Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T17:04:02.525Z Has data issue: false hasContentIssue false
  • English
  • Français

APPROXIMATING THE STIELTJES INTEGRAL FOR (φ,Φ)-LIPSCHITZIAN INTEGRATORS

Part of: Inequalities Real functions Approximations and expansions

Published online by Cambridge University Press:  01 February 2008

S. S. DRAGOMIR
S. S. DRAGOMIR*
Affiliation:
School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, VIC 8001, Australia (email: sever.dragomir@vu.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Approximations for the Stieltjes integral with (φ,Φ)-Lipschitzian integrators are given. Applications for the Riemann integral of a product and for the generalized trapezoid and Ostrowski inequalities are also provided.

Keywords

Stieltjes integralRiemann integraltrapezoid rulemidpoint ruleOstrowski inequality

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals 26A42: Integrals of Riemann, Stieltjes and Lebesgue type 41A55: Approximate quadratures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 1 , February 2008 , pp. 73 - 90
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Barnett, N. S., Cheung, W. S., Dragomir, S. S. and Sofo, A., ‘Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators’, RGMIA Res. Rep. Coll. 9(4) (2006) (Article 9).Google Scholar
[2]Cerone, P., Cheung, W. S. and Dragomir, S. S., ‘On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation’, RGMIA Res. Rep. Coll. 9(2) (2006) (Article 14).Google Scholar
[3]Cerone, P. and Dragomir, S. S., ‘Trapezoid type rules from an inequalities point of view’, in: Handbook of analytic computational methods in applied mathematics (ed. G. Anastassiou) (CRC Press, New York, 2000), pp. 65–134.CrossRefGoogle Scholar
[4]Cerone, P. and Dragomir, S. S., ‘A refinement of the Grüss inequality and applications’, Tamkang J. Math. 38(1) (2007), 37–49.CrossRefGoogle Scholar
[5]Cerone, P., Dragomir, S. S. and Pearce, C. E. M., ‘A generalised trapezoid inequality for functions of bounded variation’, Turkish J. Math. 24 (2000), 147–163.Google Scholar
[6]Cheng, X. L. and Sun, J., ‘A note on the perturbed trapezoid inequality’, J. Ineq. Pure Appl. Math. 3(2) (2002) (Article 29).Google Scholar
[7]Cheung, W. S. and Dragomir, S. S., ‘Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions’, Bull. Austral. Math. Soc. 75 (2007), 299–311.CrossRefGoogle Scholar
[8]Dragomir, S. S., ‘Ostrowski’s inequality for monotonous mappings and applications’, J. KSIAM 3 (1999), 127–135.Google Scholar
[9]Dragomir, S. S., ‘On the Ostrowski’s inequality for Riemann–Stieltjes integral’, Korean J. Comput. Appl. Math. 7 (2000), 477–485.CrossRefGoogle Scholar
[10]Dragomir, S. S., ‘On the Ostrowski’s inequality for Riemann–Stieltjes integral where f is of Hölder type and u is of bounded variation and applications’, J. KSIAM 5 (2001), 35–45.Google Scholar
[11]Dragomir, S. S., ‘A companion of the Grüss inequality and applications’, Appl. Math. Lett. 17 (2004), 429–435.CrossRefGoogle Scholar
[12]Dragomir, S. S., ‘Inequalities of Grüss type for the Stieltjes integral’, Kragujevac J. Math. 26 (2004), 89–122.Google Scholar
[13]Dragomir, S. S., ‘A generalisation of Cerone’s identity and applications’, Tamsui Oxford J. Math. Sci. 23(1) (2007), 79–90.Google Scholar
[14]Dragomir, S. S., ‘Inequalities for Stieltjes integrals with convex integrators and applications’, Appl. Math. Lett. 20 (2007), 123–130.CrossRefGoogle Scholar
[15]Dragomir, S. S., Buşe, C., Boldea, M. V. and Braescu, L., ‘A generalisation of the trapezoidal rule for the Riemann–Stieltjes integral and applications’, Nonlinear Anal. Forum 6 (2001), 337–351.Google Scholar
[16]Dragomir, S. S. and Fedotov, I., ‘An inequality of Grüss type for the Riemann–Stieltjes integral and applications for special means’, Tamkang J. Math. 29 (1998), 287–292.CrossRefGoogle Scholar
[17]Dragomir, S. S. and Fedotov, I., ‘A Grüss type inequality for mappings of bounded variation and applications for numerical analysis’, Nonlinear Funct. Anal. Appl. 6 (2001), 425–433.Google Scholar
[18]Liu, Z., ‘Refinement of an inequality of Grüss type for Riemann–Stieltjes integral’, Soochow J. Math. 30 (2004), 483–489.Google Scholar
[19]Ostrowski, A., ‘Uber die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert’, Comment. Math. Helv. 10 (1938), 226–227.CrossRefGoogle Scholar