermite-Hadamard (HH) type inequalities for fractional integrals with exponential kernels. Secondly, based on the discovered fractional identity by separating [a; b] to n equal subintervals, and the fact that the twice derivative in absolute value is exponential type convex, we present multipoint-based HH inequalities, which cover the trapezoid- and Bullen-type inequalities for n = 1 and 2, correspondingly. During the period, some numerical examples with graphs are provided to show the validity of the deduced inequalities. [ABSTRACT FROM AUTHOR]

In this paper we present a very short proof of inequalities of Hermite-Hadamard type obtained by M. Bessenyei and Zs. Páles. This proof is based on the recently developed method connected with use of stochastic orderings of random variables. In the second part of the paper we present a way to extend these known inequalities. Namely, we describe completely the possible inequalities of Hermite-Hadamard type for longer expression than it was the case in the results of Bessenyei and Páles. [ABSTRACT FROM AUTHOR]

Fractional q -calculus is considered to be the fractional analogs of q -calculus. In this paper, the fuzzy interval-valued Riemann–Liouville fractional (RLF) q -integral operator is introduced. Also new fuzzy variants of Hermite–Hadamard (HH) type and HH–Fejér inequalities, involving ℏ -convex fuzzy interval-valued functions (FIVFs), are presented by making use of the RLF q -integral. The results not only generalize existing findings in the literature but also lay a solid foundation for research on inequalities concerning FIVFs. Moreover, to verify our theoretical findings, numerical examples and imperative graphical illustrations are provided. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper contains a variety of new integral inequalities for (s , m) -convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are (s , m) -convex. Inequalities involving the digamma function and special means are deduced as applications. [ABSTRACT FROM AUTHOR]

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, we address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued fractional inequalities for a new class of preinvexity, i.e., cr -h-preinvexity, are estimated. The fractional operator is used for the first time to prove such inequalities involving center–radius-ordered functions. Some numerical examples are also provided to validate the presented inequalities. [ABSTRACT FROM AUTHOR]

Intensive studies aiming to extend the Hermite-Hadamard inequalities and to explore some properties and applications of these inequalities have been carried out recently. The contribution of this paper falls within this framework. We investigate here some refinements of the Hermite-Hadamard inequalities for operator convex maps involving multiple operator arguments. [ABSTRACT FROM AUTHOR]

This paper is devoted to proving some new fractional inequalities via recent generalized fractional operators. These inequalities are in the Hermite–Hadamard and Minkowski settings. Many previously documented inequalities may clearly be deduced as specific examples from our findings. Moreover, we give some comparative remarks to show the advantage and novelty of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Hermite-Hadamard inequality for the Riemann-Liouville integral transformations for uniformly convex functions from a geodesic perspective in the Hadamard spaces. This inequality is widely used in some fractional integral approximations. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new versions of Hermite–Hadamard type inequalities for co-ordinated convex functions via q 1 , q 2 -integrals. Since the inequalities are newly proved, we therefore consider some examples of co-ordinated convex functions and show their validity for particular choices of q 1 , q 2 ∈ (0 , 1) . We hope that the readers show their interest in these results. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

As convexity plays an important role in many aspects of mathematical programming, e.g., for obtaining sufficient optimality conditions and in duality theorems, and one of the most important inequalities for convex functions is the Hermite–Hadamard inequality, the importance of this paper lies in providing some new improvements for convex functions and new directions in studying new variants of the Hermite–Hadamard inequality. The first part of the article includes some known concepts regarding convex functions and related inequalities. In the second part of the study, a derivation of the Hermite–Hadamard inequality for convex functions of higher order is given, emphasizing the purpose and importance of some quadrature formulas. In the third section, the applications of the main results are presented by obtaining Hermite–Hadamard-type estimates for various classical quadrature formulas such as the Gauss–Legendre two-point quadrature formula and the Gauss–Chebyshev two-point quadrature formulas of the first and second kind. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain Fejér type inequalities for higher order convex functions and the general weighted integral formula involving w-harmonic sequences of functions. Also, Fejér type inequalities are obtained for the general three, four and five point quadrature formulae. Further, Fejér type inequalities for the corrected three, corrected four and corrected five point quadrature formulae are considered. In special cases, Fejér type estimates for Simpson, Maclaurin, corrected Simpson and corrected Maclaurin quadrature rules are derived. [ABSTRACT FROM AUTHOR]

The connection between generalized convexity and symmetry has been studied by many authors in recent years. Due to this strong connection, generalized convexity and symmetry have arisen as a new topic in the subject of inequalities. In this paper, we introduce the concept of interval-valued preinvex functions on the coordinates in a rectangle from the plane and prove Hermite–Hadamard type inclusions for interval-valued preinvex functions on coordinates. Further, we establish Hermite–Hadamard type inclusions for the product of two interval-valued coordinated preinvex functions. These results are motivated by the symmetric results obtained in the recent article by Kara et al. in 2021 on weighted Hermite–Hadamard type inclusions for products of coordinated convex interval-valued functions. Our established results generalize and extend some recent results obtained in the existing literature. Moreover, we provide suitable examples in the support of our theoretical results. [ABSTRACT FROM AUTHOR]

The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameterized inequalities for the functions whose derivatives in absolute value are convex. Finally, we show that our main inequalities reduce to Ostrowski type inequalities, Simpson type inequalities and trapezoid type inequalities which are proved in earlier published papers. [ABSTRACT FROM AUTHOR]

The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work. [ABSTRACT FROM AUTHOR]

In this paper, we tend to establish some new Hermite–Hadamard type integral inequalities for multiplicatively convex function on coordinates and for product of two multiplicatively convex functions on coordinates. [ABSTRACT FROM AUTHOR]

In this paper, a new identity is established for differentiable mappings. By using the mathematical analysis techniques, some new integral inequalities of Hermite-Hadamard type for differentiable p-convex functions are proved. A comparison of the established results with previously obtained results is demonstrated to show that the results presented in this paper are better than those already exist in literature. [ABSTRACT FROM AUTHOR]

Abstract Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field.

In the paper, the authors establish some new Hermite–Hadamard type inequalities for harmonically convex functions via generalized fractional integrals. Moreover, the authors prove extensions of the Hermite–Hadamard inequality for harmonically convex functions via generalized fractional integrals without using the harmonic convexity property for the functions. The results offered here are the refinements of the existing results for harmonically convex functions. [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 paper, we establish Hermite-Hadamard type inequalities for uniformly p-convex functions. Also, a new fractional Hermite-Hadamard type inequality for convex functions is obtained by using only the left Riemann-Liouville fractional integral. Finally some estimation of left fractional integration studies for Hermite-Hadamard type inequalities. [ABSTRACT FROM AUTHOR]

The main ob jective of this paper is to study the classical Hermite- Hadamard-type inequalities for Riemann-liouville fractional integrals. The Hermite-Hadamard inequalities are derived for k -Riemann-Liouville fractional integrals by using the definitions of different types of convex functions such as s-convex function, m-convex function, (s, m)- convex function and twice differentiable geometric-arithmetically s-convex function. Our results generalize the results of [8] and [18]. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a notion of the $$\mathbb {K}$$ -Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite-Hadamard inequalities to the case when the usual Riemann integral is replaced by the $$\mathbb {K}$$ -Riemann integral and the convexity notion is replaced by $$\mathbb {K}$$ -convexity. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some Hermite-Hadamard type inequalities for uniformly convex functions with respect to geodesic in Hadamard space. Also, we give some application of these functions in means. [ABSTRACT FROM AUTHOR]

The main object of this paper is to present a new inequalities by using of the Hermite-Hadamard inequalities. Later, in some special cases, Bullen, Simpson, Midpoint and Trapezoid type inequalities and many new results related to these inequalities will be given. [ABSTRACT FROM AUTHOR]

In this paper, we present a new definition of preinvex functions depending on the given function which is called as approximately harmonic h-preinvex functions. We establish some inequalities of Hermite-Hadamard, Simpson and midpoint type for newly define class of functions. We show that this class contains several other new and known extensions of harmonic preinvexity. Special cases which can be deduced from the main results are also discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we give new definitons related to Katugampola fractional integral for two variables functions. We are interested in giving the Hermite{Hadamard inequality for a rectangle in plane via convex functions on co-ordinates involving Katugampola fractional integral. [ABSTRACT FROM AUTHOR]

In this paper, with a new approach, a new fractional Hermite-Hadamard type inequalities for convex functions is obtained by using only the left Riemann-Liouville fractional integral. Also, to have new fractional trapezoid and midpoint type inequalities for the differentiable convex functions, two new equalities are proved. Our results generalize earlier studies. We expect that this study will be lead to the new fractional integration studies for Hermite-Hadamard type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, a new identity with parameters involving local fractional integrals is derived. Using this identity, some general local fractional integral inequalities for generalized preinvex functions are established. A parallel development is deduced for generalized preconcave functions. Taking special values for the parameters, some generalized midpoint inequalities, trapezoidal inequalities and Simpson inequalities are obtained. Finally, as some applications, error estimates of numerical integration for local fractional integrals are given. [ABSTRACT FROM AUTHOR]

In this paper, we prove three new Katugampola fractional Hermite-Hadamard type inequalities for harmonically convex functions by using the left and the right fractional integrals independently. One of our Katugampola fractional Hermite-Hadamard type inequalities is better than given in [17]. Also, we give two new Katugampola fractional identities for differentiable functions. By using these identities, we obtain some new trapezoidal type inequalities for harmonically convex functions. Our results generalize many results from earlier papers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of convex functions which is called strongly (p; h)-convex functions. We show that this class includes several other new classes of convex functions. We also establish some new results for Hermite-Hadamard type inequalities via strongly (p; h)-convex functions. Some special cases which can be deduced from our main results are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we gave adjustments for some results in the paper [1], and proved three new Katugampola fractional Hermite-Hadamard type inequalities for convex functions by using the left and the right fractional integrals independently. One of our Katugampola fractional Hermite-Hadamard type inequalities is better than given by Chen and Katugampola. Also, we gave two new Katugampola fractional identities for differentiable functions. By using these identities, we obtained some new trapezoidal type inequalities for convex functions. Our results generalize earlier results. [ABSTRACT FROM AUTHOR]

In this paper, a new weighted identity involving a differentiable mapping and a non-negative p-symmetric mapping is established. By using the mathematical analysis techniques, some new integral inequalities of Hermite-Hadamard and Fejer type for differentiable p-convex functions are proved. A comparison of the established results is presented with the help of suitable graphs by using the software Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, we prove three new conformable fractional Hermite-Hadamard type inequalities for harmonically convex functions by using the left and the right fractional integrals independently. Also, we give two new conformable fractional identities for differentiable functions. By using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalizes the results given by İşcan in [5] and Sanli et al. in [18]. [ABSTRACT FROM AUTHOR]

This paper’s main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics.