Refined Inequalities of Perturbed Simpsons Type for Higher-Order Absolutely Continuous Functions and Applications

In this paper, we first establish a new identity for higher-order differentiable functions. By employing this identity, we derive new integral inequalities for different classes of functions whose derivatives satisfy various regularity conditions. In addition, based on the obtained results, error estimates for illustrative examples of functions from different classes are examined with respect to their derivative orders, and the corresponding approximation graphs are constructed and analyzed to provide further insights. The obtained results not only provide new perspectives on Simpson-type inequalities but also extend their applicability to a broader range of functions.