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**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.

Reviewer: James Adedayo Oguntuase (Abeokuta)

### MSC:

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 |

### Keywords:

Hermite-Hadamard-Fejer inequality; Simpson inequality; co-ordinated convex; integral inequalities
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\textit{H. Budak} et al., J. Math. Ext. 15, No. 1, 149--177 (2021; Zbl 1483.26014)

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### References:

[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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