New Forms of the Open Newton-Cotes-Type Inequalities for a Family of the Quantum Differentiable Convex Functions

The main objective of this paper is to establish some new inequalities related to the open Newton-Cotes formulas in the setting of q-calculus. We establish a quantum integral identity first and then prove the desired inequalities for $q$-differentiable convex functions. These inequalities are useful for determining error bounds for the open Newton-Cotes formulas in both classical and $q$-calculus. This work distinguishes itself from existing studies by employing quantum operators, leading to sharper and more precise error estimates. These results extend the applicability of Newton-Cotes methods to quantum calculus, offering a novel contribution to the numerical analysis of convex functions. Finally, we provide mathematical examples and computational analysis to validate the newly established inequalities.