Weighted Montgomery's identities for higher order differentiable functions of two variables

Asif R. Khan; Josip Pečarić; Marjan Praljak

doi:10.33993/jnaat421-982

Authors

Asif R. Khan University of Karachi and Abdus Salam School of Mathematical Sciences, Pakistan
Josip Pečarić Abdus Salam School of Mathematical Sciences and Zagreb University, Pakistan
Marjan Praljak Abdus Salam School of Mathematical Sciences and Zagreb University, Pakistan

DOI:

https://doi.org/10.33993/jnaat421-982

Keywords:

Montgomery's identities, Ostrowski-type inequalities, Grüss-type inequalities

Abstract views: 327

Abstract

We give weighted Montgomery's identities for higher order differentiable functions of two variables and by using these identities we obtain generalized Ostrowski-type and Grüss-type inequalities for double weighted integrals of higher order differentiable functions of two independent variables.

Downloads

Download data is not yet available.

References

M. Alomari and M. Darus, Some Ostrowski type inequalities for convex functions with applications, RGMIA13 (1) (2010) No. 3, Preprint.

M. Alomari and M. Darus, Some Ostrowski type inequalities for quasi-convex functions with applications to special means, RGMIA13 (2) (2010) No. 3, Preprint.

N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27(1) (2001), pp. 1-10.

C. Buse, P. Cerone, S. S. Dragomir and J. Roumeliotis, A refinement of Grüss type inequality for the Bochner integral of vector-valued functions in Hilbert spaces and applications, J. Korean Math. Soc., 43 (2006), pp. 911-929, https://doi.org/10.4134/jkms.2006.43.5.911 DOI: https://doi.org/10.4134/JKMS.2006.43.5.911

P. Cerone and S. S. Dragomir, A refinment of the Grüss inequality and applications, Tamkang J. Math., 38 (2007), pp. 37-49, https://doi.org/10.5556/j.tkjm.38.2007.92 DOI: https://doi.org/10.5556/j.tkjm.38.2007.92

S. S. Dragomir, New Grüss' type Inequalities for functions of bounded variation and applications, Appl. Math. Lett., 25 (2012), pp. 1475-1479, https://doi.org/10.1016/j.aml.2011.12.027 DOI: https://doi.org/10.1016/j.aml.2011.12.027

S. S. Dragomir, Y. J. Cho, S. S. Kim and Y. Kim, On Bessel's and Grüss inequalities for orthonormal families in 2-inner product spaces and applications, Kyungpook Math. J., 48 (2008), pp. 207-222, https://doi.org/10.5666/kmj.2008.48.2.207 DOI: https://doi.org/10.5666/KMJ.2008.48.2.207

S. S. Dragomir, N. S. Barnett and P. Cerone, An Ostrowski type inequality for double integrals in terms of L_{p}-norms and applications in numerical integration, Rev. Anal. Numér. Théor. Approx., 32(2), (2003), pp. 161-169, http://ictp.acad.ro/jnaat/journal/article/view/2003-vol32-no2-art4

S. S. Dragomir, P. Cerone, N. S.Barnett and J. Roumeliotis, An inequality of the Ostrowski type for double integrals and applications for cubature formulae, Tamsui Oxf. J. Math. Sci., 16 (2000), pp. 1-16.

S. S. Dragomir and T. M. Rassias, Ostrowski type inequalities and applications in numerical integration, Kluwer Academics Publishers, The Netherlands, 2002. DOI: https://doi.org/10.1007/978-94-017-2519-4

G. Grüss, Über das maximum des absoluten betrages von [1/(b-a)]∫abf(x)g(x)dx-[1/(b-a)²]∫abf(x)dx∫abg(x)dx, Math. Z., 39 (1935), pp. 215-226, https://doi.org/10.1007/bf01201355 DOI: https://doi.org/10.1007/BF01201355

A. Guezane-Lakoud and F. Aissaoui, New Cebysev's type inequalitities for double integrals, J. Math. Ineq., 5 (2011), pp. 453-462,https://doi.org/10.7153/jmi-05-40 DOI: https://doi.org/10.7153/jmi-05-40

A. R. Khan, J. Pečari c and S. Varosanec, Popoviciu type characterization of positivity of sums and integrals for convex functions of higher order, J. Math. Ineq., 7 (2013), pp. 195-212,https://doi.org/10.7153/jmi-07-19 DOI: https://doi.org/10.7153/jmi-07-19

W. J. Liu, New bounds for the companions of Ostrowski's inequality and applications, submitted (http://arxiv.org/abs/1205.4549 for preprints).

D. S. Mitrinovic, J. Pecaric and A. M. Fink, Inequalities for functions and their integrals and derivatives, Kluwer Academic Publishers, Dordrecht, 1994.

A. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihren integralmittelwert, Comment. Math. Helv., 10 (1938), pp. 226-227, https://doi.org/10.1007/bf01214290 DOI: https://doi.org/10.1007/BF01214290

J. Pecaric, On the Cebysev's inequality, Bull. Inst. Politeh. Timisoara, 25(39) (1980), pp.10-11.

J. Pecaric, On the Ostrowski's generalization of Cebysev's inequality, J. Math. Anal. Appl., 102 (1984), pp. 479-487, https://doi.org/10.1016/0022-247x(84)90187-2 DOI: https://doi.org/10.1016/0022-247X(84)90187-2

J. Pecaric, Some further remarks on the Ostrowski's generalization of Cebysev's inequality, J. Math. Anal. Appl., 123 (1987), pp. 18-33,https://doi.org/10.1016/0022-247x(87)90290-3 DOI: https://doi.org/10.1016/0022-247X(87)90290-3

J. Pecaric and A. Vukelic, Montgomery's identities for functions of two variables, J. Math. Anal. Appl., 332 (2007), pp. 617-630, https://doi.org/10.1016/j.jmaa.2006.10.028 DOI: https://doi.org/10.1016/j.jmaa.2006.10.028