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    Error estimates for perturbed Milne-type inequalities by twice-differentiable functions using conformable fractional integrals
    (Springer, 2025) Unes, Esra; Demir, Izzettin
    Milne's inequality provides an upper bound for the error in definite integral approximations using Milne's rule, making it a useful tool for evaluating the rule's precision. For this reason, this inequality is widely applied in engineering, physics, and applied mathematics. Additionally, conformable fractional integral operators establish a stronger relationship between classical and fractional calculus, enhancing the modeling, analysis, and resolution of complex problems. Therefore, we focus on the study of conformable fractional integral operators and Milne-type inequalities, which have significant applications in various fields. In this study, we first obtain an integral identity involving conformable fractional integral operators and twice-differentiable functions. Building on this new identity, we develop various perturbed Milne-type integral inequalities for twice-differentiable convex functions. We also validate them numerically through examples, computational analysis, and visual representations. In conclusion, it is evident that our findings significantly enhance and expand upon prior findings regarding integral inequalities. In addition to improving the scope of previous discoveries, the obtained results offer meaningful approaches and methods for tackling mathematical and scientific issues.
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    Fractional integral approaches to weighted corrected Euler-Maclaurin-type inequalities for different classes of functions
    (Pergamon-Elsevier Science Ltd, 2025) Demir, Izzettin; Unes, Esra
    In recent years, a wide variety of integral inequalities, including Newton-type, Simpson-type, and corrected Euler-Maclaurin-type inequalities, have been extensively studied, particularly in the framework of fractional calculus using Riemann-Liouville or conformable fractional integrals. Among these, fractional corrected Euler-Maclaurin-type inequalities have emerged as a valuable tool due to their improved approximation capabilities. In this study, we focus on developing weighted corrected Euler-Maclaurin-type inequalities for different classes of functions using Riemann-Liouville fractional integrals. To achieve this, we first derive a key integral equality with the aid of a positive weighted function, providing the foundation for the primary outcomes. Through the use of this integral equality, we prove new inequalities for differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. Also, for better explanation, we offer some examples together with their matching graphs. Moreover, these findings extend previous results. Consequently, the study clarifies the significance of corrected Euler-Maclaurin-type inequalities and suggests opportunities for further exploration.