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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. [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]

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]

A new auxiliary result pertaining to twice ( q 1 , q 2 )-differentiable functions is derived. Using this new auxiliary result, some new versions of Hermite–Hadamard's inequality involving the class of generalized m -convex functions are obtained. Finally, to demonstrate the significance of the main outcomes, some applications are discussed for hypergeometric functions, Mittag–Leffler functions, and bounded functions. [ABSTRACT FROM AUTHOR]