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In mathematical analysis theory of inequalities has considerable influence due to its massive utility in various fields of physical sciences. These are investigated via multiple approaches to acquire more precise and rectified forms of already celebrated consequences. Integral inequalities are investigated to compute the error bounds for quadrature schemes. Among all of them, one is Hermite-Hadamard inequality, which has mighty efficacy. Numerous generalizations have been proposed in the literature based on different novel and innovative procedures. In recent years, Bullen inequality has been very commonly studied inequality. The main objective of our progressive study is to establish a new set of Bullen-type inequalities concerning the Jensen-Mecer inequality. For the completion of the current investigation, we derive a new general Bullen-Mecer equality, which is beneficial to achieve our primary consequences. Furthermore, Considering the Bullen-Mecer equation, we employ the convexity property together with famous Hölder's type and Young's inequalities, bounding, and Lipschitz characteristics of functions to conclude new variants of generalized upper bounds of Bullen inequality. Also, we deliver some applications of outcomes to means, special functions, error bounds, and iterative methods to solve non-linear problems. Lastly, we verify our findings through various simulations. The advantage of the current study is that several results concerning Bullen's inequality can be retrieved from our proposed results and various new results can be achieved by choosing the values for γ and δ. By utilizing the similar technique that we have adopted new iterative schemes can be established from integral inequalities. [ABSTRACT FROM AUTHOR]

Integral inequalities are the proficient aspect of mathematical analysis. Various techniques have been deployed to acquire to fresh inequalities which are beneficial in various area problems. The aim of this paper is to derive some new analytic inequalities involving generalized weighted exponential beta functions. To attain our primary objectives, we introduce the generalized exponential function X and weighted form of exponential beta functions Furthermore, we briefly discuss their properties. we derive several inequalities in association with As the applications of these new developments,we conclude some error estimates of Ostrwoski's type inequalities, which show the significance of the obtained results. [ABSTRACT FROM AUTHOR]

This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s -convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson's, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcomes to special means of real numbers have been explored. Moreover, we have derived a new generic numerical scheme for solving non-linear equations, demonstrating an application of our main results in numerical analysis. [ABSTRACT FROM AUTHOR]

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR - γ -convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen's inequalities using CR - γ -convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard's, Hermite-Hadamard-Fejer, and Pachpatte's type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations. [ABSTRACT FROM AUTHOR]

Being a critical part of classical analysis, some of the convex functions and inequalities have drawn much attention recently because both concepts establish a strong relationship. As a familiar extension of classical one, the interval-valued analysis is frequently used to the research of control theory, mathematical economy and so on. Motivated by the importance of convexity and inequality, our aim is to consider new class of convex interval-valued functions is known as LR-(p, h)-convex interval-valued functions through pseudo order relation(≤p). This order relation is defined on interval space. By using this concept, firstly we obtain Hermite- Hadamard (HH-) and Hermite- Hadamard-Fejer (HH-Fejer) type inequalities through pseudo order relation. Secondly, we present some new versions of discrete Jensen and Schur type inequalities via LR-(p, h)-convex interval-valued functions. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-h-convex-IVFs and their variant forms as special cases. Under some mild restrictions, we have proved that the inclusion relation "⊆" coincident to pseudo order relation "≤p" when the interval-valued function is LR-(p, h)-convex or LR-(p, h)-concave. Results obtained in this paper can be viewed as improvement and refinement of previously known results. [ABSTRACT FROM AUTHOR]

Quantum calculus provides a significant generalization of classical concepts and overcomes the limitations of classical calculus in tackling non-differentiable functions. Implementing the q-concepts to obtain fresh variants of classical outcomes is a very intriguing aspect of research in mathematical analysis. The objective of this article is to establish novel Milne-type integral inequalities through the utilization of the Mercer inequality for q-differentiable convex mappings. In order to accomplish this task, we begin by demonstrating a new quantum identity of the Milne type linked to left and right q derivatives. This serves as a supporting result for our primary findings. Our approach involves using the q-equality, well-known inequalities, and convex mappings to obtain new error bounds of the Milne–Mercer type. We also provide some special cases, numerical examples, and graphical analysis to evaluate the efficacy of our results. To the best of our knowledge, this is the first article to focus on quantum Milne–Mercer-type inequalities and we hope that our methods and findings inspire readers to conduct further investigation into this problem. [ABSTRACT FROM AUTHOR]

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 E μ , α , l γ , 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 E μ , α , l γ , 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. [ABSTRACT FROM AUTHOR]

We propose the concept of up and down harmonically convex mapping for fuzzy-number-valued mapping as our main goal in this work. With the help of up and down harmonically fuzzy-number convexity and the fuzzy fractional integral operator, we also show the results for the Hermite–Hadamard ( H – H ) inequality, the Fejér type inequality, and some other related versions of inequalities. Moreover, some examples are also presented to discuss the validity of the main results. The results from the new technique show how the suggested scheme is accurate, adaptable, efficient, and user-friendly. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notion of aT¯p;q-integrals. Using the definition of a T¯p;q-integrals, we derive some new post quantum analogues of some classical results of Young's inequality, H¨older's inequality, Minkowski's inequality, Ostrowski's inequality and Hermite-Hadamard's inequality. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In recent years, the theory of convex mappings has gained much more attention due to its massive utility in different fields of mathematics. It has been characterized by different approaches. In 1929, G. H. Hardy, J. E. Littlewood, and G. Polya established another characterization of convex mappings involving an ordering relationship defined over R n known as majorization theory. Using this theory many inequalities have been obtained in the literature. In this paper, we study Hermite–Hadamard type inequalities using the Jensen–Mercer inequality in the frame of q ˙ -calculus and majorized l-tuples. Firstly we derive q ˙ -Hermite–Hadamard–Jensen–Mercer (H.H.J.M) type inequalities with the help of Mercer's inequality and its weighted form. To obtain some new generalized (H.H.J.M)-type inequalities, we prove a generalized quantum identity for q ˙ -differentiable mappings. Next, we obtain some estimation-type results; for this purpose, we consider q ˙ -identity, fundamental inequalities and the convexity property of mappings. Later on, We offer some applications to special means that demonstrate the importance of our main results. With the help of numerical examples, we also check the validity of our main outcomes. Along with this, we present some graphical analyses of our main results so that readers may easily grasp the results of this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive from an auxiliary identity some new generalizations of fractional integral inequalities of Simpson, midpoint, and trapezoid using the class of harmonic convex functions. To show that our results are quite unifying, we discuss several new special cases. Finally, some applications regarding error estimations for the Simpson quadrature formula are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]

Convexity performs its due role in the theoretical field of inequalities according to the nature and conduct of the properties it displays. A correlation connectivity, which is visible between the two variables symmetry and convexity, enhances its importance. In this paper, we derive a new multi-parameter quantum integral identity involving Raina's function. Applying this generic identity as an auxiliary result, we establish some new generalized quantum estimates of certain integral inequalities pertaining to the class of R s -convex functions. Moreover, we give quantum integral inequalities for the product of R s 1 - and R s 2 -convex functions as well as another quantum result for a function that satisfies a special condition. In order to demonstrate the efficiency of our main results, we offer many important special cases for suitable choices of parameters and finally for R s -convex functions that are absolute-value bounded. [ABSTRACT FROM AUTHOR]

We introduce the class of R γ -convex functions and discuss that it relates to some other classes of convexity. We study the class of R γ -convex functions in the perspective of trapezium-like inequalities, for which we also derive a new integral identity involving a k -th order differentiable function. In order to show the significance of our results, we also discuss several special cases and offer some interesting applications. [ABSTRACT FROM AUTHOR]

In both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and integral inequality is strong. By the importance of these concepts, we have introduced the new class of generalized convex function is known as LR‐ h‐convex interval‐valued function (LR‐ h‐convex‐IVF) by means of pseudo‐order relation (≤p). This order relation is defined on interval space. Under the new concept, first, both discrete and continuous new versions of Jensen‐type inequalities are presented by means of pseudo‐order relation. Second, several new Hermite–Hadamard (HH)‐ and Hermite–Hadamard–Fejér (HH‐Fejér)‐type inequalities are also derived for LR‐ h‐convex‐IVFs. Moreover, we have shown that our results include a wide class of new and known inequalities for LR‐ h‐convex‐IVFs and their variant forms as special cases. Useful examples that verify the applicability of the theory developed in this study are presented. It the end, we have proved that the set inclusion "⊆" coincident to pseudo‐order relation "≤p." The concepts and techniques of this paper may be the starting point for further research in this area and used as a tool to investigate the research of probability and optimization, among others. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of uniform harmonic χ -convex functions. We show that this class relates several other unrelated classes of uniform harmonic convex functions. We derive a new version of Hermite-Hadamard's inequality and its fractional analogue. We also derive a new fractional integral identity using Caputo-Fabrizio fractional integrals. Utilizing this integral identity as an auxiliary result, we obtain new fractional Dragomir-Agarwal type of inequalities involving differentiable uniform harmonic χ -convex functions. We discuss numerous new special cases which show that our results are quite unifying. Finally, in order to show the significance of the main results, we discuss some applications to means of positive real numbers. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the several generalizations by newly proposed generalized K-fractional conformable integral operator. Based on these novel ideas, we derived a novel framework to study for Čebysšev and Pólya-Szegö type inequalities by generalized K-fractional conformable integral operator. Several special cases are apprehended in the light of generalized fractional conformable integral. This novel strategy captures several existing results in the relative literature. We also aim at showing important connections of the results here with those including Riemann-Liouville fractional integral operator. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called -convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the -convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results. [ABSTRACT FROM AUTHOR]

In the article, we introduce the generalized exponentially μ-preinvex function, derive a new q-integral identity for second order q-differentiable function, and establish several new q-trapezoidal type integral inequalities for the function whose absolute value of second q-derivative is exponentially μ-preinvex. [ABSTRACT FROM AUTHOR]

Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial P -convexity, we present two applications for the modified Bessel function and q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem. [ABSTRACT FROM AUTHOR]

We establish several basic inequalities versions of the Hermite-Hadamard type inequalities for GA- and GG-convexity for conformable fractional integrals. Several special cases are also discussed, which can be deduced from our main result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the generalized K -fractional integral in the frame of a new parameter K > 0 . This paper offers some new important inequalities of Grüss type using the generalized K -fractional integral and associated integral inequalities. Our results with this new integral operator have the abilities to be implemented for the evaluation of many mathematical problems related to the real world applications. [ABSTRACT FROM AUTHOR]

In the article, we present several new Hermite-Hadamard type inequalities for the exponentially GA and GG-convex functions by use of an integral identity. Our results are the refinements and improvements of some previously results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Trapezoid type inequalities for generalized fractional integral and related inequalities via exponentially convex functions. A novel and new approach is used to obtain these results for general Riemann Liouville fractional integrals. Various special cases are briefly discussed as applications of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we present new classes of ϕ-geometrically log hconvex mappings in the first sense and in the second sense and establish Hermite-Hadamard(H-H) type inequality for these classes. It reveals that the class of generalized geometrically log h-convex mappings in both senses include several new and known classes of log h- convex mappings. Results obtained in this article can be viewed as a new contributions in this area of research. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new class of generalized convex functions involving two arbitrary auxiliary functions h1; h2 : I = [a; b] ⊆ ℝ ￫ ℝ; which is called generalized (h1; h2) convex functions. We obtain several new classes of convex functions as special cases. We derive some new integral inequalities for generalized convex functions. Several special cases are also discussed. Results proved in this paper can be viewed as new significant contributions in this area. [ABSTRACT FROM AUTHOR]

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In this paper, we consider a newly introduced class of convex functions that is η-convex functions. We give some new quantum analogues for Hermite-Hadamard, Iynger and Ostrowski type inequalities via η-convex functions. Some special cases are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new quantum analogues of Hermite–Hadamard and Iyengar type inequalities for some classes of preinvex functions. Some special cases are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we derive some Newton's type of integral inequalities for geometrically relative convex functions. Some special cases are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of coordinated harmonically convex functions. We derive some new integral inequalities of Hermite–Hadamard type for coordinated Harmonically convex functions. The interested readers are encouraged to find the applications of harmonically convex functions in pure and applied sciences. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some new Hermite–Hadamard type inequalities for harmonic h -convex functions involving hypergeometric functions. We also discuss some new and known special cases, which can be deduced from our results. The ideas and techniques of this paper may inspire further research in this field. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce the notion "logarithmically h-preinvex functions", reveal that the class of h-preinvex functions include several new and known classes of preinvex functions, and establish several integral inequalities of Hermite-Hadamard type. [ABSTRACT FROM AUTHOR]

In this paper, some new classes of harmonically convex functions are introduced and investigated. We derive several Hermite-Hadamard inequalities for these new classes of harmonically convex functions. The ideas and techniques of this paper may be extended for other classes of convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider φ-convex functions which were introduced and studied by Noor [Some new classes of nonconvex functions, Nonl. Funct. Anal. Appl., 11(2006), No. 1, 165-171.]. We prove several Hermite-Hadamard type integral inequalities for φ-convex functions via fractional integrals. Our results include several new and known results as special cases. Results proved in this paper represent refinement and improvement of the previously known results. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce a notion 'log - h-convex functions' and establish several Hermite-Hadamard type integral inequalities for this kind of functions. [ABSTRACT FROM AUTHOR]

It is well known that the nonconvex variational inequalities are equivalent to the fixed point problems. We use this equivalent alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational inequalities. We discuss the convergence of the iterative method under suitable conditions. We also introduce a new class of Wiener -- Hopf equations. We establish the equivalence between the nonconvex variational inequalities and the Wiener -- Hopf equations. This alternative equivalent formulation is used to suggest some iterative methods. We also consider the convergence analysis of these iterative methods. Our method of proofs is very simple compared to other techniques. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and consider a new class of quasi variational inequalities, which is called the nonconvex quasi variational inequality. We show that the projection method can be used to establish the equivalence between the nonconvex quasi variational inequalities and the fixed points problems and the Wiener-Hopf equations. We use this equivalent alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational inequalities. We discuss the convergence of the iterative method under suitable conditions conditions. Our method of proofs is very simple as compared with other techniques. [ABSTRACT FROM AUTHOR]

The main motivation of this article is derive a new post-quantum integral identity using twice (p, q)-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard's inequality essentially via the class of ψ-preinvex functions. To support our results, we offer some applications to a special means of positive real numbers and twice (p, q)-differentiable functions that are in absolute value bounded as well. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to derive some new generalizations of Ostrowski type inequalities for the functions whose first derivatives absolute value are harmonically convex. We also discuss some special cases of the obtained results. In the last section, we present some applications of the obtained results. [ABSTRACT FROM AUTHOR]

We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, (k , s) -fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes. [ABSTRACT FROM AUTHOR]

In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function Ψ. The authors prove several inequalities for newly defined GPF-integral with respect to another function Ψ. Our consequences will give noted outcomes for a suitable variation to the GPF-integral in the sense of another function Ψ and the proportionality index ς. Furthermore, we present the application of the novel operator with several integral inequalities. A few new properties are exhibited, and the numerical approximation of these new operators is introduced with certain utilities to real-world problems. [ABSTRACT FROM AUTHOR]

In the article, we establish some new general fractional integral inequalities for exponentially m-convex functions involving an extended Mittag-Leffler function, provide several kinds of fractional integral operator inequalities and give certain special cases for our obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of relative harmonic m-convex functions with respect to an arbitrary non-negative function. New integral inequalities are obtained via the relative harmonic m-convex functions. Some special cases are considered which can be obtained from our main results. The ideas and technique of this paper motivate further research in this fascinating are. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to derive some new integral inequalities related to Hermite-Hadamard type by using Riemann-Liouville fractional integral operator for the class of exponentially harmonically convex functions. The formal technique of this paper may enhance further research in this field. [ABSTRACT FROM AUTHOR]