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In this paper, the generalization of Simpson’s identity has been derived. This generalized identity has been used to obtain new Hermite-Hadamard inequalities for differentiable convex and quasi-convex functions. Also, the validation of the derived inequalities has been established using suitable examples. [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]

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of h$$ h $$‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of h$$ h $$‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research. [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]

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]

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting α = 0 = φ , γ = 1 , and w = 0 , σ (0) = 1 , λ = 1 , are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field. [ABSTRACT FROM AUTHOR]

In the present note we are interested in proving the counterpart of the (right-hand side of the) celebrated Hermite–Hadamard inequality for φ -convex functions. In particular, we prove that the only φ -convex function for which the Hermite–Hadamard inequality holds with the Lagrangian mean on the right-hand side is (up to an affine transformation) the log -convex function. [ABSTRACT FROM AUTHOR]

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 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 article, we implemented the idea of a fuzzy interval-valued function with the well-known generalized fuzzy fractional operators, associated with different types of convexities and preinvexities. We developed the Prabhakar fuzzy fractional operators using the fuzzy interval-valued function. We presented the novel extensions of Hermite-Hadamard fuzzy-type and trapezoidal fuzzy-type inequalities, based on the h-Godunova-Levin convex and h-Godunova preinvex fuzzy interval-valued functions. [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.

In this study, some mappings related to the Fejér-type inequalities for GAGA-convex functions are defined over the interval [0,1]{[}0,1]. Some Fejér-type inequalities for GAGA-convex functions are proved using these mappings. Properties of these mappings are considered and consequently we obtain refinements of some known results.

For a given sequence a=(a1,…,an)∈Rna=\left({a}_{1},\ldots ,{a}_{n})\in {{\mathbb{R}}}^{n}, our aim is to obtain an estimate of En≔a1+an2−1n∑i=1nai{E}_{n}:= \left|\hspace{-0.33em},\frac{{a}_{1}+{a}_{n}}{2}-\frac{1}{n}{\sum }_{i=1}^{n}{a}_{i},\hspace{-0.33em}\right|. Several classes of sequences are studied. For each class, an estimate of En{E}_{n} is obtained. We also introduce the class of convex matrices, which is a discrete version of the class of convex functions on the coordinates. For this set of matrices, new discrete Hermite-Hadamard-type inequalities are proved. Our obtained results are extensions of known results from the continuous case to the discrete case.

Abstract We use a new function class called B-function to establish a novel version of Hermite–Hadamard inequality for weighted ψ-Hilfer operators. Additionally, we prove two new identities involving weighted ψ-Hilfer operators for differentiable functions. Moreover, by employing these equalities and the properties of the B-function, we derive several trapezoid- and midpoint-type inequalities for h-convex functions. Furthermore, the obtained results are reduced to several well-known and some new inequalities by making specific choices of the function h.

In this article, we implemented the idea of a fuzzy interval-valued function with the well-known generalized fuzzy fractional operators, associated with different types of convexities and preinvexities. We developed the Prabhakar fuzzy fractional operators using the fuzzy interval-valued function. We presented the novel extensions of Hermite-Hadamard fuzzy-type and trapezoidal fuzzy-type inequalities, based on the $ h $-Godunova-Levin convex and $ h $-Godunova preinvex fuzzy interval-valued functions.

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.

In this paper, we introduce a new convexity notion for inter-valued functions, known as Geometrically–Arithmetically Cr-h-convex functions (abbreviated as GA-Cr-h-CFs) and explore its properties. The family of GA-Cr-h-CFs simultaneously covers the family of GA-CFs, GA-h-CFs and GA-Cr-CFs. On the other hand, we demonstrate its equivalence with convex functions (CFs), h-CFs, Cr-CFs and Cr-h-CFs via exponential function. We also investigate necessary and sufficient conditions for GA-Cr-h-CFs via two corresponding real-valued GA-h-CFs. Due to significance and applications of mathematical inequalities, we apply our findings and propose several inequalities such as Jensen–Mercer inequality (JMI), Hermite–Hadamard inequality (HHI) and weighted Hermite–Hadamard-type inequalities (wHHIs) for GA-Cr-h-CFs. Through our results, we re-capture many known results and inequalities for subclasses of GA-Cr-h-CFs studied in the recent literature. [ABSTRACT FROM AUTHOR]

We use a new function class called B-function to establish a novel version of Hermite–Hadamard inequality for weighted ψ-Hilfer operators. Additionally, we prove two new identities involving weighted ψ-Hilfer operators for differentiable functions. Moreover, by employing these equalities and the properties of the B-function, we derive several trapezoid- and midpoint-type inequalities for h-convex functions. Furthermore, the obtained results are reduced to several well-known and some new inequalities by making specific choices of the function h. [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 this research, we introduce a new and generalized family of convex functions, entitled the α-convex functions in the second sense with respect to a parameter and examine their important algebraic properties. Based on this novel convexity concept, we explore a new class of fractional integral inequalities for functions that are twice differentiable. These results are derived from fundamental identities obtained using classical Riemann–Liouville fractional integrals. To validate our findings, we provide 2D and 3D graphs of the main results. Furthermore, as an additional aspect of our study, we explore error estimates for differences of generalized means. [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]

There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that are related to interval-valued cr-log-h-convex functions, and some known results are recaptured. To support our main results, we offer three examples and two applications related to modified Bessel functions and special means as well. [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. [ABSTRACT FROM AUTHOR]

The main identity of midpoint type inequalities is generalized and using this identity some composite midpoint type inequalities are estimated. Moreover, applications to some special means and some error estimates for the composite midpoint formula are discussed. [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]

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]

We employ a new function class called B-function to create a new version of fractional Hermite–Hadamard and trapezoid type inequalities on the right-hand side that involves h-convex and ψ -Hilfer operators. We also provide new midpoint-type inequalities using h-convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce four new generalized quantum integrals of two variables functions which are called T q -integrals. We also give some examples of each of these given integrals. Then we establish corresponding T q -Hermite–Hadamard type inequalities for co-ordinated convex functions. In order to show correctness of our results, we apply our main theorems to several co-ordinated convex functions. [ABSTRACT FROM AUTHOR]

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

We present an adaptive method of approximate integration of convex (as well as concave) functions based on a certain refinement of the celebrated Hermite–Hadamard inequality. Numerical experiments are performed and the role of harmonic numbers is shown.

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.

We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.

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.

The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval $ [\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $, we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at $ \mathrm{b_{0}}\in[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point $ \mathrm{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 [ 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]

In this paper the extension of the weighted Montgomery identity is established by using the integral formula of Pecaric, Matic and Ujevc. Further, by using this extended weighted Montgomery identity for functions whose derivatives of order n-1 are absolutely continunous functions, new inequalities of the weighted Hermite-Hadamard type are obtained. Also, applications of these results are given for various types of weight function. [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 we prove a new variant of q-Hermite–Hadamard–Mercer-type inequality for the functions that satisfy the Jensen–Mercer inequality (JMI). Moreover, we establish some new midpoint- and trapezoidal-type inequalities for differentiable functions using the JMI. The newly developed inequalities are also shown to be extensions of preexisting inequalities in the literature. [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, 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]