A Novel Variant of Milne’s Rule Inequalities on Quantum Calculus for Convex Functions with Their Computational Analysis

Abstract

In this investigation, we introduce a novel approach for establishing Milne’s type inequalities in the context of quantum calculus for differentiable convex functions. First, we prove a quantum integral identity. We derive numerous new Milne’s rule inequalities for quantum differentiable convex functions. These inequalities are relevant in open Newton-Cotes formulas, as they facilitate the determination of bounds for Milne’s rule applicable to differentiable convex functions in both classical and $q$-calculus. In addition, we conduct a computational analysis of these inequalities for convex functions and provide mathematical examples to demonstrate the validity of the newly established results within the framework of $q$-calculus.