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<p>In this study, we propose a novel method for establishing Milne’s rule-type inequalities within the context of quantum calculus applied to differentiable convex functions. Initially, we obtain a quantum integral identity, which serves as the foundation for deriving several new Milne’s rule inequalities tailored for quantum differentiable convex functions. These inequalities are particularly relevant in Open-Newton’s Cotes formulas, facilitating the determination of bounds for Milne’s rule in both classical and <i>q</i>-calculus domains. Additionally, we conduct computational analysis on these inequalities for convex functions and present mathematical examples and graphical representation to demonstrate the validity of our newly established results within the realm of <i>q</i>-calculus.</p>

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New perspectives on Milne’s rule inequalities with their computational analysis via quantum calculus

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Wali Haider,
  • Hüseyin Budak,
  • Asia Shehzadi,
  • Fatih Hezenci,
  • Hai-bo Chen

摘要

In this study, we propose a novel method for establishing Milne’s rule-type inequalities within the context of quantum calculus applied to differentiable convex functions. Initially, we obtain a quantum integral identity, which serves as the foundation for deriving several new Milne’s rule inequalities tailored for quantum differentiable convex functions. These inequalities are particularly relevant in Open-Newton’s Cotes formulas, facilitating the determination of bounds for Milne’s rule in both classical and q-calculus domains. Additionally, we conduct computational analysis on these inequalities for convex functions and present mathematical examples and graphical representation to demonstrate the validity of our newly established results within the realm of q-calculus.