A Parameter-Based Ostrowski Type Inequality for Functions whose Derivatives Belongs to $L_p([a,b])$ Involving Multiple Points


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Authors: S. KERMAUSUOR

DOI: 10.46793/KgJMat2303.445K

Abstract:

A new generalization of Ostrowski’s inequality for functions whose derivatives belong to Lp([a,b]) (1 p < ) for k points via a parameter is provided. Some particular integral inequalities are derived as by products. Our results generalize some results in the literature.



Keywords:

Ostrowski’s inequality, midpoint inequality, Simpson’s inequality, Montgomery identity, Hölder’s inequality, parameter.



References:

[1]   G. A. Anastassiou, Ostrowski type inequalities, Proc. Amer. Math. Soc. 123(12) (1995), 3775–3781. https://doi.org/10.2307/2161906

[2]   S. S. Dragomir, A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L1[a,b] and applications in numerical integration, J. Comput. Anal. Appl. 3 (2001), 343–360. https://doi.org/10.1023/A:1012050307412

[3]   S. S. Dragomir, A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to Lp[a,b] and applications in numerical integration, J. Math. Anal. Appl. 255(2) (2001), 605–626. https://doi.org/10.1006/jmaa.2000.7300

[4]   S. S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation, Arch. Math. 91 (2008), 450–460. https://doi.org/10.1007/s00013-008-2879-2

[5]   S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in L1-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. Math. 28 (1997), 239–244.

[6]   S. S. Dragomir and S. Wang, Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett. 11 (1998), 105–109. https://doi.org/10.1016/S0893-9659(97)00142-0

[7]   S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in Lp-norm, Indian J. Math. 40(3) (1998), 299–304.

[8]   M. A. Khan, S. Begum, Y. Khurshid and Y.-M. Chu, Ostrowski type inequalities involving conformable fractional integrals, J. Inequal. Appl. 2018(70) (2018). https://doi.org/10.1186/s13660-018-1664-4

[9]   S. Kermausuor, A generalization of Ostrowski’s inequality for functions of bounded variation via a parameter, Australian J. Math. Anal. Appl. 16(1) (2019), Article ID 16, 1–12.

[10]   S. Kermausuor and E. R. Nwaeze, A parameter-based Ostrowski type inequality on time scales for k points for functions with bounded second derivatives, J. Math. Inequal. 12(4) (2018), 1159–1172. https://dx.doi.org/10.7153/jmi-2018-12-88

[11]   S. Kermausuor, E. R. Nwaeze and D. F. M. Torres, Generalized weighted Ostrowski and Ostrowski-Grüss type inequalities on time scales via a parameter function, J. Math. Inequal. 11(4) (2017), 1185–1199. https://dx.doi.org/10.7153/jmi-2017-11-88

[12]   Y. Khurshid, M. A. Khan and Y.-M. Chu, Ostrowski type inequalities involving conformable integrals via preinvex functions, AIP Advances 10 (2020), Article ID 055204.

[13]   W. J. Liu, Q. A. Ngô and W. Chen, A new generalization of Ostrowski type inequality on time scales, An. St. Univ. Ovidius Constanta 17(2) (2009), 101–114.

[14]   W. J. Liu and Q. A. Ngô, A generalization of Ostrowski inequality on time scales for k points, Appl. Math. Comput. 203(2) (2008), 754–760. https://doi.org/10.1016/j.amc.2008.05.124

[15]   E. R. Nwaeze and S. Kermausuor, New Bounds of Ostrowski–Grüss type inequality for (k + 1) points on time scales, Int. J. Anal. Appl. 15(2) (2017), 211–221. https://doi.org/10.28924/2291-8639-15-2017-211

[16]   E. R. Nwaeze, S. Kermausuor and A. M. Tameru, New time scale generalizations of the Ostrowski–Grüss type inequality for k points, J. Inequal. Appl. 2017 (2017), Article ID 245. https://doi.org/10.1186/s13660-017-1525-6

[17]    A. M. Ostrowski, Über die Absolutabweichung einer differentiebaren funktion von ihrem Integralmitelwert, Comment. Math. Helv. 10 (1938), 226–227.

[18]   G. Xu and Z. B. Fang, A generalization of Ostrowski type inequality on time scales with k points, J. Math. Inequal. 11(1) (2017), 41–48. https://dx.doi.org/10.7153/jmi-11-04