Generalized weighted Ostrowski, Trapezoid and Grüss type inequalities on time scales

• Autorzy: El-Deeb A. A. , Elsennary H. A. , Nwaeze E. R.

Identyfikatory

DOI

10.1515/fascmath-2018-0008

• Języki publikacji: EN

Abstrakty

EN

In this article, using two parameters, we obtain generalizations of a weighted Ostrowski type inequality and its companion inequalities on an arbitrary time scale for functions whose first delta derivatives are bounded. Our work unifies the continuous and discrete versions and can also be applied to the quantum calculus case.

Słowa kluczowe

EN

Montgomery identity; Ostrowski inequality; Trapezoid inequality; Gruss inequality; time scales

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2018
• Tom: Nr 60
• Strony: 123--144
• Opis fizyczny: Bibliogr. 50 poz.

Twórcy

autor

El-Deeb A. A.

• Department of Mathematics Faculty of Science Al-Azhar University Nasr City (11884), Cairo, Egypt

autor

Elsennary H. A.

• Department of Mathematics Faculty of Engineering and Sciences Sinai University Arish (45615), North Sinai, Egypt

autor

Nwaeze E. R.

• Department of Mathematics Tuskegee University Tuskegee (36088), AL, USA

Bibliografia

• [1] Agarwal R., Bohner M., Peterson A., Inequalities on time scales: a survey, Mathematical Inequalities and Applications, 4(2001), 535-558.
• [2] Agarwal R., O’Regan D., Saker S., Dynamic inequalities on time scales, vol. 2014, Springer, 2014.
• [3] Ahmad F., Cerone P., Dragomir S.S., Mir N.A., On some bounds of Ostrowski and Čebyšev type, J. Math. Inequal., 4(1)(2010), 53-65.
• [4] Ammi M.R.S., Torres D.F.M., Combined dynamic Griüss inequalities on time scales, Journal of Mathematical Sciences, 161(6)(2009), 792-802.
• [5] Anastassiou G.A., Representations and Ostrowski type inequalities on time scales, Computers & Mathematics with Applications, 62(10)(2011), 3933-3958.
• [6] Bohner M., Erbe L., Peterson A., Oscillation for nonlinear second order dynamic equations on a time scale, J. Math. Anal. Appl., 301(2)(2005), 491-507.
• [7] Bohner M., Matthews T., The Grüss inequality on time scales, Communications in Mathematical Analysis, 3(1)(2007).
• [8] Bohner M., Matthews T., Ostrowski inequalities on time scales, J. Inequal. Pure Appl. Math., 9(1)(2008), pages 8.
• [9] Bohner M., Matthews T., Tuna A., Diamond-alpha Grüss type inequalities on time scales, International Journal of Dynamical Systems and Differential Equations, 3(1-2)(2011), 234-247.
• [10] Bohner M., Peterson A., Dynamic equations on time scales, Birkhaüser Boston, Inc., Boston, MA, 2001. An introduction with applications.
• [11] Bohner M., Peterson A., Advances in dynamic equations on time scales, Birkhaüser Boston, Inc., Boston, MA, 2003.
• [12] Cerone P., Dragomir S.S., Roumeliotis J., An inequality of Ostrowski-Grüss type for twice differentiable mappings and applications in numerical integration, RGMIA research report collection, 1(2)(1998).
• [13] Dinu C., Ostrowski type inequalities on time scales, Annals of the University of Craiova-Mathematics and Computer Science Series, 34(2007), 43-58.
• [14] Dragomir S.S., A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L1 [a, b] and applications in numerical integration, Journal of Computational Analysis and Applications, 3(4)(2001), 343-360.
• [15] Dragomir S.S., A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to Lp [a, b] and applications in numerical integration, Journal of Mathematical Analysis and Applications, 255(2)(2001), 605-626.
• [16] Dragomir S.S., Agarwal R.P., Cerone P., On Simpson’s inequality and applications, RGMIA research report collection, 2(3)(1999).
• [17] Dragomir S.S., Cerone P., Roumeliotis J., A new generalization of Ostrowski’s integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Applied Mathematics Letters, 13(1)(2000), 19-25.
• [18] Feng Q., Meng F., Generalized Ostrowski type inequalities for multiple points on time scales involving functions of two independent variables, Journal of inequalities and applications, 2012(1)(2012), 74 pages.
• [19] Hilger S., Ein makettenkalkül mit Anwendung auf Zentrumsmannig-faltigkeiten Ph. D, PhD thisis, thesis, 1988.
• [20] Hilger S., Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math., 18(1-2)(1990), 18-56.
• [21] Hilscher R., A time scales version of a Wirtinger-type inequality and applications, Journal of Computational and Applied Mathematics, 141(1)(2002), 219-226.
• [22] Hussain S., Amer L.M., Alomari M., Generalized double-integral Ostrowski type inequalities on time scales, Applied Mathematics Letters, 24(8)(2011), 1461-1467.
• [23] Kac V., Cheung P., Quantum calculus, Universitext, Springer-Verlag,, New York, 2002.
• [24] Kermausuor S., Nwaeze Eze R., Torres D.F.M., Generalized weighted Ostrowski and Ostrowski-Grüss type inequalities on time scale via a parameter function, Journal of Mathematical Inequalities, 11(4)(2017), 1185-1199.
• [25] Li W.N., Some delay integral inequalities on time scales, Computers & Mathematics with Applications, 59(6)(2010), 1929-1936.
• [26] Li W.N., Sheng W., Some Gronwall type inequalities on time scales, J. Math. Inequal., 4(1)(2010), 67-76.
• [27] Li W.N., Sheng W., Some nonlinear dynamic inequalities on time scales, Proceedings Mathematical Sciences, 117(4)(2007), 545-554.
• [28] Liu W., Some weighted integral inequalities with a parameter and applications, Acta Applicandae Mathematicae, 109(2)(2010), 389-400.
• [29] Liu W., Yu H., Pan X., New weighted Ostrowski-Grüss-Čebyšev type inequalities, Bulletin of the Korean Mathematical Society, 45(3)(2008), 477-483.
• [30] Liu We., Xue Q., Wang S.-F., Several new perturbed Ostrowski-like type inequalities, J. Inequal. Pure Appl. Math., 8(4)(2007), pp. 6.
• [31] Liu W., Several error inequalities for a quadrature formula with a parameter and applications, Computers & Mathematics with Applications, 56(7)(2008), 1766-1772.
• [32] Liu W., Tuna A., Weighted Ostrowski, Trapezoid and Grüss type inequalities on time scales, J. Math. Inequal., 6(3)(2012), 381-399.
• [33] Liu W., Tuna A., Diamond-a weighted Ostrowski type and Grüss type inequalities on time scales, Applied Mathematics and Computation, 270(2015), 251-260.
• [34] Liu W., Tuna A., Jiang Y., On weighted Ostrowski type, Trapezoid type, Grüss type and Ostrowski-Grass like inequalities on time scales, Applicable Analysis, 93(3), (2014), 551-571.
• [35] Mitrinović D.S., Peĉarić J., Fink A.M., Classical and New Inequalities in Analysis, volume 61, Springer Science & Business Media, 2013.
• [36] Mitrinović D.S., Peĉarić J., Fink A.M., Inequalities Involving Functions and Their Integrals and Derivatives, volume 53, Springer Science & Business Media, 2012.
• [37] Ngô Q.A., Liu W.J., A sharp Grüss type inequality on time scales and application to the sharp Ostrowski-Grüss inequality, Commun. Math. Anal., 6(20)(2009), 33-41.
• [38] Nwaeze Eze R., Generalized weighted trapezoid and Grüss type inequalities on time scales, Aust. J. Math. Anal. Appl., 14(1)(2017), 13.
• [39] Nwaeze Eze R., A new weighted Ostrowski type inequality on arbitrary time scale, Journal of King Saud University-Science, 29(2)(2017), 230-234.
• [40] Nwaeze E.R., New integral inequalities on time scales with applications to the continuous and discrete calculus, Communications in Applied Analysis, 22(1)(2018), 1-17.
• [41] Nwaeze E.R., Kermausuor S., New Bounds of Ostrowski-Grüss type inequality for (k + 1) points on time scales, International Journal of Analysis and Applications, 15(2)(2017), 211-221.
• [42] Nwaeze E.R., Kermausuor S., Tameru A.M., New time scale generalizations of the Ostrowski-Grüss type inequality for k points, Journal of Inequalities and Applications, 201(245)(2017).
• [43] Nwaeze E.R., Tameru A.M., On weighted Montgomery identity for k points and its associates on time scales, Abstract and Applied Analysis, (2017). Art. ID: 5234181.
• [44] Ostrowski A., Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Commentarii Mathematici Helvetici, 10(1)(1937), 226-227.
• [45] Sarikaya M.Z., New weighted Ostrowski and Čebyšev type inequalities on time scales, Computers & Mathematics with Applications, 60(5)(2010), 1510-1514.
• [46] Srivastava H.M., Tseng K.-L., Tseng S.-J., Lo J.-C., Some weighted Opial-type inequalities on time scales, Taiwanese Journal of Mathematics, (2010), 107-122.
• [47] Tuna A., Daghan D., Generalization of Ostrowski and Ostrowski-Grüss type inequalities on time scales, Computers & Mathematics with Applications, 60(3)(2010), 803-811.
• [48] Tuna A., Jiang Y., Liu W., Weighted Ostrowski, Ostrowski-Grüss and Ostrowski-Čebyšev type inequalities on time scales, Publ. Math. Debrecen, 81(2012), 81-102.
• [49] Yeh C.-C., Ostrowski’s inequality on time scales, Applied Mathematics Letters, 21(4)(2008), 404-409.
• [50] Zheng B., Feng Q., Generalized dimensional Ostrowski type and Grüss type inequalities on time scales, Journal of Applied Mathematics, 2014(2014).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).