On Ostrowski Type Inequalities for Functions of Two Variables with Bounded Variation

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Hüseyin Budak, Mehmet Zeki Sarikaya

Abstract

In this paper, we establish a new generalization of Ostrowski type inequalities for functions of two independent variables with bounded variation and apply it for qubature formulae. Some connections with the rectangle, the midpoint and Simpson's rule are also given.

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References

  1. M. W. Alomari, A generalization of weighted companion of Ostrowski integral inequality for mappings of bounded variation, RGMIA Research Report Collection, 14 (2011), Art. ID 87.
  2. M. W. Alomari and M.A. Latif, Weighted companion for the Ostrowski and the generalized trapezoid inequalities for mappings of bounded variation, RGMIA Research Report Collection, 14 (2011), Art. ID 92.
  3. M. W. Alomari, Approximating the Riemann-Stieltjes integral by a three-point quadrature rule and applications, Konuralp Journal of Mathematics, 2(2014), 22-34.
  4. M. W. Alomari and S. S. Dragomir, Some Grus type inequalities for the Riemann-Stieltjes integral with lipshitzian integrators, Konuralp Journal of Mathematics, 2 (2014), 36-44
  5. P. Cerone, W.S. Cheung, and S.S. Dragomir, On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation, Computers and Mathematics with Applications 54 (2007), 183-191.
  6. P. Cerone, S. S. Dragomir, and C. E. M. Pearce, A generalized trapezoid inequality for functions of bounded variation, Turk J Math, 24 (2000), 147-163.
  7. J.A. Clarkson and C. R. Adams , On definitions of bounded variation for functions of two variables, Bull. Amer. Math. Soc. 35 (1933), 824-854.
  8. J.A. Clarkson, On double Riemann-Stieltjes integrals, Bull. Amer. Math. Soc. 39 (1933), 929-936.
  9. S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull.Austral. Math. Soc., 60(1) (1999), 495-508.
  10. S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications, Kragu- jevac J. Math. 22(2000), 13-19.
  11. S. S. Dragomir, On the Ostrowski's integral inequality for mappings with bounded variation and applications, Math. Inequal. Appl. 4 (2001), no. 1, 59-66.
  12. S. S. Dragomir, A companion of Ostrowski's inequality for functions of bounded variation and applications, Int. J. Nonlinear Anal. Appl. 5 (2014) No. 1, 89-97
  13. S. S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded vari- ation, Arch. Math. (Basel) 91 (2008), no. 5, 450-460.
  14. S.S. Dragomir and E. Momoniat, A Three Point quadrature rule for functions of bounded variation and applications, Mathematical and Computer Modelling 57 (2013), 612622.
  15. S. S. Dragomir, Some perturbed Ostrowski type inequalities for functions of bounded variation, Asian-European Journal of Mathematics, 8 (2015), No. 04, doi: 10.1142/S1793557115500692.
  16. Fr ´ echet, M., Extension au cas des int ´ egrals multiples d'une d ´ efinition de l'int ´ egrale due á Stieltjes, Nouvelles Annales de Math ematiques 10 (1910), 241-256.
  17. Y. Jawarneh and M.S.M Noorani, Inequalities of Ostrowski and Simpson type for mappings of two variables with bounded variation and applications, TJMM, 3 (2011), No. 2, 81-94.
  18. W. Liu and Y. Sun, A refinement of the companion of Ostrowski inequality for functions of bounded variation and applications, arXiv:1207.3861v1
  19. [math.FA], 2012.
  20. A. M. Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10 (1938), 226-227.
  21. K-L Tseng, G-S Yang, and S. S. Dragomir, Generalizations of weighted trapezoidal inequality for mappings of bounded variation and their applications, Mathematical and Computer Modelling, 40 (2004), 77-84.
  22. K-L Tseng, Improvements of some inequalites of Ostrowski type and their applications, Taiwan. J. Math. 12 (9) (2008), 2427-2441.
  23. K-L Tseng, S-R Hwang, G-S Yang, and Y-M Chou, Improvements of the Ostrowski integral inequality for mappings of bounded variation I, Applied Mathematics and Computation 217 (2010), 2348-2355.
  24. K-L Tseng, S-R Hwang, G-S Yang, and Y-M Chou, Weighted Ostrowski integral inequality for mappings of bounded variation, Taiwanese J. of Math., 15 (2011), No. 2, 573-585.
  25. K-L Tseng, Improvements of the Ostrowski integral inequality for mappings of bounded variation II, Applied Math- ematics and Computation, 218 (2012), 5841-5847.