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.