Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis

This paper presents new integral inequalities for differentiable generalized convex functions in the second sense, with a focus on improving the accuracy of Weddle’s formula for numerical integration. The study is motivated by the following three key factors: the generalization of convexity through s-convex functions, the enhancement of the approximation quality, particularly as s→0+, and the effectiveness of Weddle’s formula in cases where Simpson’s 1/3 rule fails. An integral identity is derived for differentiable functions, which is then used to establish sharp error bounds for Weddle’s formula under s-convexity. Numerical examples and comparative tables demonstrate that the proposed inequalities yield significantly tighter bounds than those based on classical convexity. Applications to numerical quadrature highlight the practical utility of the results in computational mathematics.