Some New Boole-Type Inequalities via Modified Convex Functions with Their Applications and Computational Analysis

In numerical analysis, the Boole’s formula serves as a pivotal tool for approximating definite integrals. The approximation of the definite integrals has a big role in numerical methods for differential equations; in particular, in the finite volume method, we need to use the best approximation of the integrals to obtain better results. This paper presents a rigorous proof of integral inequalities for first-time differentiable s-convex functions in the second sense. This paper has two main goals. The first is that the use of s-convex function extends the results for convex functions which cover a large class of functions and the second is the best approximation. To prove the main inequalities, we drive integral identity for differentiable functions. Then, with the help of this identity, we prove the error bounds of Boole’s formula for differentiable s-convex functions in the second sense. Some new midpoint-type inequalities for generalized convex functions are also given which can help us in finding better error bounds for midpoint integration formulas compared to the existing ones. Moreover, we provide some applications to quadrature formulas and special means for the real numbers of these newly established inequalities. Furthermore, we present numerical examples and computational analysis that show that these newly established inequalities are numerically valid.