Generalized Ostrowski inequalities and computational integration

Nazia Irshad; Asif R. Khan; Hina Musharraf

doi:10.33993/jnaat492-1224

Authors

Nazia Irshad Dawood University of Engineering and Technology, Pakistan https://orcid.org/0000-0001-6711-3358
Asif R. Khan University of Karachi, Pakistan https://orcid.org/0000-0002-4700-4987
Hina Musharraf University of Karachi, Pakistan

DOI:

https://doi.org/10.33993/jnaat492-1224

Keywords:

Ostrowski Inequality, Quadrature Rules

Abstract views: 206

Abstract

We state and prove three generalized results related to Ostrowski inequality by using differentiable functions which are bounded, bounded below only and bounded above only, respectively. From our proposed results we get number of established results as our special cases.

Some applications in numerical integration are also given which gives us some standard and nonstandard quadrature rules.

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References

W. G. Alshanti and G. V. Milovanovic, Double-sided inequalities of Ostrowski’s Type and some Applications, J. Computational Analysis and Applications, 28[4], (2020), pp. 724–736.

E. F. Beckenbach and R. Bellman, Springer-Verlag, Berlin-Gottinggon-Heidelberg,1961, https://doi.org/10.1007/978-3-642-64971-4 DOI: https://doi.org/10.1007/978-3-642-64971-4

S. S. Dragomir, P. Cerone and J. Roumeliotis, A new generalization of Ostrowski integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Appl. Math. lett. 13 No. 1, (2000), pp. 19–25, https://doi.org/10.1007/978-3-642-64971-4 DOI: https://doi.org/10.1016/S0893-9659(99)00139-1

W. Gautschi, Numerical Analysis: An Introduction, Birkahauser, Boston 1997

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1934.

M. Imtiaz, N. Irshad, and A. R. Khan, Generalization of weighted Ostrowski integral inequality for twice differentiable mappings, Adv. Inequal. Appl., 2016 (2016), pp. 1–17, Article No. 20.

N. Irshad, A. R. Khan and A. Nazir, Extension of Ostrowki type inequality via moment generating function, Adv. Inequal. Appl.,2020(2) (2020), pp. 1–15, https://doi.org/10.28919/aia/3600 DOI: https://doi.org/10.28919/aia/3600

N. Irshad, A. R. Khan, And M. A. Shaikh, Generalized Weighted Ostrowski-Gruss Type Inequality with Applications, Global J. Pure Appl. Math., 15 (5) (2019), pp. 675–692, https://doi.org/10.28919/aia/3998 DOI: https://doi.org/10.28919/aia/3998

N. Irshad, A. R. Khan, M. A. Shaikh, Generalization of Weighted Ostrowski Inequality with Applications in Numerical Integration, Adv. Ineq. Appl., 2019 (7) (2019),pp. 1–14.

N. Irshad and A. R. Khan, On Weighted Ostrowski Gruss Inequality with Applications, TJMM, 10(1) (2018), pp. 15–22

N. Irshad and A. R. Khan, Some Applications of Quadrature Rules for Mappings on Lp[u, v] Space via Ostrowski-type Inequality, J. Num. Anal. Approx. Theory,46(2)(2017), pp. 141-149, https://ictp.acad.ro/jnaat/journal/article/view/1107

N. Irshad and A. R. Khan, Generalization of Ostrowski inequality for differentiable functions and its applications to numerical quadrature rules, J. Math. Anal.,8(1)(2017), pp. 79–102.

M. Masjed-Jamei and S. S. Dragomir, Generalization of Ostrowski Gruss Inequality, World Scientific 12(2) (2014), pp. 117–130, https://doi.org/10.1142/s0219530513500309 DOI: https://doi.org/10.1142/S0219530513500309

M. Masjed-Jamei and S. S. Dragomir, An analogue of the Ostrowski Inequality and applications, Filomat 28(2) (2014), pp. 373–381, https://doi.org/10.2298/fil1402373m DOI: https://doi.org/10.2298/FIL1402373M

G. V. Milovanovi c, On some Integral Inequalities, Univ. Beograd. Publ. Elektrotehn.Fak. Ser. Mat. Fiz., pp. 498–541 (1975), pp. 97–106.

G. V. Milovanovic and J. E. Pecaric, On Generalization of the Inequality of A.Ostrowski and Some related Applications, Univ. Beograd. Publ. Elektrotehn. Fak. Ser.Mat. Fiz., pp. 544–576 (1976), pp. 155–158.

D. S. Mitrinovi c , J. E. Pecaric , and A. M. Fink, “Classical and New Inequalities in Analysis” Kluwer Academic Publishers, Dordrecht (1991).

D. S. Mitrinovic , J. E. Pecaric, and A. M. Fink, “Inequalities Involving Functions and their Integrals and Derivatives”. Kluwer Academic Publishers, Dordrecht (1991). DOI: https://doi.org/10.1007/978-94-011-3562-7_15

A. M. Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihrenintegralmittelwert, Comment. Math. Helv., 10(1938), pp. 226–227, https://doi.org/10.1007/bf01214290 DOI: https://doi.org/10.1007/BF01214290

M. A. Shaikh, A. R. Khan and N. Irshad, Generalized Ostrowski Inequality with Applications in Numerical Integration and Special Means, Adv. Inequal Appl., 2018 (2018):7, pp. 1–22, https://doi.org/10.28919/aia/3553 DOI: https://doi.org/10.28919/aia/3553

N. Ujevic, A generalization of Ostrowski’s inequality and applications in numerical integration, Appl. Math. Lett.,17, (2004), pp. 133–137, https://doi.org/10.1016/S0893-9659(04)90023-7 DOI: https://doi.org/10.1016/S0893-9659(04)90023-7

M. J. Vivas-Cortez, A. Kashuri, R. Liko and J. E. Hernandez Hernandez, Quantum estimates of Ostrowski inequalities for generalized φ-convex functions, Symmetry, 11(12) (2019), pp. 1–16, https://doi.org/10.3390/sym11121513 DOI: https://doi.org/10.3390/sym11121513

M. J. Vivas-Cortez, A. Kashuri, R. Liko and J. E. Hernandez Hernandez, Some inequalities using generalized convex functions in quantum analysis, Symmetry, 11(11) (2019), pp. 1-14, https://doi.org/10.3390/sym11111402 DOI: https://doi.org/10.3390/sym11111402