Abstract

Quantum calculus is a powerful extension of classical calculus, providing novel tools for deriving sharper and more efficient analytical results without relying on limits. This study investigates error estimations for Milne formula-type inequalities within the framework of quantum calculus, offering a fresh perspective on numerical integration theory. New variants of Milne’s formula-type inequalities are established for q-differentiable convex functions by first deriving a key quantum integral identity. The primary aim of this work is to obtain sharper and more accurate bounds for Milne’s formula compared to existing results in the literature. The validity of the proposed results is demonstrated through illustrative examples and graphical analysis. Furthermore, applications to special means of real numbers, the Mittag–Leffler function, and numerical integration formulas are presented to emphasize the practical significance of the findings. This study contributes to advancing the theoretical foundations of both classical and quantum calculus and enhances the understanding of integral inequality theory.