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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]

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]

The main aim of the present article is to introduce some new ∇-conformable dynamic inequalities of Hardy type on time scales. We present and prove several results using chain rule and Fubini's theorem on time scales. Our results generalize, complement, and extend existing results in the literature. Many special cases of the proposed results, such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities, and new classical conformable fractional integral inequalities, are obtained and analyzed. [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]

We introduce a new class of functions called strongly (η , ω) -convex functions. This class of functions generalizes some recently introduced notions of convexity, namely, the η-convex functions and strongly η-convex functions. We also establish inequalities of the Hermite–Hadamard–Fejér's type, which generalize results of Delavar and Dragomir (Math. Inequal. Appl. 20(1):203–216, 2017) and Awan et al. (Filomat 31(18):5783–5790, 2017). In addition, we obtain some new results for this class of functions. Finally, we apply our results to the k-Riemann–Liouville fractional integral operators to obtain more results in this direction. [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]

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]

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]

The notion of m-polynomial convex interval-valued function Ψ = [ ψ − , ψ + ] is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions ψ − and ψ + . For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, ρ , ϵ > 0 and ζ , η ∈ S , then m m + 2 − m − 1 Ψ (ζ + η 2) ⊇ Γ ρ (ϵ + ρ) (η − ζ) ϵ ρ [ ρ J ζ + ϵ Ψ (η) + ρ J η − ϵ Ψ (ζ) ] ⊇ Ψ (ζ) + Ψ (η) m ∑ p = 1 m S p (ϵ ; ρ) , where Ψ is Lebesgue integrable on [ ζ , η ] , S p (ϵ ; ρ) = 2 − ϵ ϵ + ρ p − ϵ ρ B (ϵ ρ , p + 1) and B is the beta function. We extend, generalize, and complement existing results in the literature. By taking m ≥ 2 , we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation. [ABSTRACT FROM AUTHOR]

Using the newly introduced fractional integral operators in (Fasc. Math. 20(4):5-27, 2016) and (East Asian Math. J. 21(2):191-203, 2005), we establish some novel inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute value are η-quasiconvex. Results obtained herein give a broader generalization to some existing results in the literature by choosing appropriate values of the parameters under consideration. We apply our results to some special means such as the arithmetic, geometric, harmonic, logarithmic, generalized logarithmic, and identric means to obtain more results in this direction. [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]