Derivation and applications of inequalities of Ostrowski type for n-times differentiable mappings for cumulative distribution function and some quadrature rules

Volume 9, Issue 4, pp 1844--1857 http://dx.doi.org/10.22436/jnsa.009.04.38

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1364 Downloads
• 1942 Views

Authors

Ather Qayyum - Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, 32610 Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia. Abdul Rehman Kashif - Department of Mathematics, University of Hail, 2440, Saudi Arabia. Muhammad Shoaib - Abu Dhabi Mens College, Higher Colleges of Technology, P. O. Box 25035, Abu Dhabi, United Arab Emirates. Ibrahima Faye - Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, 32610 Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia.

Abstract

In this paper new integral inequalities of Ostrowski type are developed for n-times differentiable mappings. Some well known inequalities become special cases of the inequalities obtained in this paper. With the help of obtained inequalities, we will derive new and efficient quadrature rules which are analyzed with the help of specific examples. We also give applications for cumulative distribution function.

Share and Cite

Keywords

• Ostrowski inequality
• numerical integration
• composite quadrature rule
• cumulative distributive function.

MSC

•  26D15
•  26D20
•  26D99

References

• [1] R. P. Agarwal, V. Čuljak, J. Pečarić, Some integral inequalities involving bounded higher order derivatives, Math. Comput. Modelling, 28 (1998), 51-57.


• [2] M. Alomari, M. Darus, Some Ostrowski type inequalities for convex functions with applications to special means, RGMIA Res. Rep. Coll., (2010)


• [3] P. L. Čebyšev , Sur less expressions approximatives des integrales definies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov, 2 (1882), 93-98.


• [4] P. Cerone, A new Ostrowski Type Inequality Involving Integral Means Over End Intervals, Tamkang J. Math., 33 (2002), 109-118.


• [5] P. Cerone, S. S. Dragomir, J. Roumeliotis, An inequality of Ostrowski type for mappings whose second derivatives are bounded and applications, RGMIA Res. Rep. Coll., (1998)


• [6] P. Cerone, S. S. Dragomir, J. Roumeliotis, Some Ostrowski type Inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32 (1999), 697-712.


• [7] S. S. Dragomir, E. Momoniat, A three point quadrature rule for functions of bounded variation and applications, Math. Comput. Modelling, 57 (2013), 612-622.


• [8] S. S. Dragomir, S. Wang, An inequality of Ostrowski-Grüs type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl., 33 (1997), 15-20.


• [9] S. S. Dragomir, S. Wang, A new inequality Ostrowski's type in \(L_1\)- norm and applications to some special means and some numerical quadrature rules, Tamkang J. Math., 28 (1997), 239-244.


• [10] S. S. Dragomir, S. Wang , A new inequality Ostrowski's type in \(L_p\)- norm, Indian J. Math., 40 (1998), 299-304.


• [11] S. S. Dragomir, S. Wang, Applications of Ostrowski's inequality to the estimation of error bounds for some special means and to some numerical quadrature rules, Appl. Math. Lett., 11 (1998), 105-109.


• [12] G. Grüss, Über das Maximum des absoluten Betrages von \(\frac{1}{ b-a} \int^a_b f (x) g (x) dx -\frac {1}{ (b-a)^2} \int^a_b f (x) dx \int^a_b g (x) dx\), Math. Z., 39 (1935), 215-226.


• [13] D. S. Mitrinović, J. E. Pecarić, A. M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht (1994)


• [14] A. Ostrowski, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralimittelwert, Comment. Math. Helv., 10 (1938), 226-227.


• [15] A. Qayyum, I. Faye, M. Shoaib, Improvement of Ostrowski Integral Type inequalities with application, Filomat , (Accepted)


• [16] A. Qayyum, S. Hussain, A new generalized Ostrowski Grüss type inequality and applications, Appl. Math. Lett., 25 (2012), 1875-1880.


• [17] A. Qayyum, M. Shoaib, I. Faye , Some new generalized results on ostrowski type integral inequalities with application , J. Comput. Anal. Appl., 19 (2015), 693-712.


• [18] A. Qayyum, M. Shoaib, I. Faye, On New Weighted Ostrowski Type inequalities Involving Integral Means over End Intervals and Application, Turk. J. Anal. Number Theory, 3 (2015), 61-67.


• [19] A. Qayyum, M. Shoaib, A. E. Matouk, M. A. Latif, On New Generalized Ostrowski Type Integral inequalities, Abstr. Appl. Anal., 2014 (2014), 8 pages.


• [20] M. Z. Sarikaya, E. Set, On New Ostrowski Type Integral Inequalities, Thai J. Math., 12 (2014), 145-154.


• [21] A. Sofo, S.S. Dragomir, An inequality of Ostrowski type for twice differentiable mappings in term of the \(L_p\)-norm and applications, Soochow J. Math., 27 (2001), 97-111.