Some Milne's Rule Type Inequalities for Convex Functions with Their Computational Analysis On Quantum Calculus

Abstract

In this paper, we establish some new Milne's rule type inequalities for the first differentiable convex functions in $q$-calculus. For this, we prove a quantum integral identity first and then we prove some new Milne's rule type inequalities for quantum differentiable convex functions. These inequalities play an important role in Open-Newton's Cotes formula because, with the help of these inequalities, we can find the error bounds of Milne's rule for differentiable convex functions in quantum or classical calculus. Furthermore, we give the computational analysis of these inequalities for convex functions and proved that the bounds of this paper are better than the existing ones. Ultimately, we provide some mathematical examples to show the validity of newly established inequalities in quantum calculus.