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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]

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]

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]

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]