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This article is mainly concerned to link the Hermite-Hadamard and the Jensen-Mercer inequalities by using majorization theory and fractional calculus. We derive the Hermite-Hadamard-Jensen-Mercer-type inequalities in conticrete form, which serve as both discrete and continuous inequalities at the same time, for majorized tuples in the framework of the famous Atangana-Baleanu fractional operators. Also, the main inequalities encompass the previously established inequalities as special cases. Using generalized Mercer’s inequality, we also investigate the weighted forms of our major inequalities for certain monotonic tuples. Furthermore, the derivation of new integral identities enables us to construct bounds for the discrepancy of the terms concerning the main results. These bounds are constructed by incorporating the convexity of ∣f′∣| f^{\prime} | and ∣f′∣q(q>1){| f^{\prime} | }^{q}\hspace{0.33em}\left(q\gt 1) and making use of power mean and Hölder inequalities along with the established identities.

This document is a summary of a special issue of the journal "Fractal & Fractional" titled "Mathematical Inequalities in Fractional Calculus and Applications." The special issue contains fifteen articles that explore mathematical inequalities involving fractional derivatives and fractional integral operators and their applications in various mathematical and related fields. The articles cover a range of topics, including convexity, quantum inequalities, fuzzy-interval-valued functions, integral operators, and symmetric derivatives. The authors of the summary express their gratitude to the contributing authors and reviewers for their valuable contributions to the special issue. [Extracted from the article]

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right ℏ -convexity ( L R - ℏ -convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results' numerical validations that examples are nontrivial. By taking the product of two left and right coordinated ℏ -convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes. [ABSTRACT FROM AUTHOR]

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article's main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as U D - Ԓ -convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down ( U D ) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for Ԓ , we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard ( H H ) and Pachpatte types of inequalities using the concepts of coordinated U D - Ԓ -convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples. [ABSTRACT FROM AUTHOR]

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

In this paper, by using Jensen–Mercer's inequality we obtain Hermite–Hadamard–Mercer's type inequalities for a convex function employing left-sided (k ,   ψ) -proportional fractional integral operators involving continuous strictly increasing function. Our findings are a generalization of some results that existed in the literature. [ABSTRACT FROM AUTHOR]

It is the purpose of this paper to propose a novel class of convex functions associated with strong η -convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance of the new class of functions, some examples are presented. Additionally, the famous Hermite–Hadamard inequality is derived for this generalized family of convex functions. Furthermore, some inequalities and results for strong η -convex function are also derived. We anticipate that this new class of convex functions will open the research door to more investigations in this direction. [ABSTRACT FROM AUTHOR]

A generalization of the Mercer type inequality, for strongly convex functions with modulus c > 0, is hereby established. Let h : [δ, ζ] → R be a strongly convex function on the interval [δ, ζ] ⊂ R. Let a = (a1, ...., as), b = (b1, ...., bs) and p = (p1, ...., ps), where ak, bk ∈ [δ, ζ], pk > 0 for each k = 1, s. If n ∈ Rs, = 0 and under some separability assumptions, then we prove that s∑ l=1 plh(bl) ≤ s∑ l=1 plh(al) − c s∑ l=1, s(al − bl) ². Using the above result, we derive loads of inequalities for similarly separable vectors. We further applied our results to different types of tuples. Our results extend, complement and generalize known results in the literature. [ABSTRACT FROM AUTHOR]

In this work, we establish some new midpoint and trapezoidal type inequalities for prequasiinvex functions via the Katugampola fractional integrals. Some of the results obtained in this paper are generalizations of some earlier results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we give new Simpson's type integral inequalities for the class of functions whose derivatives of absolute values are s-convex via generalized proportional fractional integrals. Some results in the literature are particular cases of our results. [ABSTRACT FROM AUTHOR]

An implicit time–fractal–fractional differential equation involving the Atangana's fractal–fractional derivative in the sense of Caputo with the Mittag–Leffler law type kernel is studied. Using the Banach fixed point theorem, the well-posedness of the solution is proved. We show that the solution exhibits an exponential growth bound, and, consequently, the long-time (asymptotic) property of the solution. We also give examples to illustrate our problem. [ABSTRACT FROM AUTHOR]

We introduce the notion of m-polynomial harmonically convex interval-valued function. A relationship between a given interval-valued function and its component real-valued functions is pointed out. Moreover, some new Hermite-Hadamard type results are established for this class of functions. In particular, we show that if a nonnegative interval-valued function F, defined on a harmonically convex set S, is m-polynomial harmonically convex with α < β and α; β ∈ S, then ... where F is Lebesgue integrable on [α; β]. Our results complement and extend existing results in the literature. By taking m ≥ 2, we derive loads of new and interesting inclusions. We anticipate that the idea outlined herein will trigger further investigations in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new generalized version of the quantum Montgomery identity. Using this identity, some new estimates, for the class of strongly preinvex functions of higher order, are established. Furthermore, novel inequalities are deduced from our main results as special cases and in addition, recapture some known results as well. We anticipate that results presented herein will trigger further interest in this direction. [ABSTRACT FROM AUTHOR]

A new identity, with two parameters for differentiable function with respect to another functions via generalized integral operators, is first obtained. By applying the established identity, the trapezium, midpoint and Simpson type integral inequalities pertaining to generalized (m, h1, h2) -preinvex functions are deduced. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of the our results to special means and new error estimates for the trapezium and midpoint quadrature formula have been presented. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we derived two Ostrowski-Grüss type inequalities on time scales involving a combination of Δ-integral means. One of the inequalities is sharp. We also obtained 2-dimensional Ostrowski type inequality involving a combination of Δ-integral means. Our results extend some known results in the literature. Furthermore, we apply our results to the continuous, discrete and quantum calculus to obtain some interesting inequalities in these directions. [ABSTRACT FROM AUTHOR]

In this paper, we use the newly developed technique of parameter function to obtain some new Ostrowski type inequalities for double integrals. For this, we first obtain a generalized Montgomery identity for double integral and then use it obtain two Ostrowski type inequalities. Our results generalize some of the results of Zheng in [27] and Liu et al. in [22]. Furthermore, we apply our results to different time scales to obtain some interesting results that may be applied in the study of difference and differential equations. [ABSTRACT FROM AUTHOR]

Some trapezoid inequalities were proved in 2004 by B. G. Pach-patte. It is our purpose in this paper to extend these results to time scales. Besides that, two more results are also established. Furthermore, we also apply our results to continuous and discrete calculus to derive some new inequalities as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we establish some novel Ostrowski-Grü ss, Čebyšev, and Trapezoid type inequalities involving functions whose second derivatives are bounded on time scales. We also give some other interesting inequalities as special cases of our results. [ABSTRACT FROM AUTHOR]

We introduced several new integral inequalities of the Hermite-Hadamard type for strongly η-convex functions via the Katugampola fractional integrals. Some results in the literature are particular cases of our results. [ABSTRACT FROM AUTHOR]

In 2016, some inequalities of the Ostrowski type for functions (of two variables) differentiable on the coordinates were established. In this paper, we extend these results to an arbitrary time scale by means of a parameter λ∈0,1. The aforementioned results are regained for the case when the time scale T=R. Besides extension, our results are employed to the continuous and discrete calculus to get some new inequalities in this direction. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a weighted Montgomery identity for k points and then use this identity to prove a new weighted Ostrowski type inequality. Our results boil down to the results of Liu and Ngô if we take the weight function to be the identity map. In addition, we also generalize an inequality of Ostrowski-Grüss type on time scales for k points. For k=2, we recapture a result of Tuna and Daghan. Finally, we apply our results to the continuous, discrete, and quantum calculus to obtain more results in this direction. [ABSTRACT FROM AUTHOR]

The theory of symmetry has a significant influence in many research areas of mathematics. The class of symmetric functions has wide connections with other classes of functions. Among these, one is the class of convex functions, which has deep relations with the concept of symmetry. In recent years, the Schur convexity, convex geometry, probability theory on convex sets, and Schur geometric and harmonic convexities of various symmetric functions have been extensively studied topics of research in inequalities. The present attempt provides novel portmanteauHermite–Hadamard–Jensen–Mercer-type inequalities for convex functions that unify continuous and discrete versions into single forms. They come as a result of using Riemann–Liouville fractional operators with the joint implementations of the notions of majorization theory and convex functions. The obtained inequalities are in compact forms, containing both weighted and unweighted results, where by fixing the parameters, new and old versions of the discrete and continuous inequalities are obtained. Moreover, some new identities are discovered, upon employing which, the bounds for the absolute difference of the two left-most and right-most sides of the main results are established. [ABSTRACT FROM AUTHOR]

Since not every quasiconvex function is convex, it is our purpose in this present paper to extend some already established inequalities of the Hermite--Hadamard--Fej´er type and its companions for convex functions to the class of η-quasiconvex functions. The new results obtained herein are in terms of the ω-Riemann--Liouville fractional integral operators and they reduce to inequalities for quasiconvex functions for a particular choice of the bifunction η. In addition, we apply some of our results to certain special means of positive real numbers to obtain more estimates in this regard. [ABSTRACT FROM AUTHOR]

In this paper, we introduced some new integral inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute values at certain powers are strongly η -convex functions via the Katugampola fractional integrals. [ABSTRACT FROM AUTHOR]

In this paper, we present some Ostrowski–Grüss-type inequalities on time scales for functions whose derivatives are bounded by functions for k points via a parameter. The 2D versions of these inequalities are also presented. Our results generalize some of the results in the literature. As a by-product, we apply our results to the continuous and discrete calculus to obtain some interesting inequalities in this direction. [ABSTRACT FROM AUTHOR]

Two Ostrowski-Grüss type inequalities for k points with a parameter [Formula: see text] are hereby presented. The first generalizes a recent result due to Nwaeze and Tameru, and the second extends the result of Liu et al . to k points. Many new interesting inequalities can be derived as special cases of our results by considering different values of λ and [Formula: see text]. In addition, we apply our results to the continuous, discrete, and quantum time scales to obtain several novel inequalities in this direction.