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    New fractional integral inequalities for preinvex functions involving Caputo-Fabrizio operator
    (AIMS Press, 2021) Tariq, Muhammad; Ahmad, Hijaz; Shaikh, Abdul Ghafoor; Sahoo, Soubhagya Kumar; Khedher, Khaled Mohamed; Gia, Tuan Nguye
    It’s undeniably true that fractional calculus has been the focus point for numerous researchers in recent couple of years. The writing of the Caputo-Fabrizio fractional operator has been on many demonstrating and real-life issues. The main objective of our article is to improve integral inequalities of Hermite-Hadamard and Pachpatte type incorporating the concept of preinvexity with the Caputo-Fabrizio fractional integral operator. To further enhance the recently presented notion, we establish a new fractional equality for differentiable preinvex functions. Then employing this as an auxiliary result, some refinements of the Hermite-Hadamard type inequality are presented. Also, some applications to special means of our main findings are presented.
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    Some Novel Fractional Integral Inequalities over a New Class of Generalized Convex Function
    (MDPI, 2022) Sahoo, Soubhagya Kumar; Tariq, Muhammad; Ahmad, Hijaz; Kodamasingh, Bibhakar; Shaikh, Asif Ali; Botmart, Thongchai; El-Shorbagy, Mohammed A.
    The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity.