On some novel Boole’s type inequalities for conformable fractional integrals with their applications

The extension of classical numerical quadrature rules, such as Boole’s rule, into the fractional calculus framework is an essential step for enhancing computational methods in applied mathematics. However, a significant challenge has been the development of sharp error bounds that are applicable under minimal differentiability conditions, particularly for the rapidly evolving field of Conformable fractional integrals. This work addresses this gap by first establishing a novel Boole-type identity within the Conformable fractional paradigm. This foundational identity serves as the critical tool to derive new and refined inequalities specifically designed for convex functions. A primary objective and achievement of this research is the determination of improved error bounds that require only a single time differentiability, significantly broadening their applicability. Our results not only generalize existing theorems but also provide sharper estimates for numerical integration. To ensure the validity and applicability of our proposed results, we demonstrate a brief numerical investigation, which not only supports the theoretical findings but also illustrates their relevance to practical and computational problems. Moreover, the theoretical inequalities are then applied to the error analysis of quadrature formulas and to derive new relationships for Mittag-Leffler function and special means for real numbers. Our findings extend classical Boole’s rule to the fractional setting and provide tighter error bounds than already existed in the literature.