Some Milne’s rule type inequalities in quantum calculus

Abstract

The main goal of the current study is to establish some new Milne?s rule type inequalities for single-time differentiable convex functions in the setting of quantum calculus. For this, we establish a quantum integral identity and then we prove some new inequalities of Milne?s rule type for quantum differentiable convex functions. These inequalities are very important in Open-Newton?s Cotes formulas because, with the help of these inequalities, we can find the bounds of Milne?s rule for differentiable convex functions in classical or quantum calculus. The method adopted in this work to prove these inequalities are very easy and less conditional compared to some existing results. Finally, we give some mathematical examples to show the validity of newly established inequalities.