A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral and applications. (English) Zbl 1354.26028

Alomari, Mohammad W.

Ann. Univ. Paedagog. Crac., Stud. Math. 182(15), 69-78 (2016).

Summary: A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral \(\int_a^b f(t)du(t) \), where \(f\) is assumed to be of \(r\)-\(H\)-Hölder type on \([a, b]\) and \(u\) is of bounded variation on \([a, b]\), is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.

MSC:

26D15	Inequalities for sums, series and integrals
26D20	Other analytical inequalities

Keywords:

Ostrowski inequality; quadrature formula; Riemann-Stieltjes integral

References:

[1]	[1] Alomari, Mohammad W. “A companion of Ostrowski’s inequality with applications.” Transylv. J. Math. Mech. 3, no. 1 (2011): 9-14. Cited on 70. · Zbl 1233.26006
[2]	[2] Alomari, Mohammad W. “A companion of Dragomir’s generalization of the Ostrowski inequality and applications to numerical integration.” Ukrainian Math. J. 64, no. 4 (2012): 491-510. Cited on 70. · Zbl 1257.26016
[3]	[3] Alomari, Mohammad W. “A companion of the generalized trapezoid inequality and applications.” J. Math. Appl. 36 (2013): 5-15. Cited on 70. · Zbl 1382.26019
[4]	[4] Alomari, Mohammad W. “New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann-Stieltjes integrals and applications.” Ukrainian Mathematical Journal 65, no. 7 (2013): 995-1018. Cited on 70. · Zbl 1298.26065
[5]	[5] Anastassiou, George A. “Univariate Ostrowski inequalities, revisited.” Monatsh. Math. 135, no. 3 (2002): 175-189. Cited on 70. · Zbl 1006.26018
[6]	[6] Barnett, Neil S., and Sever S. Dragomir, and Ian Gomm. “A companion for the Ostrowski and the generalised trapezoid inequalities.” Math. Comput. Modelling 50, no. 1-2 (2009): 179-187. Cited on 70. · Zbl 1185.26038
[7]	[7] Barnett, Neil S., et al. “Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators.” Comput. Math. Appl. 57, no. 2 (2009): 195-201. Cited on 70. · Zbl 1165.26322
[8]	[8] Cerone, Pietro, and Wing-Sum Cheung, and Sever S. Dragomir. “On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation.” Comput. Math. Appl. 54, no. 2 (2007): 183-191. Cited on 70. · Zbl 1144.26025
[9]	[9] Cerone, Pietro, and Sever S. Dragomir. “New bounds for the three-point rule involving the Riemann-Stieltjes integral.” In Advances in statistics, combinatorics and related areas, ed. C. Gulati, et al., 53-62. River Edge, NJ: World Sci. Publ., 2002. Cited on 70. · Zbl 1285.26034
[10]	[10] Cerone, Pietro, and Sever S. Dragomir. “Approximating the Riemann-Stieltjes integral via some moments of the integrand.” Math. Comput. Modelling 49, no. 1-2 (2009): 242-248. Cited on 70. · Zbl 1165.65323
[11]	[11] Dragomir, Sever S., and Themistocles M. Rassias, eds. “Ostrowski type inequalities and applications in numerical integration.” Dordrecht: Kluwer Academic Publishers, 2002. Cited on 70. · Zbl 0992.26002
[12]	[12] Dragomir, Sever S. “On the Ostrowski inequality for Riemann-Stieltjes integral \(∫_a^b {f(t)\;du(t)} \) where f is of Hölder type and u is of bounded variation and applications.” J. Korean Soc. Ind. Appl. Math. 5, no. 1 (2001): 35-45. Cited on 69.
[13]	[13] Dragomir, Sever S. “On the Ostrowski’s inequality for Riemann-Stieltjes integral and applications.” Korean J. Comput. Appl. Math. 7, no. 3 (2000): 611-627. Cited on 70. · Zbl 0969.26017
[14]	[14] Dragomir, Sever S. “Some companions of Ostrowski’s inequality for absolutely continuous functions and applications.” Bull. Korean Math. Soc. 42, no. 2 (2005): 213-230. Cited on 70. · Zbl 1099.26015
[15]	[15] Dragomir, Sever S. “A companion of Ostrowski’s inequality for functions of bounded variation and applications.” RGMIA Preprint Vol. 5 Supp. (2002): article no. 28. Cited on 70. · Zbl 1315.26020
[16]	[16] Dragomir, Sever S., et al. “A generalisation of the trapezoid rule for the Riemann-Stieltjes integral and applications.” Nonlinear Anal. Forum 6, no. 2 (2001): 337-351. Cited on 70.
[17]	[17] Dragomir, Sever S. “Some inequalities of midpoint and trapezoid type for the Riemann-Stieltjes integral.” Nonlinear Anal. 47, no. 4 (2001): 2333-2340. Cited on 70. · Zbl 1042.26508
[18]	[18] Dragomir, Sever S. “Approximating the Riemann-Stieltjes integral in terms of generalised trapezoidal rules.” Nonlinear Anal. 71, no. 12 (2009): e62-e72. Cited on 70.
[19]	[19] Dragomir, Sever S. “Approximating the Riemann-Stieltjes integral by a trapezoidal quadrature rule with applications.” Math. Comput. Modelling 54, no. 1-2 (2011): 243-260. Cited on 70. · Zbl 1225.65030
[20]	[20] Guessab, Allal, and Gerhard Schmeisser. “Sharp integral inequalities of the Hermite-Hadamard type.” J. Approx. Theory 115, no. 2 (2002): 260-288. Cited on 70. · Zbl 1012.26013
[21]	[21] Kumar, Pranesh. “The Ostrowski type moment integral inequalities and moment-bounds for continuous random variables.” Comput. Math. Appl. 49, no. 11-12 (2005): 1929-1940. Cited on 70. · Zbl 1083.60505
[22]	[22] Liu, Zheng. “Some companions of an Ostrowski type inequality and applications.” JIPAM. J. Inequal. Pure Appl. Math. 10, no. 2 (2009): Article 52, 12 pp. Cited on 70. · Zbl 1168.26310
[23]	[23] Liu, Zheng. “Refinement of an inequality of Grüss type for Riemann-Stieltjes integral.” Soochow J. Math. 30, no. 4 (2004): 483-489. Cited on 70. · Zbl 1061.26019
[24]	[24] Mercer, Peter R. “Hadamard’s inequality and trapezoid rules for the Riemann-Stieltjes integral.” J. Math. Anal. Appl. 344, no. 2 (2008): 921-926. Cited on 70. · Zbl 1147.26013