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This paper derives the sharp bounds for Hermite–Hadamard inequalities in the context of Riemann–Liouville fractional integrals. A generalization of Jensen's inequality called the Jensen–Mercer inequality is used for general points to find the new and refined bounds of fractional Hermite–Hadamard inequalities. The existing Hermite–Hadamard inequalities in classical or fractional calculus have been proved for convex functions, typically involving only two points as in Jensen's inequality. By applying the general points in Jensen–Mercer inequalities, we extend the scope of the existing results, which were previously proved for two points in the Jensen's inequality or the Jensen–Mercer inequality. The use of left and right Riemann–Liouville fractional integrals in inequalities is challenging because of the general values involved in the Jensen–Mercer inequality, which we overcame by considering different cases. The use of the Jensen–Mercer inequality for general points to prove the refined bounds is a very interesting finding of this work, because it simultaneously generalizes many existing results in fractional and classical calculus. The application of these new results is demonstrated through error analysis of numerical integration formulas. To show the validity and significance of the findings, various numerical examples are tested. The numerical examples clearly demonstrate the significance of this new approach, as using more points in the Jensen–Mercer inequality leads to sharper bounds. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and investigating the class of generalized n-fractional polynomial s-type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowskitype inequalities are presented for generalized n-fractional polynomial s-type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized n-fractional polynomial s-type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the RiemannLiouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. [ABSTRACT FROM AUTHOR]

Establishing a new integral inequality for the Riemann–Liouville fractional integral operator is the main objective of this paper. For twice differentiable s- (κ , H) -convex functions, we present a number of new inequalities that are connected to the Hermite–Hadamard integral inequality. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of (η 1 , η 2) -convex stochastic processes on coordinates and establish Hermite-Hadamard-type inequality for these stochastic processes. Moreover, we prove new integral inequality of Hermite-Hadamard-Fejér type for newly defined coordinated η -convex stochastic processes on a rectangle. The results presented in this article would provide extensions of those given in earlier works. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and investigating the class of generalized $ n $-fractional polynomial $ s $-type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowski-type inequalities are presented for generalized $ n $-fractional polynomial $ s $-type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized $ n $-fractional polynomial $ s $-type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the Riemann-Liouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields.

This article explores integral inequalities within the framework of local fractional calculus, focusing on the class of generalized α$$ \alpha $$‐convex functions. It introduces a novel extension of the Hermite‐Hadamard inequality and derives numerous fractal inequalities through a novel multiparameterized identity. The primary aim is to generalize existing inequalities, highlighting that previously established results can be obtained by setting specific parameters within the main inequalities. The validity of the derived results is demonstrated through an illustrative example, accompanied by 2D and 3D graphical representations. Lastly, the paper discusses potential practical applications of these findings. [ABSTRACT FROM AUTHOR]

In this paper, we present Katugampola fractional parameterized inequalities for s-convex function in the fourth sense. In the first place, with the help of integration by parts method, we provide a very important identity that will be of great significance for the next research. In addition, by comprehensively making use of some important concepts such as s-convex function in the fourth sense, Hölder's integral inequality, and the well-known power-mean inequality, we establish three new parameterized inequalities. Finally, we provide some corollaries to help readers gain a deeper understanding of the theorems and the relevant theories. [ABSTRACT FROM AUTHOR]

In this manuscript, we obtain refinements of the Hermite-Hadamard type inequalites for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is trigonometrically ρ-convex. We also show that our results coincide with previous results. [ABSTRACT FROM AUTHOR]

In this study, we introduce some new mappings in connection with Hermite-Hadamard and Fej'er type integral inequalities which have been proved using the GA-convex functions. As a consequence, we obtain certain new inequalities of the Fej'er type that provide refinements of the Hermite-Hadamard and Fej'er type integral inequalities that have already been obtained. [ABSTRACT FROM AUTHOR]

In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four examples with graphics which illustrate the validity of the results. Moreover, several Hermite–Hadamard–Mercer-type inequalities for fractional integrals for functions whose powers of the absolute values of the third derivatives are convex are presented. [ABSTRACT FROM AUTHOR]

Unfortunately, eleven of our provinces were severely affected due to two severe earthquakes that occurred in our country, the Republic of Turkey, on February 6, 2023. As a result, thousands of buildings were destroyed and tens of thousands of our citizens lost their lives. From past to present, such disasters have occurred in many parts of our world and will continue to happen. In order to raise awareness for researchers and academicians reading our article, we will give a new definition of convexity in this article, and we will call it "earthquake convexity". In this paper, we study some algebraic properties of the earthquake convexity. Then we compare the results obtained with both Hölder, Hölder–İşcan inequalities and power-mean, improved power-mean integral inequalities and show that the results obtained with Hölder–İşcan and improved power-mean inequalities are better than the others. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order β approaches 1, in addition to the fractional integrals we examined. [ABSTRACT FROM AUTHOR]

This work presents new versions of the Hermite-Hadamard Inequality, for (m-F)-convex functions, defined on fractal sets R ς (0 < ς ≤ 1). So, we show some new results for twice differentiable functions using local fractional calculus, as well as some new definitions. We will construct these new integral inequality using the generalized Hölder-integral inequality and the power mean integral inequality. Furthermore, we present some new inequalities for the midpoint and trapezoid formulas in a novel type of fractal calculus. The conclusions in this paper are substantial advancements and generalizations of prior research reported in the literature. 2020 Mathematics Subject Classification. [ABSTRACT FROM AUTHOR]

In this paper, by using Hölder-İşcan, Hölder integral inequality and an general identity for differentiable functions we can get new estimates on generalization of Hadamard, Ostrowski and Simpson type integral inequalities for functions whose derivatives in absolute value at certain power are quasi-convex functions. It is proved that the result obtained Hölder-İşcan integral inequality is better than the result obtained Hölder inequality. [ABSTRACT FROM AUTHOR]

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s = 1 , λ = α ,   σ (0) = 1 , and w = 0 . In conclusion, the methodology described in this article is expected to stimulate further research in this area. [ABSTRACT FROM AUTHOR]

In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities. [ABSTRACT FROM AUTHOR]

The class of geometrically convex functions is a rich class that contains some important functions. In this paper, we further explore this class and present many interesting new properties, including fundamental inequalities, supermultiplicative type inequalities, Jensen-Mercer inequality, integral inequalities, and refined forms. The obtained results extend some celebrated results from the context of convexity to geometric convexity, with interesting applications to numerical inequalities for the hyperbolic and exponential functions. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided.

In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When p = − 1 , our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. [ABSTRACT FROM AUTHOR]

In this study, some new Hermite-Hadamard type inequalities for co-ordinated convex functions were obtained with the help of conformable fractional integrals. We have presented some remarks to give the relation between our results and earlier obtained results. Moreover, an identity for partial differentiable functions has been established. By using this equality and concept of co-ordinated convexity, we have proven a trapezoid type inequality for conformable fractional integrals. [ABSTRACT FROM AUTHOR]

In this paper, we studied the generalizations of quantum calculus on finite intervals. We presented the new definitions of the quantum derivative and quantum integral of a function with respect to another function and studied their basic properties. We gave an application of these newly defined quantum calculi by obtaining a new Hermite-Hadamard inequality for a convex function. Moreover, an impulsive boundary value problem involving quantum derivative, with respect to another function, was studied via the Banach contraction mapping principle. [ABSTRACT FROM AUTHOR]

The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval [ b 0 , b 1 ] ⊂ ℜ , we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at b 0 ∈ [ b 0 , b 1 ] ⊂ ℜ. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point b 1 , and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus. The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval , we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at . We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point , and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided. [ABSTRACT FROM AUTHOR]

In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain pre-existing inequalities found in the literature. Finally, we provide applications to the newly established inequalities to verify their usefulness. [ABSTRACT FROM AUTHOR]

In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using G A -convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions. [ABSTRACT FROM AUTHOR]

In this paper, several applications of the Hermite-Hadamard inequality for fractional integrals using harmonic convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Beta function. [ABSTRACT FROM AUTHOR]

In this paper, we use Riemann-Liouville k-fractional and k-fractional confomable integrals to prove Hermite-Hadamard inequality, an identity and Hermite-Hadamard type inequality for p-convex functions. Some special cases are also discussed. Our work is extensions of other related previous results. [ABSTRACT FROM AUTHOR]

The concept of convexity of functions is a useful instrument that is used to solve a wide range of pure and applied scientific issues. The Hermite-Hadamard inequality which is also used frequently in many other parts of practical mathematics notably in optimization and probability is one of the most important mathematical inequalities relevant to convex maps. The fractional calculus, a calculus of non-integer order has applications in diverse fields of physical sciences. In this paper, we have established Hermite-Hadamard's inequalities via Riemann-Liouville fractional integral for the case of harmonically convex function as well as the products of two harmonically convex functions via Riemann-Liouville fractional integrals. [ABSTRACT FROM AUTHOR]

In this paper, we present a new generalized class of harmonically convex functions and discuss some of their algebraic properties. Furthermore, we establish some Hermite–Hadamard‐type inequalities via new generalized harmonically convexity. As applications of the findings in this study, some particular cases are described. [ABSTRACT FROM AUTHOR]

The Hermite-Hadamard type inequalities involving fractional integral operations for p-convex functions with respect to another function are studied. Then, the inequalities via Riemann-Liouville and Hadamard fractional integrals are presented specially. Using the obtained results, some inequality relations among special functions including beta and incomplete beta functions, gamma and incomplete gamma functions, and hypergeometric functions are presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we aim to introduce and explore a new class of preinvex functions called npolynomial 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. [ABSTRACT FROM AUTHOR]

We propose the definitions of some fractional integral operators called k-Atangana–Baleanu fractional integral operators. These newly proposed operators are generalizations of the well-known Atangana–Baleanu fractional integral operators. As an application, we establish a generalization of the Hermite–Hadamard inequality. Additionally, we establish some new identities involving these new integral operators and obtained new fractional integral inequalities of the midpoint and trapezoidal type for functions whose derivatives are bounded or convex. [ABSTRACT FROM AUTHOR]

The theory of integral inequality has gained considerable attention due to its influential impact on several fields of mathematics and applied sciences. Over the years, numerous refinements, generalizations, and extensions of convexity have been explored to achieve more precise variants of already established results. The principal idea of this article is to establish some interval-valued integral inequalities of the Hermite–Hadamard type in the fractional domain. First, we propose the idea of generalized interval-valued convexity with respect to the continuous monotonic functions ⋎, bifunction ζ , and based on the containment ordering relation, which is termed as (⋎ , h) pre-invex functions. This class is innovative due to its generic characteristics. We generate numerous known and new classes of convexity by considering various values for ⋎ and h. Moreover, we use the notion of (⋎ , h) -pre-invexity and Atangana–Baleanu (AB) fractional operators to develop some fresh fractional variants of the Hermite–Hadamard (HH), Pachpatte, and Hermite–Hadamard–Fejer (HHF) types of inequalities. The outcomes obtained here are the most unified forms of existing results. We provide several specific cases, as well as a numerical and graphical study, to show the significance of the major results. [ABSTRACT FROM AUTHOR]

This article investigates inequalities for certain types of strongly convex functions by applying q-h-integrals. These inequalities provide the refinements of some well-known results that hold for (α , m) - and (ℏ - m) -convex and related functions. Inequalities for q-integrals are deducible by vanishing the parameter h. Some particular cases are discussed after proving the main results. [ABSTRACT FROM AUTHOR]

In this research, some novel Hermite–Hadamard–Fejér-type inequalities using Raina fractional integrals for the class of ϑ -convex functions are obtained. These inequalities are more comprehensive and inclusive than the corresponding ones present in the literature. [ABSTRACT FROM AUTHOR]

The application of fractional calculus to interval analysis is vital for the precise derivation of integral inequalities on set-valued mappings. The objective of this article is to reformulated the well-known Hermite–Hadamard inequality into various new variants via fractional integral operator (Riemann–Liouville) and generalize the various previously published results on set-valued mappings via center and radius order relations using harmonical h-convex functions. First, using these notions, we developed the Hermite–Hadamard ( H – H ) inequality, and then constructed some product form of these inequalities for harmonically convex functions. Moreover, to demonstrate the correctness of these results, we constructed some interesting non-trivial examples. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a more generalized class of convex stochastic processes and explore some of their algebraic properties. This new class of stochastic processes is called the n -polynomial s -type convex stochastic process. We demonstrate that this new class of stochastic processes leads to the discovery of novel Hermite–Hadamard type inequalities. These inequalities provide upper bounds on the integral of a convex function over an interval in terms of the moments of the stochastic process and the convexity parameter s. To compare the effectiveness of the newly discovered Hermite–Hadamard type inequalities, we also consider other commonly used integral inequalities, such as Hölder, Hölder–Ïşcan, and power-mean, as well as improved power-mean integral inequalities. We show that the Hölder–Ïşcan and improved power-mean integral inequalities provide a better approach for the n -polynomial s -type convex stochastic process than the other integral inequalities. Finally, we provide some applications of the Hermite–Hadamard type inequalities to special means of real numbers. Our findings provide a useful tool for the analysis of stochastic processes in various fields, including finance, economics, and engineering. [ABSTRACT FROM AUTHOR]

This paper introduces some new Hermite-Hadamard type integral inequalities for s-convex function via Caputo-Fabrizio fractional integral. At the end, applicability of obtained results to special means of real numbers has been discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the concept of multiplicatively p-convex functions and established Hermite-Hadamard type integral inequalities in the setting of multiplicative calculus for this newly created class of functions. We also gave integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively p-convex functions. Furthermore, we obtained novel multiplicative integral-based inequalities for the product and quotient of convex and multiplicatively p-convex functions. Additionally, we derived certain upper limits for this new class of functions. The findings we proved are generalizations of the results in the literature. The results obtained in this study may inspire further research in various scientific areas. [ABSTRACT FROM AUTHOR]

In this work, the notion of a multiplicative harmonic convex function is examined, and Hermite–Hadamard inequalities for this class of functions are established. Many inequalities of Hermite–Hadamard type are also taken into account for the product and quotient of multiplicative harmonic convex functions. In addition, new multiplicative integral-based inequalities are found for the quotient and product of multiplicative harmonic convex and harmonic convex functions. In addition, we provide certain upper limits for such classes of functions. The obtained results have been verified by providing examples with included graphs. The findings of this study may encourage more research in several scientific areas. [ABSTRACT FROM AUTHOR]

Fractional calculus is used to examine and enhance the concept of calculus in diverse fields of science. In this paper, we establish Hermite‐Hadamard inequalities for composite (α,m)$$ \left(\alpha, m\right) $$‐convex function. The generalized identities are established for Riemann‐type fractional integrals. The explored identities are used to examine error estimates of Hermite‐Hadamard inequalities for (α,m)$$ \left(\alpha, m\right) $$‐convex function concerning a strictly monotone function. Some results that exist in literature are obtained as special cases to our general results. The conclusions of this article may be useful in determining the uniqueness of partial differential equations and fractional boundary value problems. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach by introducing a set of operators known as the left and right generalized tempered fractional integral operators. These operators are utilized to establish new Hermite–Hadamard inequalities for convex functions as well as the multiplication of two convex functions. Additionally, this paper gives two useful identities involving the generalized tempered fractional integral operator for differentiable functions. By leveraging these identities, our results consist of integral inequalities of the Hermite–Hadamard type, which are specifically designed to accommodate convex functions. Furthermore, this study encompasses the identification of several special cases and the recovery of specific known results through comprehensive research. Lastly, this paper offers a range of applications in areas such as matrices, modified Bessel functions and q -digamma functions. [ABSTRACT FROM AUTHOR]

Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many authors have just recently introduced various generalized inequalities that involved the fractional integral operators. The main goal of the present study is to incorporate the concept of strongly -convex functions and Hermite-Hadamard inequality with Caputo-Fabrizio integral operator. Also, we consider a new identity for twice differentiable mapping in the context of Caputo-Fabrizio fractional integral operator. Then, considering this identity as an auxiliary result, new mid-point version using well known inequalities like Hölder, power-mean, Young are presented. Moreover, some graphs of obtained inequalities are given for better understanding by the reader. Finally, we discussed some applications to matrix inequalities and spacial means. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When p=−1, our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields.

This article presents new developments and applications of Jensen’s inequality. We explore various variations of Jensen’s inequality related to modified (h, m)-convex functions. The obtained results extend the applicability of the inequality to a broader class of functions and contexts. In addition to the fact that the results presented in the article provide additional results available in the literature, we also give examples of their application. [ABSTRACT FROM AUTHOR]

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings ( U D E T - convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for U D E T - convex   F N V M s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems. [ABSTRACT FROM AUTHOR]

The well-known Hermite-Hadamard inequality for convex functions is extensively studied for different kinds of integrals and derivatives. This paper investigates some of its variants for q − h -integrals using properties of convex functions. Inequalities for q -integrals that have been published in recent years can be extracted from the main results of this paper. [ABSTRACT FROM AUTHOR]

In this study, we present new variants of the Hermite–Hadamard inequality via non-conformable fractional integrals. These inequalities are proven for convex functions and differentiable functions whose derivatives in absolute value are generally convex. Our main results are established using the classical Jensen–Mercer inequality and its variants for (h , m) -convex modified functions proven in this paper. In addition to showing that our results support previously known results from the literature, we provide examples of their application. [ABSTRACT FROM AUTHOR]

In this paper, a new class of convex functions, which is called exponential trigonometric convex functions is studied. We obtain some refinements of the Hermite-Hadamard type inequalities via functions whose second derivatives in absolute value at certain power are exponential trigonometric convex functions. The results of the research by the means of Holder-İşcan and improved-power-mean integral inequalities. [ABSTRACT FROM AUTHOR]