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Öğe Advancements in Hermite-Hadamard inequalities via conformable fractional integrals for subadditive functions(World Scientific Publ Co Pte Ltd, 2025) Haider, Wali; Budak, Huseyin; Shehzadi, Asia; Hezenci, Fatih; Chen, HaiboThis study advances Hermite-Hadamard inequalities for subadditive functions using conformable fractional integrals. It establishes and explores numerous versions of these inequalities, as well as fractional integral inequalities for the product of two subadditive functions via conformable fractional integrals. The findings indicate that these inequalities improve and extend prior results for convex and subadditive functions, significantly enhancing the theoretical framework of fractional calculus and inequality theory. Moreover, computational analysis is conducted on these inequalities for subadditive functions, and mathematical examples are given to validate the newly established results within the framework of conformable fractional calculus.Öğe Analysing Milne-type inequalities by using tempered fractional integrals(Springer Basel Ag, 2024) Haider, Wali; Budak, Huseyin; Shehzadi, Asia; Hezenci, Fatih; Chen, HaiboIn this research, we define an essential identity for differentiable functions in the framework of tempered fractional integral. By utilizing this identity, we deduce several modifications of fractional Milne-type inequalities. We provide novel expansions of Milne-type inequalities in the domain of tempered fractional integrals. The investigation emphasises important functional categories, including convex functions, bounded functions, Lipschitzian functions, and functions with bounded variation.Öğe A comprehensive study on Milne-type inequalities with tempered fractional integrals(Springer, 2024) Haider, Wali; Budak, Hüseyin; Shehzadi, Asia; Hezenci, Fatih; Chen, HaiboIn the framework of tempered fractional integrals, we obtain a fundamental identity for differentiable convex functions. By employing this identity, we derive several modifications of fractional Milne inequalities, providing novel extensions to the domain of tempered fractional integrals. The research comprehensively examines significant functional classes, including convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation.Öğe Fractional Euler-Maclaurin-type inequalities for twice-differentiable functions(Springer, 2025) Shehzadi, Asia; Budak, Huseyin; Haider, Wali; Hezenci, Fatih; Chen, HaiboThis article establishes a novel equality for twice-differentiable functions with convex absolute values in their second derivatives. This equality is used to establish Euler-Maclaurin-type inequalities through Riemann-Liouville fractional integrals. By utilizing convexity, the power mean inequality, and the H & ouml;lder inequality, several significant fractional inequalities can be derived. Moreover, the recently derived inequalities are not only grounded in theory but are also accompanied by concrete instances to further solidify their validity.Öğe Fractional Milne-type inequalities for twice differentiable functions for Riemann-Liouville fractional integrals(Springer Basel Ag, 2024) Haider, Wali; Budak, Huseyin; Shehzadi, AsiaIn this research, we investigate the error bounds associated with Milne's formula, a well-known open Newton-Cotes approach, initially focused on differentiable convex functions within the frameworks of fractional calculus. Building on this work, we investigate fractional Milne-type inequalities, focusing on their application to the more refined class of twice-differentiable convex functions. This study effectively presents an identity involving twice differentiable functions and Riemann-Liouville fractional integrals. Using this newly established identity, we established error bounds for Milne's formula in fractional and classical calculus. This study emphasizes the significance of convexity principles and incorporates the use of the H & ouml;lder inequality in formulating novel inequalities. In addition, we present precise mathematical illustrations to showcase the accuracy of the recently established bounds for Milne's formula.Öğe Generalizations Euler-Maclaurin-type inequalities for conformable fractional integrals(Univ Nis, Fac Sci Math, 2025) Haider, Wali; Budak, Huseyin; Shehzadi, Asia; Hezenci, Fatih; Chen, HaiboIn this study, we obtain a unique insight into differentiable convex functions by employing newly defined conformable fractional integrals. With this innovative approach, we unveil fresh Euler-Maclaurintype inequalities designed specifically for these integrals. Our proofs draw on fundamental mathematical principles, including convexity, Holder's inequality, and power mean inequality. Furthermore, we delve into new inequalities applicable to bounded functions, Lipschitzian functions, and functions of bounded variation. Notably, our findings align with established results under particular circumstances.Öğe Milne-type inequalities for co-ordinated convex functions(Univ Nis, Fac Sci Math, 2024) Shehzadi, Asia; Budak, Huseyin; Haider, Wali; Chen, HaiboIn this research, our objective is to formulate a unique identity for Milne-type inequalities involving for functions of two variables having convexity on co-ordinates over [mu, v] x [omega, kappa]. By employing this identity, we establish some new inequalities of the Milne-type for co-ordinated convex functions. Furthermore, the propose identity strengthens the theoretical basis of mathematical inequalities showcasing its significance in various fields.Öğe Novel Fractional Boole's-Type Integral Inequalities via Caputo Fractional Operator and Their Implications in Numerical Analysis(Mdpi, 2025) Haider, Wali; Mateen, Abdul; Budak, Hueseyin; Shehzadi, Asia; Ciurdariu, LoredanaThe advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n-times fractional Boole's-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Boole's-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Boole's formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations.Öğe Numerical Approximations and Fractional Calculus: Extending Boole's Rule with Riemann-LiouvilleFractional Integral Inequalities(Mdpi, 2025) Mateen, Abdul; Haider, Wali; Shehzadi, Asia; Budak, Huseyin; Bin-Mohsin, BandarThis paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann-Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann-Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole's formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings.Öğe On new versions of Milne-type inequalities based on tempered fractional integrals(World Scientific Publ Co Pte Ltd, 2025) Shehzadi, Asia; Budak, Huseyin; Haider, Wali; Hezenci, Fatih; Chen, HaiboThis investigation reveals significant identity related to the Milne-type inequalities. Utilizing this identity, we derive Milne-type inequalities by incorporating differentiable convex mappings, including tempered fractional integrals. Our strategy involves delving into notable functional categories such as convex, bounded, Lipschitzian, and functions with bounded variation. What's more, new findings are achieved through special choices.
