Some new inequalities of the Ostrowski-Grüss, Čebyšev, and Trapezoid types on time scales
Authors
Eze R. Nwaeze
- Department of Mathematics, Tuskegee University, Tuskegee, AL 36088, USA.
Nurhan Kaplan
- Art and Science Faculty, Mathematics Department, Niğde Ömer Halisdemir University, Niğde, Turkey.
Fatma Gozde Tuna
- Art and Science Faculty, Mathematics Department, Niğde Ömer Halisdemir University, Niğde, Turkey.
Adnan Tuna
- Art and Science Faculty, Mathematics Department, Niğde Ömer Halisdemir University, Niğde, Turkey.
Abstract
In this paper, we establish some novel Ostrowski-Grüss, Čebyšev, and
Trapezoid type inequalities involving functions whose second derivatives are
bounded on time scales. We also give some other interesting inequalities as
special cases of our results.
Keywords
- Ostrowski's inequality
- Čebyšev inequality
- Ostrowski-Grüss
- Trapezoid inequality
- time scales
MSC
References
-
[1]
R. Agarwal, M. Bohner, A. Peterson , Inequalities on time scales: a survey, Math. Inequal. Appl., 4 (2001), 535–557.
-
[2]
E. Akin-Bohner, M. Bohner, T. Matthews, Time scales Ostrowski and Grüss type inequalities involving three functions, Nonlinear Dyn. Syst. Theory, 12 (2012), 119–135.
-
[3]
M. Bohner, T. Matthews, Ostrowski inequalities on time scales , JIPAM. J. Inequal. Pure Appl. Math., 9 (2008), 8 pages.
-
[4]
M. Bohner, E. R. Nwaeze, A. Tuna, Trapezoid-Type Inequalities on Time Scales, , (Submitted),
-
[5]
M. Bohner, A. Peterson, Advances in dynamic equations on time scales , Birkhäuser Boston, Boston (2003)
-
[6]
M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser Boston, Boston (2001)
-
[7]
P. L. Čebyšev , Sue les expressions approxmatives des intégrales définies par les autres prises entre les mêmes limites , Proc. Math. Soc. Charkov, 2 (1882), 93–98.
-
[8]
A. A. El-Deeb, H. A. Elsennary, E. R. Nwaeze, Generalized Weighted Ostrowski, Trapezoid and Grüss Type Inequalities on Time Scales, Fasc. Math., 60 (2018), 123–144.
-
[9]
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universität Würzburg, Würzburg, Germany (1988)
-
[10]
V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan , Dynamic systems on measure chains, Kluwer Academic Publishers Group, Dordrecht (1996)
-
[11]
W. J. Liu, Q.-A. Ngô, W. B. Chen, A new generalization of Ostrowski type inequality on time scales, An. St. Univ. Ovidius Constanta, 17 (2009), 101–114.
-
[12]
W. J. Liu, A. Tuna, Diamond weighted Ostrowski type and Grüss type inequalities on time scales, Appl. Math. Comput., 270 (2015), 251–260.
-
[13]
W. J. Liu, A. Tuna, Weighted Ostrowski, Trapezoid and Grüss type inequalities on time scales, J. Math. Inequal., 6 (2012), 381–399.
-
[14]
W. J. Liu, A. Tuna, Y. Jiang, New weighted Ostrowski and Ostrowski-Grüss type inequalities on time scales , An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), 60 (2014), 57–76.
-
[15]
W. J. Liu, A. Tuna, Y. Jiang, On weighted Ostrowski type, Trapezoid type, Grüss type and Ostrowski-Grüss like inequalities on time scales, Appl. Anal., 93 (2014), 551–571.
-
[16]
E. R. Nwaeze , Generalized weighted trapezoid and Grüss type inequalities on time scales, Aust. J. Math. Anal. Appl., 14 (2017), 13 pages.
-
[17]
A. Ostrowski , Uber die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1937), 226–227.
-
[18]
B. G. Pachpatte , On trapezoid and Grüss-like integral inequalities , Tamkang J. Math., 34 (2003), 365–369.
-
[19]
A. Tuna, Y. Jiang, W. J. Liu , Weighted Ostrowski, Ostrowski-Grüss and Ostrowski-Čebyšev Type Inequalities on Time Scales , Publ. Math. Debrecen, 81 (2012), 81–102.
-
[20]
A. Tuna, W. Liu, New weighted Čebyšev-Ostrowski type integral inequalities on time scales, J. Math. Inequal., 10 (2016), 327–356.
-
[21]
G. P. Xu, Z. B. Fang, A New Ostrowski type inequality on time scales, J. Math. Inequal., 10 (2016), 751–760.