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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

We let υ be a convex function on an interval [ι1,ι2]⊂R. If ζ∈C([ι1,ι2]), ζ≥0 and ζ is symmetric with respect to ι1+ι22, then υ12∑j=12ιj∫ι1ι2ζ(s)ds≤∫ι1ι2υ(s)ζ(s)ds≤12∑j=12υ(ιj)∫ι1ι2ζ(s)ds. The above estimates were obtained by Fejér in 1906 as a generalization of the Hermite–Hadamard inequality (the above inequality with ζ≡1). This work is focused on the study of right-side Fejér-type inequalities in one- and two-dimensional cases for new classes of differentiable functions υ. In the one-dimensional case, the obtained results hold without any symmetry condition imposed on the weight function ζ. In the two-dimensional case, the right side of Fejer’s inequality is extended to the class of subharmonic functions υ on a disk.

This note generalizes several existing results related to Hermite–Hadamard inequality using h-Godunova–Levin and (h1,h2)-convex functions using a fractional integral operator associated with the Caputo–Fabrizio fractional derivative. This study uses a non-singular kernel and constructs some new theorems associated with fractional order integrals. Furthermore, we demonstrate that the obtained results are a generalization of the existing ones. To demonstrate the correctness of these results, we developed a few interesting non-trivial examples. Finally, we discuss some applications of our findings associated with special means.

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.

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.

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.

The goal of this study is to refine some numerical radius inequalities in a novel way. The new improvements and refinements purify some famous inequalities pertaining to Hilbert space operators numerical radii. The inequalities that have been demonstrated in this work are not only an improvement over old inequalities but also stronger than them. Several examples supporting the validity of our results are provided as well.

In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results.

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 (UDET-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 UDET-convex FNVMs. 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.

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.

The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, the theory of convexity and inequalities have recently attracted a lot of attention from historians and modern researchers. This article explores the concept of a new group of modified harmonic exponential s-convex functions. Some of its significant algebraic properties are elegantly elaborated to maintain the newly described idea. A new sort of Hermite–Hadamard-type integral inequality using this new concept of the function is investigated. In addition, several new estimates of Hermite–Hadamard inequality are presented to improve the study. These new results illustrate some generalizations of prior findings in the literature.

The aim of this research is to explore fractional integral inequalities that involve interval-valued preinvex functions. Initially, a new set of fractional operators is introduced that uses the extended generalized Mittag-Leffler function Eμ,α,lγ,δ,k,c(τ;p) as a kernel in the interval domain. Additionally, a new form of Atangana–Baleanu operator is defined using the same kernel, which unifies multiple existing integral operators. By varying the parameters in Eμ,α,lγ,δ,k,c(τ;p), several new fractional operators are obtained. This study then utilizes the generalized AB integral operators and the preinvex interval-valued property of functions to establish new Hermite–Hadamard, Pachapatte, and Hermite–Hadamard–Fejer inequalities. The results are supported by numerical examples, graphical illustrations, and special cases.

The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of Hermite–Hadamard and Pachpatte-type integral inequalities involving the idea of the preinvex function in the frame of a fractional integral operator, namely the Caputo–Fabrizio fractional operator. By employing our approach, a new fractional integral identity that correlates with preinvex functions for first-order differentiable mappings is presented. Moreover, we derive some refinements of the Hermite–Hadamard-type inequality for mappings, whose first-order derivatives are generalized preinvex functions in the Caputo–Fabrizio fractional sense. From an application viewpoint, to represent the usability of the concerning results, we presented several inequalities by using special means of real numbers. Integral inequalities in association with convexity in the frame of fractional calculus have a strong relationship with symmetry. Our investigation provides a better image of convex analysis in the frame of fractional calculus.

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.

In this paper, we provide different variants of the Hermite–Hadamard (H⋅H) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically s-convex fuzzy-number-valued functions (UDH s-convex FNVM) in the second sense based on the up and down fuzzy inclusion relation. The findings are confirmed with certain numerical calculations that take a few appropriate examples into account. The results deal with various integrals of the 2ρσρ+σ type and are innovative in the setting of up and down harmonically s-convex fuzzy-number-valued functions. Moreover, we acquire classical and new exceptional cases that can be seen as applications of our main outcomes. In our opinion, this will make a significant contribution to encouraging more research.

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α,m)-convex functions, (α,m)-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ,m)-convex functions, m-harmonic harmonically convex functions, (s,r)-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, (α,m)-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable ϕ-convex functions, MT-convex functions, (s,m)-convex functions, p-convex functions, h-convex functions, σ-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, σ,s-convex functions, modified (p,h)-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, (α,h−m)−p-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, (k,s)-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator.

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

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn from one and apply it to the other. The objective of this note is to convert ordinary stochastic processes into interval stochastic processes due to the wide range of applications in various disciplines. We have developed Hermite–Hadamard (H.H), Ostrowski-, and Jensen-type inequalities using interval h-convex stochastic processes. Our main results can be applied to a variety of new and well-known outcomes as specific situations. The results of this study are expected to stimulate future research on inequalities using fractional and fuzzy integral operators. Furthermore, we validate our main findings by providing some non-trivial examples. To demonstrate their general properties, we illustrate the connections between the examined results and those that have already been published. The results discussed in this article can be seen as improvements and refinements to results that have already been published. This is a fascinating subject that can be investigated in the future to identify equivalent inequalities for various convexity types.

In this paper, we concentrate on and investigate the idea of a novel family of modified p-convex functions. We elaborate on some of this newly proposed idea’s attractive algebraic characteristics to support it. This is used to study some novel integral inequalities in the frame of the Hermite–Hadamard type. A unique equality is established for differentiable mappings. The Hermite–Hadamard inequality is extended and estimated in a number of new ways with the help of this equality to strengthen the findings. Finally, we investigate and explore some applications for some special functions. We think the approach examined in this work will further pique the interest of curious researchers.

For k-Riemann–Liouville fractional integral operators, the Hermite–Hadamard inequality is already well-known in the literature. In this regard, this paper presents the Hermite–Hadamard inequalities for k-Riemann–Liouville fractional integral operators by using a novel method based on Green’s function. Additionally, applying these identities to the convex and monotone functions, new Hermite–Hadamard type inequalities are established. Furthermore, a different form of the Hermite–Hadamard inequality is also obtained by using this novel approach. In conclusion, we believe that the approach presented in this paper will inspire more research in this area.

In this paper, we derive an identity of the q-definite integral of a continuous function f on a finite interval. We then use such identity to prove some new quantum integral inequalities for (p,h)-convex function. The results obtained in this paper generalize previous work in the literature.

In this article, we demonstrated various Hermite–Hadamard and Fejér type inequalities for modified h-convex functions. We showed several inequalities for the products of two modified h-convex functions. New identities related to inequalities in various forms are also established for different values of the h(φt) function. We believe that the approach presented in this paper will inspire more research in this area.

The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity (UD-convexity), we increase this mileage even further. Additionally, with the help of definitions of lower UD-concave (lower UD-concave) and upper UD-convex (concave) fuzzy number valued mappings (FNVMs), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued UD-convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study.

Convex bodies are symmetric in nature. Between the two variables of symmetry and convexity, a correlation connection is also perceptible. Due to the interchangeable analogous properties, the application on either of them has been practicable in these modern years. The current analysis sheds insight on a general new identity involving a number of parameters for a twice partial quantum differentiable function. We find several unique quantum integral inequalities by using the new identity and a twice partial quantum differentiable function whose absolute value is coordinated convex. In addition, we present several novel and interesting error estimation-like results related to the well-known quantum Hermite–Hadamard inequality. Some examples are provided at the end to support and demonstrate the effectiveness of the new outcomes.

In this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q-differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities.

In this work, we use the idea of interval-valued convex functions of Center-Radius (cr)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in the context of cr-convex interval-valued functions and deal with differintegrals of the p+s2 type. We believe this will be an important contribution to spurring additional research.

In this paper, we present some new and novel mappings defined over 0,1 with the help of GA-convex functions. As a consequence, we obtain companions of Fejér-type inequalities for GA-convex functions with the help of these mappings, which provide refinements of some known results. The properties of these mappings are discussed as well.

In this paper, a new type of convexity is defined, namely, the left–right-(k,h-m)-p IVM (set-valued function) convexity. Utilizing the definition of this new convexity, we prove the Hadamard inequalities for noninteger Katugampola integrals. These inequalities generalize the noninteger Hadamard inequalities for a convex IVM, (p,h)-convex IVM, p-convex IVM, h-convex, s-convex in the second sense and many other related well-known classes of functions implicitly. An apt number of numerical examples are provided as supplements to the derived results.

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯+ω̲2,ω¯−ω̲2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples.

This paper aims to find Hermite–Hadamard-type inequalities for a generalized notion of integrals called q−h-integrals. Inequalities for q-integrals can be deduced by taking h=0 and are connected with several known results of q-Hermite–Hadamard inequalities. In addition, we analyzed q−h-integrals, q-integrals, and the corresponding inequalities for symmetric functions.

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators.

Numerous applications of the theory of convex and nonconvex mapping exist in the fields of applied mathematics and engineering. In this paper, we have defined a new class of nonconvex functions which is known as up and down pre-invex (pre-incave) fuzzy number valued mappings (F-N-V∙Ms). The well-known fuzzy Hermite–Hadamard (𝐻𝐻)-type and related inequalities are taken into account in this work. We extend this mileage further using fuzzy Riemann integrals and the fuzzy number up and down pre-invexity. Additionally, by imposing some light restrictions on pre-invex (pre-incave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down pre-invexity (pre-incavity), which are referred to as lower up and down pre-invex (pre-incave) and upper up and down pre-invex (pre-incave) fuzzy number valued mappings. By using these definitions, we have amassed a large number of both established and novel exceptional situations that serve as implementations of the key findings. To support the validity of the fuzzy inclusion relations put out in this research, we also provide a few examples of fuzzy numbers valued up and down pre-invexity.

It is well known that the Hermite–Hadamard inequality (called the HH inequality) refines the definition of convexity of function f(x) defined on [a,b] by using the integral of f(x) from a to b. There are many generalizations or refinements of HH inequality. Furthermore HH inequality has many applications to several fields of mathematics, including numerical analysis, functional analysis, and operator inequality. Recently, we gave several types of refined HH inequalities and obtained inequalities which were satisfied by weighted logarithmic means. In this article, we give an N-variable Hermite–Hadamard inequality and apply to some norm inequalities under certain conditions. As applications, we obtain several inequalities which are satisfied by means defined by symmetry. Finally, we obtain detailed integral values.

In this study, on the subject of harmonic convex functions, we introduce some new functionals linked with weighted integral inequalities for harmonic convex functions. In addition, certain new inequalities of the Fejér type are discovered.

The main objective of this paper is to introduce a new class of convexity called left-right–bi-convex fuzzy interval-valued functions. We study this class from the perspective of fractional Hermite–Hadamard inequalities, involving a new fractional integral called the left-right–AB fractional integral. We discuss several special cases that demonstrate that our results are quite unifying. We provide non-trivial numerical examples regarding special means for positive real numbers in order to check the validity of our outcomes.

Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the attention of several mathematicians because of its real-life applications. More importantly, it has turned into a valuable tool for handling elements from perplexing frameworks within different parts of the pure and applied sciences. Integral inequalities, in association with convexity, have a strong relationship with symmetry. The objective of this article is to introduce the notion of operator refined exponential type convexity. Fractional versions of the Hermite–Hadamard type inequality employing generalized R–L fractional operators are established. Additionally, some novel refinements of Hermite–Hadamard type inequalities are also discussed using some established identities. Finally, we present some applications of the probability density function and special means of real numbers.

In this paper, we present some mappings defined over 0,1 related to the Fejér-type inequalities that have been established for harmonically convex functions. As a consequence, we obtain companions of Fejér-type inequalities for harmonically convex functions by using these mappings. Properties of these mappings are discussed, and consequently, we obtain refinement inequalities of some known results.