Weighted Inequalities of Milne‐Type and Applications: Bridging Classical and Fractional Integrals

In this study, weighted Milne‐type inequalities are derived for classical integrals, and by selecting special weight functions, several Milne‐type inequalities are presented for well‐known fractional integrals, such as Riemann–Liouville fractional integrals, conformable fractional integrals, and tempered fractional integrals. This is one of the first studies in the literature in this regard. In particular, we first present integral equalities essential to establishing the main results, using nonnegative weighted functions. By using these identities, we establish several weighted Milne‐type inequalities for diverse function classes, such as convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. We consider the applications of these results in error estimates for the weighted Milne integral formula and in deriving inequalities that involve higher moments of random variables. The findings contribute to extending existing inequalities in the literature and provide a deeper understanding of their applications in fractional calculus. This work also explores the potential for future research directions based on the established inequalities and their broader implications in mathematical analysis and its applications.