Some inequalities for double integrals and applications for cubature formula

Samet Erden; Mehmet Zeki Sarikaya

doi:10.2478/ausm-2019-0021

Abstract

We establish two Ostrowski type inequalities for double integrals of second order partial derivable functions which are bounded. Then, we deduce some inequalities of Hermite-Hadamard type for double integrals of functions whose partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Finally, some applications in Numerical Analysis in connection with cubature formula are given.

References

[1] F. Ahmad, N. S. Barnett and S. S. Dragomir, New Weighted Ostrowski and Cebysev Type Inequalities, Nonlinear Analysis: Theory, Methods & Appl., 71 (12) (2009), 1408–1412.10.1016/j.na.2009.01.178
Search in Google Scholar Back to article
[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), 109–114.
Search in Google Scholar Back to article
[3] M. K. Bakula, M. E.Özdemir and J. Pečarić, Hadamard type inequalities for m–convex and (α, m)–convex functions, J. Inequal. Pure and Appl. Math. 9 (4) (2008), Art. 96.
Search in Google Scholar Back to article
[4] F. Chen, On Hermite–Hadamard type inequalities for s–convex functions on the coordinates via Riemann–Liouville fractional integrals, J. of Appl. Math., Volume 2014, Article ID 248710, 8 pages.10.1155/2014/248710
Search in Google Scholar Back to article
[5] S. S. Dragomir, On Hadamard’s inequality for convex functions on the co–ordinates in a rectangle from the plane, Taiwanese J. of Math., 4 (2001), 775–788.10.11650/twjm/1500574995
Search in Google Scholar Back to article
[6] S. S. Dragomir, N. S. Barnett and P. Cerone, An Ostrowski type inequality for double integrals in term of Lp–norms and Applications in numerical integrations, Anal. Num. Theor. Approx., 2 (12) (1998), 1–10.
Search in Google Scholar Back to article
[7] S. Erden and M. Z. Sarikaya, On the Hermite–Hadamard’s and Ostrowski’s inequalities for the co–ordinated convex functions, RGMIA Research Report Collection, Vol. 18 (2015). Art. 57.
Search in Google Scholar Back to article
[8] S. Erden and M. Z. Sarikaya, On the Hermite–Hadamard–Type and Ostrowski Type inequalities for the co–ordinated convex functions, Palestine J. of Math., in press.
Search in Google Scholar Back to article
[9] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction considree par, Riemann, J. Math. Pures. et Appl. 58 (1893), 171–215.
Search in Google Scholar Back to article
[10] M. A. Latif and M. Alomari, Hadamard–type inequalities for product two convex functions on the co–ordinetes, Int. Math. Forum, 4 (47) (2009), 2327–2338.
Search in Google Scholar Back to article
[11] M. A. Latif and M. Alomari, On the Hadamard–type inequalities for h–convex functions on the co–ordinetes, Int. J. of Math. Analysis, 3 (33) (2009) 1645–1656.
Search in Google Scholar Back to article
[12] M. A. Latif and S.S. Dragomir, On some new inequalities for differentiable co–ordinated convex functions, J. of Inequalities and Appl., (2012), 2012:28.10.1186/1029-242X-2012-28
Search in Google Scholar Back to article
[13] M. A. Latif, S. Hussain and S. S. Dragomir, New Ostrowski type inequalities for co–ordinated convex functions, TJMM, 4 (2) (2012), 125–136.10.1186/1029-242X-2012-28
Search in Google Scholar Back to article
[14] M. E. Ozdemir, E. Set and M.Z. Sarikaya, New some Hadamard’s type inequalities for coordinated m–convex and (α, m)n–convex functions, Hacettepe J. of Math. and Statistics, 40 (2) (2011), 219–229.
Search in Google Scholar Back to article
[15] A. M. Ostrowski, “Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10 (1938), 226–227.10.1007/BF01214290
Search in Google Scholar Back to article
[16] B. G. Pachpatte, On a new Ostrowski type inequality in two independent variables, Tamkang J. Math., 32 (1) (2001), 45–49.10.5556/j.tkjm.32.2001.367
Search in Google Scholar Back to article
[17] J. Park, Some Hadamard’s type inequalities for co–ordinated (s;m)–convex mappings in the second sense, Far East J. of Mathematical Sciences, 51 (2) (2011), 205–216.
Search in Google Scholar Back to article
[18] A. Rafiq and F. Ahmad, Another weighted Ostrowski–Grüss type inequality for twice differentiable mappings, Kragujevac J. of Math., 31 (2008), 43–51.
Search in Google Scholar Back to article
[19] M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The Arabian J. for Science and Eng. (AJSE)–Math., (2011) 36: 1153–1160.10.1007/s13369-011-0102-4
Search in Google Scholar Back to article
[20] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. LXXIX, 1 (2010), 129–134.
Search in Google Scholar Back to article
[21] M. Z. Sarikaya, E. Set, M. E.Özdemir and S. S. Dragomir, New some Hadamard’s type inequalities for co–ordineted convex functions, Tamsui Oxford J. of Information and Mathematical Sciences, 28 (2) (2012), 137–152.
Search in Google Scholar Back to article
[22] M. Z. Sarikaya, On the Hermite–Hadamard–type inequalities for co–ordinated convex function via fractional integrals, Integral Transforms and Special Functions, Vol. 25, No. 2 (2014), 134–147.10.1080/10652469.2013.824436
Search in Google Scholar Back to article
[23] M. Z. Sarikaya, Some inequalities for differentiable co–ordinated convex mappings, Asian–European J. of Math., Vol: 08, 1550058 (2015), 21 pages.10.1142/S1793557115500588
Search in Google Scholar Back to article
[24] M. Z. Sarikaya, On the generalized weighted integral inequality for double integrals, Annals of the Alexandru Ioan Cuza University – Math., 61 (1) (2015), 169–179.10.2478/aicu-2014-0008
Search in Google Scholar Back to article

Some inequalities for double integrals and applications for cubature formula

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