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    Generalizations of Simpson type inequality for (?, m)-convex functions
    (Univ Nis, Fac Sci Math, 2024) Munir, Arslan; Budak, Hüseyin; Faiz, Irza; Qaisar, Shahid
    Several scholars are interested in fractional operators with integral inequalities. Due to its characteristics and wide range of applications in science, engineering fields, artificial intelligence and fractional inequalities should be employed in mathematical investigations. In this paper, we establish the new identity for the Caputo-Fabrizio fractional integral operator. By utilizing this identity, the generalization of Simpson type inequality for ( alpha, m ) -convex functions via the Caputo-Fabrizio fractional integral operator. Furthermore, we also include the applications to special means, q -digamma functions, Simpson formula, Matrix inequalities, Modified Bessel function, and mind -point formula. These applications have given a new dimension to scholars.
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    A NOTE ON SIMPSON 3/8 RULE FOR FUNCTION WHOSE MODULUS OF FIRST DERIVATIVES ARE s-CONVEX FUNCTION WITH APPLICATION
    (Kangwon-Kyungki Mathematical Soc, 2024) Munir, Arslan; Budak, Huseyin; Kara, Hasan; Rathour, Laxmi; Faiz, Irza
    Researchers continue to explore and introduce new operators, methods, and applications related to fractional integrals and inequalities. In recent years, fractional integrals and inequalities have gained a lot of attention. In this paper, firstly we established the new identity for the case of differentiable function through the fractional operator (Caputo-Fabrizio). By utilizing this novel identity, the obtained results are improved for Simpson second formula-type inequality. Based on this identity the Simpson second formula-type inequality is proved for the s-convex functions. Furthermore, we also include the applications to special means.
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    Some new fractional corrected Euler-Maclaurin type inequalities for function whose second derivatives are s-convex function
    (Taylor & Francis Inc, 2024) Munir, Arslan; Vivas-Cortez, Miguel; Qayyum, Ather; Budak, Hüseyin; Faiz, Irza; Supadi, Siti Suzlin
    Fractional integrals and inequalities have gained a lot of attention in recent years. By introducing innovative analytical approaches and applications, and by applying these approaches, numerous forms of inequalities have been examined. In this paper, we establish new identity for the twice differentiable function where the absolute value is convex. By utilizing this identity, numerous Corrected Euler-Maclaurin-type inequalities are developed for the Caputo-Fabrizio fractional integral operator. Based on this identity, the Corrected Euler-Maclaurin-type inequalities for $s$s-convex function are obtained. By employing well-known inequalities such as H & ouml;lder's and Power -Mean, we are introduced several new error bounds and estimates for Corrected Euler-Maclaurin-type inequalities. Additionally, special cases of the present results are applied to obtain the previous well-known results.
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    Some New Improvements of Hermite-Hadamard Type Inequalities Using Strongly (s, m)-Convex Function with Applications
    (Univ Maragheh, 2025) Munir, Arslan; Budak, Huseyin; Kashuri, Artion; Faiz, Irza; Kara, Hasan; Qayyum, Ather
    The trapezoidal-type inequalities are discovered in this study using the fractional operator, which produces powerful results. We established a general identity for Caputo-Fabrizio integral operators and the second derivative function. Using this identity new error bounds and estimates for strongly (s, m)-convex functions are obtained. Moreover, some novel trapezoidal-type inequalities are offered taking this identity into account using the known inequalities like Young, Jensen, Holder and power-mean inequalities. Finally, we present some applications for matrix inequality, estimation error regarding trapezoidal formulas and special means for real numbers.
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    A study of Milne-type inequalities for several convex function classes with applications
    (Univ Nis, Fac Sci Math, 2024) Munir, Arslan; Qayyum, Ather; Budak, Huseyin; Faiz, Irza; Kara, Hasan; Supadi, Siti Suzlin
    Fractional integral operators have indeed been the subject of significant research in various mathematical and scientific disciplines over the past few decades. The main aim of this article is to establish a new identity employing the Atangana Baleanu fractional integral operator for the case of differentiable functions. Moreover, we present several fractional Milne-type inequalities for bounded function by fractional integrals. Furthermore, we obtain fractional Milne-type inequalities for the case of Lipschitzian functions. Lastly, we explore applications related to special means, and quadrature formulas.