New Error Bounds for Boole‐Type Inequalities via Third‐Order Derivatives With Applications in Special Means

In mathematical analysis, inequalities are essential tools for evaluating and approximating the behavior of functions, particularly in the study of convex functions. In this work, we present, for the first time, an analysis of error bounds associated with Boole‐type inequalities for three‐times‐differentiable convex functions in the context of Riemann integrals. Building on this foundation, we derive a new family of Boole‐type inequalities for a broader class of third‐time differentiable ‐convex functions using the Riemann integral. Using a newly established integral identity, further Boole‐type inequalities are obtained by incorporating a summation parameter for ‐convex, convex, and ‐function classes. We also consider cases where the third derivative is bounded, Lipschitz, and a function of bounded variation. The practical relevance of the obtained inequalities is illustrated through their application to special functions and modified Bessel functions, highlighting their usefulness in applied mathematical analysis. Moreover, a computational study is carried out to validate the theoretical findings. In particular, two‐dimensional graphical visualizations are presented to compare the classical first‐, second‐, and third‐order error bounds, clearly demonstrating the improved sharpness achieved through higher‐order differentiability and convexity assumptions. These results extend and generalize existing inequalities, offering both theoretical insights and practical significance in the study of integral inequalities.