New Ostrowski and Ostrowski-Gruss type inequalities for double integrals on time scales involving a combination of $\Delta$-integral means

Seth Kermausuor, Eze Raymond Nwaeze

Abstract


In 2014, some Ostrowski type inequalities for functions of a single variable were obtained in [Y. Jiang, H. R\"uzgar, W. J. Liu and A. Tuna: Some new generalizations of Ostrowski type inequalities on time scales involving combination of $\Delta$-integral means, J. Nonlinear Sci. Appl., {\bf{7}} (2014), 311--324]. In this paper, we extend some of the inequalities obtained in the above paper for double integrals. One of our results generalizes a result in the article [W. J. Liu, Q. A. Ng\^o and W. Chen: On new Ostrowski type inequalities for double integrals on time scales, Dyn. Syst. Appl., {\bf 19} (2010), 189--198]. We also apply our results to the continuous, discrete and quantum time scales to obtain some interesting inequalities.

Keywords


Ostrowski inequality, Gr\"uss inequality, double integrals, $\Delta$-integral means, time scales

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References


R. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl., 4(4) (2001), 535--557.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser Boston Inc., Boston, MA, An Introduction with Applications, 2001.

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser Boston, Boston, MA, 2003.

M. Bohner and G. S. Guseinov, Partial differentiation on time scales. Dyn. Syst. Appl., 13(3--4) (2004), 351--379.

M. Bohner and G. S. Guseinov, Multiple integration on time scales, Dyn. Syst. Appl., 14(3--4) (2005), 579--606.

M. Bohner and T. Matthews, Ostrowski inequalities on time scales, J. Inequal. Pure Appl. Math., 9(6) (2008), 8pp .

P. Cerone, A new Ostrowski type inequality involving integral means over end intervals, Tamkang J. Math., 33(2002), 109-118.

S. S. Dragomir, N. S. Barnett, An Ostrowski type inequality for mappings whose second derivatives are bounded and applications, Indian J. of Math., 66(1-4) (1999), 237-245.

S. Hilger, Ein Mass kettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universitat Wurzburg, Wurzburg, Germany, (1988).

S. Hussain, M. A. Latif and M. Alomari, Generalized double integral Ostrowski type inequalities on time scales, Appl. Math. Lett., 24}(8) (2011), 1461--1467.

W. Irshad, M. I. Bhatti and M. Muddassar, Some Ostrowski type integral inequalities for double integral on time scales,J. Comput. Anal. Appl., 20(5) (2016).

Y. Jiang, H. Ruzgar, W. J. Liu and A. Tuna, Some new generalizations of Ostrowski type inequalities on time scales involving combination of $Delta$-integral means, J. Nonlinear Sci. Appl., 7(2014), 311--324.

S. Kermausuor, E. R. Nwaeze and D. F. M. Torres, Generalized weighted Ostrowski and Ostrowski--Gruss type inequalities on time scale via a parameter function, J. Math. Inequal., 11(2017), 1185--1199.

V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Mathematics and its Applications, 370, Kluwer Academic Publishers Group, Dordrecht, 1996

W. J. Liu and Q. A Ngo, W. Chen, Ostrowski type inequalities on time scales for double integrals, Acta Appl.Math., 110(1) (2010), 477--497.

[ W. J. Liu, Q. A. Ngo and W. Chen, On new Ostrowski type inequalities for double integrals on time scales, Dyn. Sys. Appl.,19(2010), 189--198.

W. J. Liu and Q. A. Ngo, W. Chen, New Generalization of Ostrowski Type Inequality on Time Scales, An. St.Univ. Ovidius Constancta, 17(2) (2009), 101--114.

E. R. Nwaeze, A new weighted Ostrowski type inequality on arbitrary time scale, J. King Saud Uni. Sci., 29(2) (2017), 230--234.

E. R. Nwaeze, New integral inequalities on time scales with applications to the continuous and discrete calculus, Communications in Applied Analysis., 22(1) (2018), 1--17.

A. M. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmitelwert, Comment. Math. Helv., 10(1938), 226--227.

U. M.Ozkan and H. Yildirim, Ostrowski type inequality for double integrals on time scales, Acta Appl. Math.,110 (1) (2010), 283--288.

A. Tuna and S. Kutukcu, New generalization of the Ostrowski inequality and Ostrowski type inequality for double integrals on time scales, J. Computat. Anal. Appl., 21(6) (2016).

A. Tuna, D. Daghan, Generalization of Ostrowski and Ostrowski-Gruss type inequalities on time scales, Comput. Math. Appl., 60(2010), 803--811.




DOI: http://dx.doi.org/10.5556/j.tkjm.49.2018.2632

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