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.