Weighted Hermite-Hadamard and Simpson type inequalities for double integrals. (English) Zbl 1483.26014

In this paper, the authors derive and prove a weighted identity for twice partially differenciable mapping. In addition, the derived identity was used to establish a weighted Hermite-Hadamard-type inequality for co-ordinated convex functions on \(\mathbb{R^2}\). Furthermore, some weighted generalizations of Hermite-Hadamard- and Simpson-type integral inequalities were established and well proved.


26D07 Inequalities involving other types of functions
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
26B15 Integration of real functions of several variables: length, area, volume
26B25 Convexity of real functions of several variables, generalizations
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[1] M. Alomari and M. Darus, The Hadamards inequality fors-convex function of 2-variables on the coordinates.Int. J. Math. Anal.,2 (13) (2008), 629-638. · Zbl 1178.26017
[2] M. Alomari and M. Darus, Fej´er inequality for double integrals, Facta Universitatis (NIˇS),Ser. Math. Inform.,24 (2009), 15-28. · Zbl 1265.26059
[3] M. Alomari, M. Darus, and S. S. Dragomir, New inequalities of Simpson´ıs type fors-convex functions with applications,RGMIA Res. Rep. Coll.,12 (4) (2009), Article 9.
[4] M. K. Bakula, An improvement of the Hermite-Hadamard inequality for functions convex on the coordinates,Australian journal of mathematical analysis and applications,11 (1) (2014), 1-7. · Zbl 1297.26054
[5] H. Budak and M. Z. Sarıkaya,Hermite-Hadamard-Fej´er inequalities for double integrals, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70 (1) (2021), 100-116. · Zbl 1489.26026
[6] F. Chen, A note on the Hermite-Hadamard inequality for convex functions on the co-ordinates,J. Math. Inequal.,8 (4) (2014), 915-923. · Zbl 1305.26043
[7] S. S. Dragomir, On Hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane.Taiwan. J. Math.,4 (2001), 775-788. · Zbl 1002.26017
[8] S. S. Dragomir and C. E. M. Pearce,Selected Topics on HermiteHadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
[9] S. S. Dragomir, R. P. Agarwal, and P. Cerone, On Simpson´ıs inequality and applications,J. of Inequal. Appl.,5 (2000), 533-579. · Zbl 0976.26012
[10] S. S. Dragomir, On Simpson’s quadrature formula for Lipschitzian mappings and applications,Soochow J. Mathematics,25 (1999), 175-180. · Zbl 0938.26014
[11] T. Du, Y. Li, and Z. Yang, A generalization of Simpson’s inequality via differentiable mapping using extended (s, m)-convex functions,Applied Mathematics and Computation,293 (2017), 358-369. · Zbl 1411.26020
[12] G. Farid, M. Marwan, and Atiq Ur Rehman, Fejer-Hadamard inequlality for convex functions on the co-ordinates in a rectangle from the plane,International Journal of Analysis and Applications,10 (1) (2016), 40-47. · Zbl 1376.26018
[13] L. Fejer, ¨Uber die Fourierreihen,II. Math. Naturwiss. Anz Ungar. Akad. Wiss.,24 (1906), 369-390. (Hungarian).
[14] S. Hussain and S. Qaisar,More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings. Springer Plus (2016), 5:77.
[15] H. Kavurmaci, A. O. Akdemir, E. Set, and M. Z. Sarikaya, Simpson’s type inequalities form−and (α, m)-geometrically convex functions,Konuralp Journal of Mathematics,2 (1) (2014), 90-101. · Zbl 1305.26046
[16] M. A. Latif, S. Hussain, and S. S. Dragomir, On some new Fejer-type inequalities for coordinated convex functions,TJMM.,3 (2) (2011), 5780. · Zbl 1275.26041
[17] M. A. Latif, On some Fejer-type inequalities for double integrals,Tamkang Journal of Mathematics,43 (3) (2012), 423-436. · Zbl 1257.26022
[18] M. A. Latif, S. S. Dragomir, and E. Momoniat,Weighted generalization of some integral inequalities for differentiable co-ordinated convex functions,Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys.,78 (4) (2016), 197-210. · Zbl 1513.26058
[19] M. A. Latif, S. S. Dragomir, and E. Momoniat, Generalization of some Inequalities for differentiable co-ordinated convex functions with applications,Moroccan J. Pure and Appl. Anal.,2 (1) (2016), 12-32. · Zbl 1492.26032
[20] L. M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions,J. Inequal. Appl.,28 (2012), 28. · Zbl 1279.26027
[21] B. Z. Liu, An inequality of Simpson type,Proc. R. Soc. A.,461 (2005), 2155-2158. · Zbl 1186.26017
[22] M. E. Ozdemir, C. Yildiz, and A. O. Akdemir, On the co-ordinated convex functions,Appl. Math. Inf. Sci.,8 (3) (2014), 1085-1091.
[23] M. E. Ozdemir, A. O. Akdemir, and H. Kavurmacı, On the Simpson’s inequality for convex functions on the co-ordinates,Turkish Journal of Analysis and Number Theory,2 (5) (2014), 165-169.
[24] J. E. Peˇcari´c, F. Proschan, and Y. L. Tong,Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992. · Zbl 0749.26004
[25] J. Pecaric and S. Varosanec, A note on Simpson’s inequality for functions of bounded variation,Tamkang Journal of Mathematics,31 (3) (2000), 239-242. · Zbl 0987.26014
[26] S. Qaisar, C. J. He, and S. Hussain, A generalizations of Simpson’s type inequality for differentiable functions using (α,m)-convex functions and applications,J. Inequal. Appl.,2013 (2013), 13, Article 158. · Zbl 1284.26035
[27] M. Z. Sarikaya, E. Set, M. E. Ozdemir, and S. S. Dragomir ,New some Hadamard’s type inequalities for co-ordinated convex functions,Tamsui Oxford Journal of Information and Mathematical Sciences,28 (2) (2012), 137-152. · Zbl 1270.26022
[28] M. Z. Sarikaya, E. Set, and M. E. Ozdemir, On new inequalities of Simpson’s type fors-convex functions,Computers and Mathematics with Applications,60 (2010), 2191-2199. · Zbl 1205.65132
[29] M. Z. Sarikaya, E. Set, and M. E. ¨Ozdemir, On new inequalities of Simpson’s type for convex functions,RGMIA Res. Rep. Coll.,13 (2) (2010), Article 2. · Zbl 1205.65132
[30] M. Z. Sarikaya, E. Set, and M. E. Ozdemir, On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex,Journal of Applied Mathematics, Statistics and Informatics,9 (1)(2013). · Zbl 1279.26051
[31] M. Z. Sarıkaya, T. Tun¸c, and H. Budak, Simpson’s type inequality for F-convex function,Facta Universitatis Ser. Math. Inform.,32 (5) (2017), 747-753. · Zbl 1474.26136
[32] E. Set, M. E. ¨Ozdemir, and S. S. Dragomir, On the Hermite-Hadamard inequality and other integral inequalities involving two functions,J. Inequal. Appl.,(2010), 9. Article ID 148102. · Zbl 1194.26037
[33] E. Set, M. E. Ozdemir and M. Z. Sarikaya, On new inequalities of Simpson’s type for quasi-convex functions with applications,Tamkang Journal of Mathematics,43 (3) (2012), 357-364. · Zbl 1257.26026
[34] E. Set, M. Z. Sarikaya, and N. Uygun, On new inequalities of Simpson’s type for generalized quasi-convex functions,Advances in Inequalities and Applications,3 (2017), 1-11.
[35] K. L. Tseng, G. S. Yang, and S. S. Dragomir, On weighted Simpson type inequalities and applications,Journal of mathematical inequalities,1 (1) (2007), 13-22. · Zbl 1145.26009
[36] D. Y. Wang, K. L. Tseng, and G. S. Yang, Some Hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane.Taiwan. J. Math.,11 (2007), 63-73. · Zbl 1132.26360
[37] N. Ujevic, Double integral inequalities of Simpson type and applications,J. Appl. Math. Comput.,14 (1-2) (2004), 213-223. · Zbl 1042.26012
[38] B. Y. Xi, J. Hua, and F. Qi, Hermite-Hadamard type inequalities for extendeds-convex functions on the co-ordinates in a rectangle.J. Appl. Anal.,20 (1) (2014), 1-17. · Zbl 1296.26105
[39] R. Xiang and F. Chen, On some integral inequalities related to HermiteHadamard-Fej´er inequalities for coordinated convex functions,Chinese Journal of Mathematics,2014, Article ID 796132, 10 pages. · Zbl 1317.26021
[40] Z. Q. Yang, Y. J. Li, and T. Du, A generalization of Simpson type inequality via differentiable functions using (s, m) -convex functions,Ital. J. Pure Appl. Math.,35 (2015), 327-338. · Zbl 1339.26071
[41] M. E. Yıldırım, A. Akkurt, and H. Yıldırım, Hermite-Hadamard type inequalities for co-ordinated (α1, m1)−(α2, m2)-convex functions via fractional integrals,Contemporary Analysis and Applied Mathematics,4 (1) (2016), 48-63. · Zbl 1350.26040
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