Some new weighted Boole’s type inequalities for differentiable generalized convex functions with their applications

This paper presents a rigorous proof of integral inequalities for first-time differentiable h-convex functions. The use of the h-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using h-convexity. Initially, we derive a weighted Boole?s formula type integral identity for differentiable functions. Utilizing this novel identity, we subsequently establish weighted Boole?s formula type inequalities specifically developed for differentiable generalized convex function. We meticulously examine numerous special cases to provide comprehensive insights. These newly derived inequalities offer valuable tools for determining error bounds in various numerical integration techniques within classical calculus. To underscore the efficacy of our principal findings, we offer practical applications to weighted Boole?s type quadrature formulas, continuous random variables, and special means for real numbers. These approximations highlight their potential impact on computational mathematics and related fields. Furthermore, we provide numerical examples of newly established inequalities to demonstrate that the results presented in this paper are numerically valid.