The present study derives novel Boole’s formula–type inequalities that address convex functions in the framework of quantum calculus, a growing domain that broadens classical analysis via a parameter \(q\) . We first formulate a unique quantum integral identity for \(q\) -differentiable convex functions, which operates as a fundamental result for deriving an extensive class of Boole’s formula–type integral inequalities in the quantum calculus setting. These inequalities not only generalize classical findings but also reveal unique fundamental aspects of convex functions via quantum differentiation and integration, thus improving the theoretical basis of both convex analysis and quantum calculus. In addition to theoretical results, we demonstrate the practical relevance of the proposed inequalities by integrating them special means of real numbers and the Mittag-Leffler function which is pivotal in generalizing the exponential functions. In order to verify the theoretical findings, we provide several numerical examples with computational and graphical insights. The interaction between abstract theory and computational practice highlights the wider significance of our results in numerical analysis and inequality theory. These examples endorse the accuracy and validity of the presented results, highlighting their potential applicability in tackling real-world problems where quantum calculus offers a more adequate platform than traditional calculus. This work strengthens the understanding of Boole-type inequalities in quantum and traditional calculus settings and provides avenues for their application in modern computational mathematics research.
