Convexity plays a crucial role in the development of fractional integral inequalities. Many fractional integral inequalities are derived based on convexity properties and techniques. These inequalities have several applications in different fields such as optimization, mathematical modeling and signal processing. The main goal of this article is to establish a novel and generalized identity for the Caputo–Fabrizio fractional operator. With the help of this specific developed identity, we derive new fractional integral inequalities via exponential convex functions. Furthermore, we give an application to some special means.

Fractional calculus provides a significant generalization of classical concepts and overcomes the limitation of classical calculus in dealing with non-differentiability function. Implementing fractional operator to obtain new versions of classical outcomes is very intriguing topic of research in the mathematical analysis. The objective of the present study is to establish novel Hermite-Hadamard integral inequalities for twice differentiable function using Caputo-Fabrizio integral operator. In order to complete task, we start by demonstrating a new identity for Hermite-Hadamard inequality that serve as supporting result for our main finding. It has been observed that the obtained Hermite-Hadamard type inequalities have a relationship with previous results. In addition, we provide application to special means and graphical analysis to evaluate the accuracy of our results.

Several scholars are interested in fractional operators with integral inequalities. Due to its characteristics and wide range of applications in science, engineering fields, artificial intelligence and frac-tional inequalities should be employed in mathematical investigations. In this paper, we establish the new identity for the Caputo-Fabrizio fractional integral operator. By utilizing this identity, the generalization of Si mpsonty pein equalityfo r(? ,m )-convex functions via the Caputo-Fabrizio fractional integral operator. Furthermore, we also include the applications to special means, q-digamma functions, Simpson formula, Matrix inequalities, Modified Bessel function, and mind-point formula. These applications have given a new dimension to scholars.

Fractional calculus has been a concept used to obtain new variants of some well-known integral inequalities. In this study, our main goal is to establish the new fractional Hermite–Hadamard, and Simpson’s type estimates by employing a differentiable function. Furthermore, a novel class of fractional integral related to prominent fractional operator (Caputo–Fabrizio) for differentiable convex functions of first order is proven. Then, taking this equality into account as an auxiliary result, some new estimation of the Hermite–Hadamard and Simpson’s type inequalities as generalization is presented. Moreover, few inequalities for concave function are obtained as well. It is observed that newly established outcomes are the extension of comparable inequalities existing in the literature. Additionally, we discuss the applications to special means, matrix inequalities, and the q-digamma function.

Fractional integral operators have been studied extensively in the last few decades, and many different types of fractional integral operators have been introduced by various mathematicians. In 1967 Michele Caputo introduced Caputo fractional derivatives, which defined one of these fractional operators, the Caputo Fabrizio fractional integral operator. The main aim of this article is to established the new integral equalities related to Caputo-Fabrizio fractional integral operator. By incorporating this identity and convexity theory to obtained a novel class of Simpson type inequality. In this paper, we present a novel generalization of Simpson type inequality via s-convex and quasi-convex functions. Then, utilizing this identity the bounds of classical Simpson type inequality is improved. Finally, we discussed some applications to Simps,on s quadrature rule.

Fractional integral operators have indeed been the subject of significant research in various mathematical and scientific disciplines over the past few decades. The main aim of this article is to establish a new identity employing the Atangana Baleanu fractional integral operator for the case of differentiable functions. Moreover, we present several fractional Milne-type inequalities for bounded function by fractional integrals. Furthermore, we obtain fractional Milne-type inequalities for the case of Lipschitzian functions. Lastly, we explore applications related to special means, and quadrature formulas.