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| A Novel Variant of Milne's Rule Inequalities on Quantum Calculus for Convex Functions with their Computational Analysis |
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| KeyWord:Milne's inequality, $q$-calculus, convex functions |
| Author Name | Affiliation | | Wali Haider | School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China | | H\"{u}seyin Budak | Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, T\"{u}rkiye | | Asia Shehzadi | School of Mathematics and Statistics, Central South University, Changsha 410083, China | | Fatih Hezenci | Department of Mathematics, Faculty of Science and Arts, D\"{u}zce University, D\"{u}zce-T\"{u}rkiye | | Haibo Chen | School of Mathematics and Statistics, Central South University, Changsha 410083, China |
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| 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. |
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