Inequalities for the Riemann-Stieltjes integral of \(S\)-dominated integrators with applications. I. (English) Zbl 1339.26047

Dragomir, S. S.

Probl. Anal. Issues Anal. 4(22), No. 1, 11-37 (2015).

Summary: Assume that \(u,v:[a,b]\to \mathbb{R}\) are monotonic nondecreasing on the interval \([a,b]\). We say that the complex-valued function \(h:[a,b]\to \mathbb{C}\) is \(S\)-dominated by the pair \((u,v)\) if \[ |h(y)-h(x)|^2 \leq [u(y)-u(x)][v(y)-v(x)] \] for any \(x,y\in [a,b]\).
In this paper we show amongst other that \[ \bigg| \int^b_a f(t)dh(t)\bigg|^2 \leq \int_a^b |f(t)| du(t) \int^b_a |f(t)|dv(t), \] for any continuous function \(f:[a,b]\to \mathbb{C}\). Applications for the trapezoidal and midpoint inequalities are given. New inequalities for some Chebyshev and (CBS)-type functionals are presented. Natural applications for continuous functions of selfadjoint and unitary operators on Hilbert spaces are provided as well.

MSC:

26D15	Inequalities for sums, series and integrals
47A63	Linear operator inequalities

Keywords:

Riemann-Stieltjes integral; functions of bounded variation; cumulative variation; selfadjoint operators; unitary operators; trapezoid and midpoint inequalities; Chebyshev and (CBS)-type functionals

References:

[1]	[1] Dragomir S. S., ”The Ostrowski inequality for mappings of bounded variation”, Bull. Austral. Math. Soc., 60 (1999), 495–826 · Zbl 0951.26011 · doi:10.1017/S0004972700036662
[2]	[2] Cerone P., Dragomir S. S., Pearce C. E. M., ”A generalised trapezoid inequality for functions of bounded variation”, Turk. J. Math., 24 (2000), 147–163 · Zbl 0974.26011
[3]	[3] Barnett N. S., Cheung W. S., Dragomir S. S., Sofo A., ”Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators”, Comput. Math. Appl., 57 (2009), 195–201 · Zbl 1165.26322 · doi:10.1016/j.camwa.2007.07.021
[4]	[4] Barnett N. S., Dragomir S. S., ”A perturbed trapezoid inequality in terms of the fourth derivative”, Korean J. Comput. Appl. Math., 9 (2002), 45–60 · Zbl 0995.26010
[5]	[5] Barnett N. S., Dragomir S. S., ”Perturbed version of a general trapezoid inequality”, Inequality theory and applications, 3 (2003), 1–12 · Zbl 1073.26013
[6]	[6] Barnett N. S., Dragomir S. S., ”A perturbed trapezoid inequality in terms of the third derivative and applications”, Inequality theory and applications, 5 (2007), 1–11 · Zbl 1208.26026
[7]	[7] Barnett N. S., Dragomir S. S., Gomm I., ”A companion for the Ostrowski and the generalised trapezoid inequalities”, Math. Comput. Modelling, 50 (2009), 179–187 · Zbl 1185.26038 · doi:10.1016/j.mcm.2009.04.005
[8]	[8] Cerone P., Dragomir S. S., ”Midpoint-type rules from an inequalities point of view”, Handbook of analytic-computational methods in applied mathematics, 2000, 135–200 · Zbl 0966.26015
[9]	[9] Cerone P., Dragomir S. S., ”Trapezoidal-type rules from an inequalities point of view”, Handbook of analytic-computational methods in applied mathematics, 2000, 65–134 · Zbl 0966.26014
[10]	[10] Cheng X. L., Sun J., ”A note on the perturbed trapezoid inequality”, J. Inequal. Pure Appl. Math., 3 (2002), 29, 7 pp. · Zbl 0994.26020
[11]	[11] Dragomir S. S., ”On the trapezoid quadrature formula and applications”, Kragujevac J. Math., 23 (2001), 25–36 · Zbl 1005.26017
[12]	[12] Dragomir S. S., ”Some inequalities of midpoint and trapezoid type for the Riemann–Stieltjes integral”, Nonlinear Anal., 47 (2001), 2333–2340 · Zbl 1042.26508 · doi:10.1016/S0362-546X(01)00357-1
[13]	[13] Dragomir S. S., ”Improvements of Ostrowski and generalised trapezoid inequality in terms of the upper and lower bounds of the first derivative”, Tamkang J. Math., 34 (2003), 213–222 · Zbl 1044.26012
[14]	[14] Dragomir S. S., ”Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation”, Arch. Math., 91 (2008), 450–460 · Zbl 1162.26005 · doi:10.1007/s00013-008-2879-2
[15]	[15] Dragomir S. S., Cho Y. J., Kim Y. H., ”On the trapezoid inequality for the Riemann–Stieltjes integral with Hölder continuous integrands and bounded variation integrators”, Inequality theory and applications, 5 (2007), 71–79 · Zbl 1208.26028
[16]	[16] Dragomir S. S., Mcandrew A., ”On trapezoid inequality via a Grüss type result and applications”, Tamkang J. Math., 31 (2000), 193–201 · Zbl 0997.26011
[17]	[17] Dragomir S. S., Pečarić J., Wang S., ”The unified treatment of trapezoid, Simpson, and Ostrowski type inequality for monotonic mappings and applications”, Math. Comput. Modelling, 31 (2000), 61–70 · Zbl 1042.26507 · doi:10.1016/S0895-7177(00)00046-7
[18]	[18] Gunawan H., ”A note on Dragomir–McAndrew’s trapezoid inequalities”, Tamkang J. Math., 33 (2002), 241–244 · Zbl 1023.26018
[19]	[19] Liu Z., ”Some inequalities of perturbed trapezoid type”, J. Inequal. Pure Appl. Math., 7 (2006), 47, 9 pp. · Zbl 1132.26363
[20]	[20] Liu W. J., Xue Q. L., Dong J. W., ”New generalization of perturbed trapezoid, mid-point inequalities and applications”, Int. J. Pure Appl. Math., 41 (2007), 761–768 · Zbl 1137.26306
[21]	[21] Kechriniotis A. I., Assimakis N. D., ”Generalizations of the trapezoid inequalities based on a new mean value theorem for the remainder in Taylor’s formula”, J. Inequal. Pure Appl. Math., 7 (2006), 90, 13 pp. · Zbl 1182.26057
[22]	[22] Mercer P. R., ”Hadamard’s inequality and trapezoid rules for the Riemann–Stieltjes integral”, J. Math. Anal. Appl., 344 (2008), 921–926 · Zbl 1147.26013 · doi:10.1016/j.jmaa.2008.03.026
[23]	[23] Mercer A. McD., ”On perturbed trapezoid inequalities”, J. Inequal. Pure Appl. Math., 7 (2006), 118, 7 pp. · Zbl 1154.26311
[24]	[24] Pachpatte B. G., ”A note on a trapezoid type integral inequality”, Bull. Greek Math. Soc., 49 (2004), 85–90 · Zbl 1286.26016
[25]	[25] Ujević N., ”Perturbed trapezoid and mid-point inequalities and applications”, Soochow J. Math., 29 (2003), 249–257 · Zbl 1042.26011
[26]	[26] Ujević N., ”On perturbed mid-point and trapezoid inequalities and applications”, Kyungpook Math. J., 43 (2003), 327–334 · Zbl 1034.26023
[27]	[27] Ujević N., ”Error inequalities for a generalized trapezoid rule”, Appl. Math. Lett., 19 (2006), 32–37 · Zbl 1080.26023 · doi:10.1016/j.aml.2005.03.005
[28]	[28] Dragomir S. S., ”On the Ostrowski’s integral inequality for mappings with bounded variation and applications”, Math. Ineq. Appl., 4 (2001), 33–40 · Zbl 1027.26020
[29]	[29] Acu A. M., Baboş A., Sofonea F., ”The mean value theorems and inequalities of Ostrowski type”, Sci. Stud. Res. Ser. Math. Inform., 21 (2001), 5–16 · Zbl 1249.26030
[30]	[30] Acu A. M., Sofonea F., ”On an inequality of Ostrowski type”, J. Sci. Arts., 16:3 (2011), 281–287 · Zbl 1247.26027
[31]	[31] Ahmad F., Barnett N. S., Dragomir S. S., ”New weighted Ostrowski and Čebyšev type inequalities”, Nonlinear Anal., 71 (2009), e1408–e1412 · Zbl 1238.26017 · doi:10.1016/j.na.2009.01.178
[32]	[32] Alomari M. W., ”A companion of Ostrowski’s inequality with applications”, Transylv. J. Math. Mech., 3 (2011), 9–14 · Zbl 1233.26006
[33]	[33] Alomari M. W., Darus M., Dragomir S. S., Cerone P., ”Ostrowski type inequalities for functions whose derivatives are \(s\)-convex in the second sense”, Appl. Math. Lett., 23 (2010), 1071–1076 · Zbl 1197.26021 · doi:10.1016/j.aml.2010.04.038
[34]	[34] Anastassiou G. A., ”Ostrowski type inequalities”, Proc. Amer. Math. Soc., 123 (1995), 3775–3781 · Zbl 0860.26009 · doi:10.1090/S0002-9939-1995-1283537-3
[35]	[35] Anastassiou G. A., ”Univariate Ostrowski inequalities, revisited”, Monatsh. Math., 135 (2002), 175–189 · Zbl 1006.26018 · doi:10.1007/s006050200015
[36]	[36] Anastassiou G. A., ”Ostrowski inequalities for cosine and sine operator functions”, Mat. Vestnik, 64 (2012), 336–346 · Zbl 1289.47088
[37]	[37] Anastassiou G. A., ”Multivariate right fractional Ostrowski inequalities”, J. Appl. Math. Inform., 30 (2012), 445–454 · Zbl 1242.26008
[38]	[38] Anastassiou G. A., ”Univariate right fractional Ostrowski inequalities”, Cubo, 14 (2012), 1–7 · Zbl 1254.26030 · doi:10.4067/S0719-06462012000100001
[39]	[39] Cerone P., Cheung W. S., Dragomir S. S., ”On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation”, Comput. Math. Appl., 54 (2007), 183–191 · Zbl 1144.26025 · doi:10.1016/j.camwa.2006.12.023
[40]	[40] Dragomir S. S., ”On the mid-point quadrature formula for mappings with bounded variation and applications”, Kragujevac J. Math., 22 (2000), 13–19 · Zbl 1012.26016
[41]	[41] Dragomir S. S., ”Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces”, Acta Math. Vietnamica, 39 (2014), 287–303 · Zbl 1320.47015 · doi:10.1007/s40306-014-0061-4
[42]	[42] Liu Z., ”Some Ostrowski type inequalities and applications”, Vietnam J. Math., 37 (2009), 15–22 · Zbl 1183.26027
[43]	[43] Liu Z., ”Some companions of an Ostrowski type inequality and applications”, J. Inequal. Pure Appl. Math., 10 (2009), 52, 12 pp. · Zbl 1168.26310
[44]	[44] Liu Z., ”A note on Ostrowski type inequalities related to some \(s\)-convex functions in the second sense”, Bull. Korean Math. Soc., 49 (2012), 775–785 · Zbl 1277.26043 · doi:10.4134/BKMS.2012.49.4.775
[45]	[45] Liu Z., ”A sharp general Ostrowski type inequality”, Bull. Aust. Math. Soc., 83 (2011), 189–209 · Zbl 1225.26049 · doi:10.1017/S1446788711001431
[46]	[46] Masjed-Jamei M., Dragomir S. S., ”A new generalization of the Ostrowski inequality and applications”, Filomat, 25 (2011), 115–123 · Zbl 1265.26098 · doi:10.2298/FIL1101115M
[47]	[47] Pachpatte B. G., ”New inequalities of Ostrowski and trapezoid type for \(n\)-time differentiable functions”, Bull. Korean Math. Soc., 41 (2004), 633–639 · Zbl 1066.26021 · doi:10.4134/BKMS.2004.41.4.633
[48]	[48] Park J., ”On the Ostrowskilike type integral inequalities for mappings whose second derivatives are s\(^*\)-convex”, Far East J. Math. Sci., 67 (2012), 21–35 · Zbl 1252.26021
[49]	[49] Park J., ”Some Ostrowskilike type inequalities for differentiable real \((\alpha, m)\)-convex mappings”, Far East J. Math. Sci., 61 (2012), 75–91 · Zbl 1252.26019
[50]	[50] Sarikaya Z., ”On the Ostrowski type integral inequality”, Acta Math. Univ. Comenian. (N. S.), 79 (2010), 129–134 · Zbl 1212.26058
[51]	[51] Sulaiman W. T., ”Some new Ostrowski type inequalities”, J. Appl. Funct. Anal., 7 (2012), 102–107 · Zbl 1260.26026
[52]	[52] Tseng K. L., ”Improvements of the Ostrowski integral inequality for mappings of bounded variation, II”, Appl. Math. Comput., 218 (2012), 5841–5847 · Zbl 1244.26054 · doi:10.1016/j.amc.2011.11.047
[53]	[53] Tseng K. L., Hwang S. R., Yang G. S., Chou Y. M., ”Improvements of the Ostrowski integral inequality for mappings of bounded variation, I”, Appl. Math. Comput., 217 (2010), 2348–2355 · Zbl 1205.26035 · doi:10.1016/j.amc.2010.07.034
[54]	[54] Vong S. W., ”A note on some Ostrowskilike type inequalities”, Comput. Math. Appl., 62 (2011), 532–535 · Zbl 1228.26034 · doi:10.1016/j.camwa.2011.05.037
[55]	[55] Wu Q., Yang S., ”A note to Ujević’s generalization of Ostrowski’s inequality”, Appl. Math. Lett., 18 (2005), 657–665 · Zbl 1068.26023 · doi:10.1016/j.aml.2004.08.010
[56]	[56] Wu Y., Wang Y., ”On the optimal constants of Ostrowskilike inequalities involving \(n\) knots”, Appl. Math. Comput., 219 (2013), 7789–7794 · Zbl 1296.26104 · doi:10.1016/j.amc.2013.02.004
[57]	[57] Xiao Y. X., ”Remarks on Ostrowskilike inequalities”, Appl. Math. Comput., 219 (2012), 1158–1162 · Zbl 1288.65030 · doi:10.1016/j.amc.2012.07.025
[58]	[58] Helmberg G., Introduction to spectral theory in Hilbert space, John Wiley & Sons, Inc., New York, 1969