J. Nonl. Mod. Anal., 7 (2025), pp. 1727-1745.
Published online: 2025-09
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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.
}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2025.1727}, url = {http://global-sci.org/intro/article_detail/jnma/24386.html} }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.