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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]

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]

This manuscript explores the idea of a multiplicatively P-superquadratic function and its properties. Utilizing these properties, we derive the inequalities of Hermite–Hadamard type for such functions in the framework of multiplicative calculus. In addition to this, we derive integral inequalities of Hermite–Hadamard type for the product and quotient of multiplicatively P-superquadratic and multiplicatively P-subquadratic functions. Moreover, we develop the fractional version of Hermite–Hadamard type inequalities involving midpoints and end points for multiplicatively P-superquadratic functions with respect to multiplicatively Riemann–Liouville (R.L) fractional integrals. Graphical illustrations based on specific relevant examples validate the credibility of the findings. The study is further stimulating by being pushed with potential applications in terms of moment of random variables, special means, and modified Bessel functions of the first kind. Regarding superquadraticity, the results presented in this work are new, therefore they clearly provide extensions and improvements of the work available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the concept of Hg-convex function is given for the first time in the literature. Some inequalities of Hadamard's type for Hg-convex functions are given. Some algebraic properties of Hg-convex functions and special cases are discussed. In addition, we establish some new integral inequalities for Hg-convex functions by using an integral identity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of generalized n-polynomial P-function. We explore some algebraic properties of this function class. Additionally, we establish a new trapezium type inequality for this generalized class of functions and derive several refinements of the trapezium type inequality for functions whose first derivative in absolute value at a certain power is generalized n-polynomial P-function. Finally, we conclude our paper by exploring some applications of the results we have obtained in the context of special means. Our novel findings generalize previously known results in the literature. [ABSTRACT FROM AUTHOR]

We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional fractional integral operators (HPFIOs) and Hadamard k-fractional integral operators (HKFIOs). Moreover, we also present the results for subclasses of GA-h-CFs and show that the inequalities proved in this paper unify the results from the recent related literature. Furthermore, we compare the two generalizations in view of the fractional operator parameters that contribute to the generalizations of the results and assess the better approximation via graphical tools. Finally, we present applications of the new inequalities via HPFIOs and HKFIOs by establishing interpolation relations between arithmetic mean and geometric mean and by proving the new upper bounds for the Tsallis relative operator entropy. [ABSTRACT FROM AUTHOR]

The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable convex functions that are connected to the right side of the H–H–M-type inequality. For these results, further remarks and observations are provided. Following that, a couple of graphical representations are shown to highlight the key findings of our study. Finally, some applications on special means are shown to demonstrate the effectiveness of our inequalities. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in investigating a weighted variant of Hermite–Hadamard type inequalities involving convex functionals. The approach undertaken makes it possible to refine and reverse certain inequalities already known in the literature. It also allows us to provide new weighted convex functional means and establish some related properties with respect to some standard means. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new versions of the Hermite–Hadamard inequality for (ϕ − h) -integrals. For this aim, we use the tangent and secant lines at the same special points. Moreover, we investigate the relations between the newly obtained results and earlier published papers. We also present some new results by special choices of ϕ and h. [ABSTRACT FROM AUTHOR]

In this manuscript, we demonstrate a graphical comparison analysis of the classical, quantum, and symmetric quantum derivatives for any continuous function, evaluated at z = 1 and q = 0.5 . We introduce the new notions of symmetric Hahn calculus and give the basic interpretations of derivative and integration of a function in Hahn symmetric quantum calculus. This enable us to to derive general Hölder's inequality in symmetric Hahn calculus and as an application, we extract corresponding symmetric Hahn Minkowski's inequality in general settings. Later, we construct an exciting Lemma in which we derive power rule of integration in symmetric Hahn calculus. An example is provided in which we give explicit analysis on four cases. Finally, these findings allow us to construct Hermite–Hadamard inequality and its related kinds in symmetric Hahn calculus. It is pertinent to mention that the particular case of our results recapture the results of symmetric quantum calculus. [ABSTRACT FROM AUTHOR]

Abstract This manuscript explores the idea of a multiplicatively P-superquadratic function and its properties. Utilizing these properties, we derive the inequalities of Hermite–Hadamard type for such functions in the framework of multiplicative calculus. In addition to this, we derive integral inequalities of Hermite–Hadamard type for the product and quotient of multiplicatively P-superquadratic and multiplicatively P-subquadratic functions. Moreover, we develop the fractional version of Hermite–Hadamard type inequalities involving midpoints and end points for multiplicatively P-superquadratic functions with respect to multiplicatively Riemann–Liouville (R.L) fractional integrals. Graphical illustrations based on specific relevant examples validate the credibility of the findings. The study is further stimulating by being pushed with potential applications in terms of moment of random variables, special means, and modified Bessel functions of the first kind. Regarding superquadraticity, the results presented in this work are new, therefore they clearly provide extensions and improvements of the work available in the literature.

Abstract In this paper, we prove some new versions of the Hermite–Hadamard inequality for ( ϕ − h ) $({\phi}-h)$ -integrals. For this aim, we use the tangent and secant lines at the same special points. Moreover, we investigate the relations between the newly obtained results and earlier published papers. We also present some new results by special choices of ϕ and h.

Abstract In this manuscript, we demonstrate a graphical comparison analysis of the classical, quantum, and symmetric quantum derivatives for any continuous function, evaluated at z = 1 $\mathfrak{z}=1$ and q = 0.5 $\mathtt{q}=0.5$ . We introduce the new notions of symmetric Hahn calculus and give the basic interpretations of derivative and integration of a function in Hahn symmetric quantum calculus. This enable us to to derive general Hölder’s inequality in symmetric Hahn calculus and as an application, we extract corresponding symmetric Hahn Minkowski’s inequality in general settings. Later, we construct an exciting Lemma in which we derive power rule of integration in symmetric Hahn calculus. An example is provided in which we give explicit analysis on four cases. Finally, these findings allow us to construct Hermite–Hadamard inequality and its related kinds in symmetric Hahn calculus. It is pertinent to mention that the particular case of our results recapture the results of symmetric quantum calculus.

Abstract In the present work we establish for the first time a class of ( P , m ) $(P,\mathrm{m})$ -superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature.

In this article, we study some Hermite-Hadamard-type inequalities for strongly hh-convex functions on co-ordinates in Rn{{\mathbb{R}}}^{n}, which extend some known results. Some mappings connected with these inequalities and related applications are also obtained.

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.

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 paper establishes some new inequalities of Hermite–Hadamard–Mercer type for s-convex functions in the framework of q-calculus and classical calculus. Some new q-midpoint-Mercer type inequalities for the q-differentiable s-convex functions are also proved. Moreover, the computational analysis of newly established inequalities for convex and s-convex functions are given to prove that the bounds of this paper are better than those of the existing ones. It is also shown with mathematical examples that the newly established inequalities are valid for s-convex functions in q-calculus. [ABSTRACT FROM AUTHOR]

We introduce the concept of exponentially (h,m)-preinvex interval-valued functions (EP-(h,m)-IVFs) by using the pseudo-order relation, and derive some interesting properties. Based on such a definition, we present some new Hermite–Hadamard and Hermite–Hadamard–Fejér-type inequalities for ϕk-Riemann–Liouville fractional integrals. These inequalities not only generalize some existing research findings in the literature, but also provide a new direction for the further study of interval integral estimation. Additionally, graphical representations and numerical estimates of some examples are given to illustrate the validity of the main results. [ABSTRACT FROM AUTHOR]

In the present work we establish for the first time a class of (P , m) -superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature. [ABSTRACT FROM AUTHOR]

The theory of inequalities is greatly influenced by interval‐valued concepts, and this contribution is explored from several perspectives and domains. The aim of this note is to develop several mathematical inequalities such as Hermite–Hadamard, Fejér, and the product version based on center radius CR‐order relations. Furthermore, we develop several nontrivial examples and remarks to support the main findings. [ABSTRACT FROM AUTHOR]

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder's inequality, unified Jensen's inequality, and Hermite–Hadamard (H - H)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of I. V - (⋏ , ℏ) generic class of convexity, which unifies several existing definitions of convexity. By utilizing Riemann–Liouville (R-L) fractional operators and I. V - (⋏ , ℏ) convexity to derive new improvements of the H - H - and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for I. V R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new inequalities of Hermite–Hadamard type for differentiable functions with h-convex derivatives. It is also shown that the newly established inequalities are extension of the existing inequalities in the literature. Finally, we give applications of the new results and outline some future problems. [ABSTRACT FROM AUTHOR]

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]

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]

The Hermite-Hadamard inequality is one of the most interesting inequalities that give lower and upper bounds of the mean value of a convex function in a way that refines the convex characteristic of the function. This paper presents a new reversed version of this outstanding result, with applications toward means of positive numbers, operator inequalities, and the Riemann-Liouville fractional integrals. [ABSTRACT FROM AUTHOR]

In the present paper, we have introduced a new class of preinvexity namely hyperbolic type harmonic preinvex functions and to support this new defi- nition, some of its algebraic properties are elaborated. By using this new class of preinvexity, we have established a few Hermite-Hadamard type integral inequalities. Some novel refinements of Hemite-Hadamard type inequalities for hyperbolic type harmonic preinvex functions are presented as well. Finally, the Riemann-Liouville fractional version of the Hermite-Hadamard Inequality is established. [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]

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]

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]

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.

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.

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 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 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]

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]

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]

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, we establish the generalized Hermite-Hadamard- and Pachpatte-type integral inequalities for local fractional integrals via the generalized subadditive functions. In particular, we put forward a refined version of the generalized Hermite-Hadamard-type inequalities in the framework of fractal space. [ABSTRACT FROM AUTHOR]

In the current investigation, we acquire the upper and lower bounds for inequalities of midpoint-type and trapezoid-type involving conformable fractional integral operators with the help of the mappings whose second derivatives are bounded. We support the established inequalities with examples. Moreover, we use graphs to demonstrate the correctness of the given examples. What's more, we prove the Hermite-Hadamard inequality, which includes conformable fractional integrals, with the aid of condition f ′(a+b-t)- f ′(t) ≥ 0, t ∈... rather than the convexity of function. [ABSTRACT FROM AUTHOR]