Some New Fractional Boole’s Inequalities with Machine Learning Validations

The present work proposes a new class of weighted Boole-type fractional inequalities formulated by integrating the Riemann–Liouville (RL) fractional integral operators with symmetric weight functions. Unlike the classical Boole error estimate which relies on the boundedness of the sixth derivative, the present approach is based on a newly derived weighted fractional identity, which is the key analytical tool of the study. Based on this identity, a family of inequalities is established for several particular classes of functions: convex functions, s-convex functions, bounded functions, Lipschitz continuous functions and bounded-variation functions, which provides meaningful error bounds under substantially weaker smoothness hypotheses. The flexibility of the proposed framework is illustrated by providing some examples of the proposed inequality limit cases, showing that the derived inequalities unify and extend the existing ones in the literature. Also, a few interesting applications to special means and a composite quadrature rule are presented. To test the inequality, we built a computational framework and generated a structured dataset of 5400 samples using six convex functions with polynomial, exponential, and hyperbolic growth. For all samples generated, a value between the left-hand-side (LHS) and right-hand-side (RHS) is numerically verified and the slack variable is numerically kept non-negative in the whole computational domain. The good consistency obtained between data-driven predictions and the analytical results confirms the applicability of the proposed framework and provides a logical bridge between the fractional inequality theory, the numerical approximation, and the explainable artificial intelligence.