Beyond Classical Milne's Scheme: Fractional Error Analysis Via Modified k-AB Integral Operators

The principal motive of the current study is to determine several new fractional bounds of Milne's inequality by utilizing different classes of functions. We initially start our investigation by proving an important identity for differentiable functions by using k-AB fractional integral operator. This identity unifies many of the existing lemmas. Relying on this identity and the invexity of functions, we generate various fractional error bounds for Milne's rule. Through the utilization of bounded functions, Lipschitzian functions, and the bounded variation characteristic of functions, we construct a few other interesting relations. In favour of our essential outcomes, we deliver interesting implications to special functions, numerical integration, and graphical and numerical procedures to highlight the efficacy of results.

کلیدواژه ها:

Milne's inequality ، preinvex functions ، Lipschitz function ، Function of bounded variation