A parameter-based Ostrowski type inequality on time scales with \(k\) points for functions having bounded second derivatives. (English) Zbl 1406.26008
Summary: In this paper, we introduce a parameter \(\lambda\in[0,1]\), and obtain a generalization of the Ostrowski type inequality on time scales for \(k\) points for functions whose second derivatives are bounded. Our results generalize some results in the literature and thereby, injects into the mathematical community some inequalities which we hope can be used to approximate the integral of a function in applied mathematics and/or mathematical physics. In addition, we apply our main theorem to the continuous, discrete, and quantum calculus to derive more inequalities in this direction.
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 26D15 | Inequalities for sums, series and integrals |
| 54C30 | Real-valued functions in general topology |
Keywords:
Montgomery identity; Ostrowski inequality; parameter; time scales; bounded second derivativesReferences:
| [1] | M. BOHNER ANDA. PETERSON, {\it Dynamic equations on time scales}, Birkh¨auser Boston, Boston,MA, (2001). · Zbl 0993.39010 |
| [2] | M. BOHNER ANDA. PETERSON, {\it Advances in Dynamic Equations on Time Series}, Birkh¨auser Boston,Boston, MA, (2003). · Zbl 1025.34001 |
| [3] | M. BOHNERANDT. MATTHEWS, {\it Ostrowski inequalities on time scales}, J. Inequal. Pure and Appl.Math. 9, (2008), Art. 6.1172S. KERMAUSUOR ANDE. R. NWAEZE · Zbl 1178.26020 |
| [4] | S. S. DRAGOMIR, {\it Gr¨uss inequality in inner product spaces}, The Aust. Math. Soc. Gazette., 26, 2(1999), 66-70. · Zbl 1021.26013 |
| [5] | S. HILGER, {\it Ein Maßkettenkalk¨ul mit Anwendung auf Zentrumsmannigfaltigkeiten}, Ph. D. thesis, Uni-versit¨at W¨urzburg, W¨urzburg, Germany, (1988). · Zbl 0695.34001 |
| [6] | B. KARPUZ ANDU. M. ¨OZKAN, {\it Ostrowski Inequality on time scales}, J. Inequal. Pure and Appl.Math. 9, 4 (2008), Art. 112. · Zbl 1175.26039 |
| [7] | S. KERMAUSUOR, {\it Ostrowski type and Ostrowski-Gruss type inequalities for vector-valued functions}{\it with k points via a parameter}, Adv. Inequal. Appl. 2018, (2018), Article ID 10. · Zbl 1425.26010 |
| [8] | S. KERMAUSUOR, E. R. NWAEZE ANDD. F. M. TORRES, {\it Generalized weighted Ostrowski and}{\it Ostrowski-Gr¨uss type inequalities on time scales via a parameter function}, J. Math. Inequal. 11, 4(2017), 1185-1199. · Zbl 1379.26019 |
| [9] | W. J. LIU ANDQ. A. NGˆO, {\it A generalization of Ostrowski inequality on time scales for k points},Appl. Math. Comput. 203, 2 (2008), 754-760. · Zbl 1169.26308 |
| [10] | W. J. LIU, Q. A. NGˆO ANDW. B. CHEN, {\it A perturbed Ostrowski-type inequality on time scales for k}{\it points for functions whose second derivatives are bounded}, J. Inequal. Appl. 2008, (2008), Article ID597241, 12 pages. · Zbl 1175.26044 |
| [11] | W. J. LIU ANDQ. A. NGˆO, {\it An Ostrowski type inequality on time scales for functions whose second}{\it derivatives are bounded}, Inequality Theory and Applications, Nova Science Pub Inc. 6, (2010), 133-141. ISBN: 978-1616686253. |
| [12] | W. J. LIU, A. TUNA ANDY. JIANG, {\it On weighted Ostrowski type, Trapezoid type, Gr¨uss type and}{\it Ostrowski-Gr¨uss like inequalities on time scales}, Appl. Anal. 93, 3 (2014), 551-571. · Zbl 1294.26026 |
| [13] | W. J. LIU, A. TUNA ANDY. JIANG, {\it New weighted Ostrowski and Ostrowski-Gr¨uss type inequalities}{\it on time scales}, An. S¸tiint¸. Univ. Al. I. Cuza las¸i. Mat. (N. S.), 60, 1 (2014), 57-76. · Zbl 1299.26053 |
| [14] | E. R. NWAEZE, {\it A new weighted Ostrowski type inequality on arbitrary time scale}, J. King Saud Uni.Sci. 29, 2 (2017), 230-234. |
| [15] | E. R. NWAEZE, {\it Time scale versions of the Ostrowski-Gr¨uss type inequality with a parameter function},J. Math. Inequal. 12, 2 (2018), 531-543. · Zbl 1395.26008 |
| [16] | E. R. NWAEZE, {\it Generalized weighted trapezoid and Gr¨uss type inequalities on time scales}, Aust. J.Math. Anal. Appl. 11, 1 (2017), Art. 4. · Zbl 1358.26021 |
| [17] | E. R. NWAEZE ANDS. KERMAUSUOR, {\it New Bounds of Ostrowski-Gr¨uss type inequality for}({\it k }+ 1){\it points on time scales}, Int. J. Anal. Appl. 15, 2 (2017), 211-221. · Zbl 1406.26009 |
| [18] | E. R. NWAEZE, S. KERMAUSUOR ANDA. M. TAMERU, {\it New time scale generalizations of the}{\it Ostrowski-Gr¨uss type inequality for k points}, J. Inequal. Appl. 2017:245, (2017). · Zbl 1373.26037 |
| [19] | E. R. NWAEZE ANDA. M. TAMERU, {\it On weighted Montgomery identity for k points and its asso-}{\it ciates on time scales}, Abstr. App. Anal. (2017), Art. ID 5234181. · Zbl 1470.26037 |
| [20] | A. TUNA ANDD. DAGHAN, {\it Generalization of Ostrowski and Ostrowski-Gr¨uss type inequalities on}{\it time scales}, Comput. Math. Appl. 60, (2010), 803-811. · Zbl 1201.26007 |
| [21] | G. XU ANDZ. B. FANG, {\it A Generalization of Ostrowski type inequality on time scales with k points},J. Math. Inequal. 11, 1 (2017), 41-48. · Zbl 1357.26046 |
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