Perturbed Trapezoid Like Inequalities
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Keywords:
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Ostrowski Inequality, Euler-Maclaurin Summation Formula, Perturbed Trapezoid Type Inequality
Abstract
In our current research article, based on a general configuration of a 3-step Peano kernel, new versions of integral inequality of Ostrowski’s type are developed for differentiable mappings that have second derivatives belong to L∞. Then we utilized these versions to generate new perturbed trapezoid like inequalities. These new perturbed trapezoid like inequalities are proposed with error bounds smaller than and similar to those reported by previous studies. Moreover, some of the obtained perturbed trapezoid like inequalities reveal the relationship between the Euler-Maclaurin summation and the trapezoidal rule. Finally, certain implementations to numerical composite quadrature rules are provided for completeness.
References
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Alshanti, W. G. (2019). Inequality of Ostrowski Type for Mappings with Bounded Fourth Order Partial Derivatives. In Abstract and Applied Analysis, 2019; 1–6
Alshanti, W. G. (2021). Riemann-Stieltjes Integrals and Some Ostrowski Type Inequalities. The Australian Journal of Mathematical Analysis and Applications, 18(1); 1–11
Alshanti, W. G., A. Alshanty, A. Zraiqat, and I. H. Jebril (2022). Cubature Formula for Double Integrals Based on Ostrowski Type Inequality. International Journal of Difference Equations, 17(2); 379–387
Alshanti, W. G. and G. V. Milovanovic (2020). Double-Sided Inequalities of Ostrowski’s Type and Some Applications. Journal of Computational Analysisand Applications, 28(4); 724–736
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Batiha, I. M. (2011). Restriction method for approximating square roots. International Journal of Open Problems in Computer Science and Mathematics, 4(3); 146–151
Batiha, I. M., S. Alshorm, A. Ouannas, S. Momani, O. Y. Ababneh, and M. Albdareen (2022). Modified Three-Point Fractional Formulas with Richardson Extrapolation. Mathematics, 10(19); 3489
Dragomir, S. S. and A. Sofo (2000). An Integral Inequality for Twice Differentiable Mappings and Applications. Tamkang Journal of Mathematics, 31(4); 257–266
Liu, W., X. Gao, and Y. Wen (2016). Approximating the Finite Hilbert Transform Via Some Companions of Ostrowski’s Inequalities. Bulletin of the Malaysian Mathematical Sciences Society, 39; 1499–1513
Liu, W. and N. Lu (2015). Approximating the Finite Hilbert Transform Via Simpson Type Inequalities and Applications. Journal Scientific Bulletin Series A Applied Mathematics and Physics, 77(3); 107–122
Liu, W. and J. Park (2017a). A Companion of Ostrowski Like Inequality and Applications to Composite Quadrature Rules. Journal of Computational Analysis and Applications, 22(1); 19–24
Liu, W. and J. Park (2017b). Some Perturbed Versions of the Generalized Trapezoid Inequality for Functions of Bounded Variation. Journal of Computational Analysis and Applications, 22(1); 11–18
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Milovanović, G. V. (2017). Summation Formulas of Euler–Maclaurin and Abel–Plana: Old and New Results and Applications. Progress in Approximation Theory and Applicable Complex Analysis: In Memory of QI Rahman, 2017; 429–461
Milovanović, G. V. and J. E. Pečarić (1976). On Generalization of the Inequality of A. Ostrowski and Some Related Applications. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, (544/576); 155–158
Ðorđević, R. Ž. and G. V. Milovanović (1975). A Generalization of E. Landau’s Theorem. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, (498/541); 97–106
Ostrowski, A. (1937). Über Die Absolutabweichung Einer Differentiierbaren Funktion Von Ihrem Integralmittelwert. Commentarii Mathematici Helvetici, 10(1); 226–227
Vasić, P. M. and G. V. Milovanović (1976). On an Inequality of Iyengar. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, (544/576); 18–24
Albadarneh, R. B., I. Batiha, A. Alomari, and N. Tahat (2021a). Numerical Approach for Approximating the Caputo Fractional-Order Derivative Operator. AIMS Mathematics, 6(11); 12743–12756
Albadarneh, R. B., I. M. Batiha, A. Adwai, N. Tahat, and A. Alomari (2021b). Numerical Approach of RiemannLiouville Fractional Derivative Operator. International Journal of Electrical and Computer Engineering, 11(6); 5367–5378
Alshanti, W. G. (2018). A Perturbed Version of General Weighted Ostrowski Type Inequality and Applications. International Journal of Analysis and Applications, 16(4); 503–517
Alshanti, W. G. (2019). Inequality of Ostrowski Type for Mappings with Bounded Fourth Order Partial Derivatives. In Abstract and Applied Analysis, 2019; 1–6
Alshanti, W. G. (2021). Riemann-Stieltjes Integrals and Some Ostrowski Type Inequalities. The Australian Journal of Mathematical Analysis and Applications, 18(1); 1–11
Alshanti, W. G., A. Alshanty, A. Zraiqat, and I. H. Jebril (2022). Cubature Formula for Double Integrals Based on Ostrowski Type Inequality. International Journal of Difference Equations, 17(2); 379–387
Alshanti, W. G. and G. V. Milovanovic (2020). Double-Sided Inequalities of Ostrowski’s Type and Some Applications. Journal of Computational Analysisand Applications, 28(4); 724–736
Alshanti, W. G. and A. Qayyum (2017). A Note on New Ostrowski Type Inequalities Using a Generalized Kernel. Bulletin of Mathematical Analysis and Applications, 9(1); 74–91
Alshanti, W. G., A. Qayyum, and M. A. Majid (2017). Ostrowski Type Inequalities by Using a Generalized Quadratic Kernel. Journal of Inequalities and Special Functions, 8(4); 111–135
Amjad, J., A. Qayyum, S. Fahad, and M. Arslan (2022). Some New Generalized Ostrowski Type Inequalities with New Error Bounds. Innovative Journal of Mathematics, 1(2); 30–43
Batiha, I. M. (2011). Restriction method for approximating square roots. International Journal of Open Problems in Computer Science and Mathematics, 4(3); 146–151
Batiha, I. M., S. Alshorm, A. Ouannas, S. Momani, O. Y. Ababneh, and M. Albdareen (2022). Modified Three-Point Fractional Formulas with Richardson Extrapolation. Mathematics, 10(19); 3489
Dragomir, S. S. and A. Sofo (2000). An Integral Inequality for Twice Differentiable Mappings and Applications. Tamkang Journal of Mathematics, 31(4); 257–266
Liu, W., X. Gao, and Y. Wen (2016). Approximating the Finite Hilbert Transform Via Some Companions of Ostrowski’s Inequalities. Bulletin of the Malaysian Mathematical Sciences Society, 39; 1499–1513
Liu, W. and N. Lu (2015). Approximating the Finite Hilbert Transform Via Simpson Type Inequalities and Applications. Journal Scientific Bulletin Series A Applied Mathematics and Physics, 77(3); 107–122
Liu, W. and J. Park (2017a). A Companion of Ostrowski Like Inequality and Applications to Composite Quadrature Rules. Journal of Computational Analysis and Applications, 22(1); 19–24
Liu, W. and J. Park (2017b). Some Perturbed Versions of the Generalized Trapezoid Inequality for Functions of Bounded Variation. Journal of Computational Analysis and Applications, 22(1); 11–18
Milovanovic, G. (1977). On Some Functional Inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz, 599; 1–59
Milovanović, G. V. (1975). On Some Integral Inequalities. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, (498/541); 119–124
Milovanović, G. V. (2017). Summation Formulas of Euler–Maclaurin and Abel–Plana: Old and New Results and Applications. Progress in Approximation Theory and Applicable Complex Analysis: In Memory of QI Rahman, 2017; 429–461
Milovanović, G. V. and J. E. Pečarić (1976). On Generalization of the Inequality of A. Ostrowski and Some Related Applications. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, (544/576); 155–158
Ðorđević, R. Ž. and G. V. Milovanović (1975). A Generalization of E. Landau’s Theorem. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, (498/541); 97–106
Ostrowski, A. (1937). Über Die Absolutabweichung Einer Differentiierbaren Funktion Von Ihrem Integralmittelwert. Commentarii Mathematici Helvetici, 10(1); 226–227
Vasić, P. M. and G. V. Milovanović (1976). On an Inequality of Iyengar. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, (544/576); 18–24
Authors
Ghazi Alshanti, W. ., Mohammad Batiha, I., Alshanty, A. ., Zraiqat, A. ., Hamzah Jebril, I., & Ahmad Abu, M. . (2023). Perturbed Trapezoid Like Inequalities. Science and Technology Indonesia, 8(2), 205–211. https://doi.org/10.26554/sti.2023.8.2.205-211
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