Novel Grüss-Type and Related Integral Inequalities via the Cotangent Fractional Integral with Respect to Another Function

Abstract

This article investigates new classes of integral inequalities within the framework of a recently introduced fractional operator: the cotangent fractional integral with respect to a rising function (RAF) ℵ. The cornerstone of this study is the derivation of a generalized Grüss-type inequality, along with several related refinements, employing this novel operator. Our methodology strategically combines the properties of the cotangent fractional integral with classical analytical tools, including Young’s inequality and weighted arithmetic–geometric mean arguments. The results obtained are highly versatile, as they not only provide new bounds for this specific operator but also, through appropriate choices of the function ℵ and the parameter r1, reduce elegantly to corresponding inequalities for well-known fractional integrals, such as those of Riemann–Liouville (RL), Hadamard, and Katugampola. Several corollaries are presented to illustrate these connections and recover existing results from the literature, thereby demonstrating the unifying nature of our approach.