Document Type : Research Paper


1 Department of Mathematics, University of Karachi, University Road, Karachi-75270 Pakistan.

2 Department of Mathematics, Federal Urdu University of Arts, Science and Technology , University Road, Karachi-75270 Pakistan.

3 Department of Basic Sciences, Muhammad Ali Jinnah University, P.E.C.H.S. Main Shahrah-e-Faisal, Karachi-75400, Pakistan.


The purpose of this article is to generalize Cebysev type inequalities for double integrals involving a weight function.
By using an integral transform that is a weighted Montgomery identity, we obtained a generalized form of weighted Cebysev type inequalities in $L_m,\, m\geq 1$ norm of differentiable functions. Also, we give some applications of the probability density function.


[1] F. Ahmad, N.S. Barnett and S.S. Dragomir, New weighted Ostrowski and Cebysev type inequalities, Nonlinear Analysis, 71 (2009), pp. 1408-1412.
[2] 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.
[3] K. Boukerrioua and A. Guezane-Lakoud, On generalization of Cebysev type inequalities, J. Inequal. Pure Appl. Math., 8(2), (2007), Art 55.
[4] P.L. Cebysev, Sur les expressions approximatives des integrales definies par les autres prises entre les m^emes limites, Proc. Math. Soc. Charkov, 2(1882), pp. 93-98.
[5] S. Sever Dragomir, N. Irshad and A.R. Khan, Generalization of weighted Ostrowski-Gruss type inequality by using Korkine's identity, Stud. Univ. Babes-Bolyai, Math., 65(2), (2020), pp. 183-198.
[6] 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(1), (2000), pp. 1-16.
[7] N. Irshad and A.R. Khan, On Weighted Ostrowski Gruss Inequality with Applications, Transylv. J. Math. Mech., 10(1), (2018), pp. 15-22.
[8] A.R. Khan, J. Pecari'c and M. Praljak, Weighted Montgomery inequalities for higher order differentiable functions of two variables, Rev. Anal. Numer. Theor. Approx., 42(1), (2013), pp. 49-71.
[9] A. Guezane-Lakoud and F. Aissaoui, New Cebysev Type Inequalities For Double Integrals, J. Math. Inequal., 5(4), (2011), pp. 453-462.
[10] D.S. Mitrinovic, J.E. Pecaric and A.M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
[11] D.S. Mitrinovic, J.E. Pecaric and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
[12] A.M. Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihren integralmittelwert, Comment. Math. Helv., 10 (1938), pp. 226-227.
[13] B.G. Pachpatte, On Cebysev-Gruss type inequalities via Pecaric’s extention of the Montgomery identity, J. Inequal. Pure Appl. Math., 7(1), (2006), Art 11.
[14] J. Pecaric and A. Vukelic, Montgomery's identities for function of two variables, J. Math. Anal. Appl., 332 (2007), pp. 617-630.
[15] G. Rahman, K. Sooppy Nisar, B. Ghanbari and T. Abdeljawad, On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals, Adv. Difference Equ., (2020), 2020:368.
[16] R.L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, The New York, 1977.