New Fractal-Fractional Trapezoidal and Simpson's Formulas with Associated Inequalities on Fractal Sets

This paper addresses the limitations in existing trapezoidal and Simpson’s-type inequalities, which are built on assumptions incompatible with fractal-fractional calculus. To resolve this, we first construct a novel Lagrange interpolating polynomial tailored to fractal domains, then use it to rigorously derive revised trapezoidal and Simpson’s formulas for fractal-fractional integrals, along with their associated inequalities. The results establish a mathematically sound framework, with comparative analysis showing that the proposed formulas yield significantly sharper error bounds and more accurate approximations than existing methods, providing a reliable foundation for future work in fractal-fractional numerical analysis and inequality theory.