A Family of Newton–Cotes‐Type Inequalities in Multiplicative Calculus With Their Applications to Quadrature Formulas and Numerical Analysis

ABSTRACT This study introduces a novel class of the Newton–Cotes‐type inequalities derived from a parameterized identity within the framework of multiplicative calculus. These inequalities provide an innovative approach to integral approximation, refining existing results for specific parameter choices, including the Midpoint, Trapezoidal, Simpson's, Newton's, Maclaurin's, and Weddle's formulas. This development is particularly significant in numerical analysis, where precise integral approximations are critical, such as solving differential equations and implementing methods like the finite volume method. The paper highlights that multiplicative calculus delivers more accurate absolute error bounds than classical calculus, especially for higher‐degree polynomials. Notably, the proposed inequalities streamline computations by eliminating the need to derive each inequality individually. By adjusting parameters, these results allow for efficient determination of error bounds for various quadrature formulas. Applications to numerical integration and special means for real numbers are explored, demonstrating the practical utility and accuracy of the proposed inequalities. A detailed mathematical example, supported by 2D and 3D visualizations, further validates the performance of these results. This work advances numerical methods and provides valuable insights into applying the Newton–Cotes‐type inequalities in mathematical research, computation, and integration techniques.