Some integral inequalities of the Hermite--Hadamard type for log-convex functions on co-ordinates

Bai, Yu-Mei¹ ; Qi, Feng²

¹ College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China
² Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China;Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 12, p. 5900-5908.

Résumé

In the paper, the authors establish some new integral inequalities for log-convex functions on co-ordinates. These newly-established inequalities are connected with integral inequalities of the Hermite-Hadamard type for log-convex functions on co-ordinates.

Détail

DOI : 10.22436/jnsa.009.12.01

Classification : 26A51, 26D15, 26D20, 26E60, 41A55
Keywords: Log-convex functions, co-ordinates, integral inequality, Hermite-Hadamard type.

Bibliographie

[1] Alomari, M.; Darus, M. On the Hadamard's inequality for log-convex functions on the coordinates, J. Inequal. Appl., Volume 2009 (2009), pp. 1-13

[2] Bai, S.-P.; Qi, F.; Wang, S.-H. Some new integral inequalities of Hermite-Hadamard type for (\(\alpha,m,P\))-convex functions on co-ordinates, J. Appl. Anal. Comput., Volume 6 (2016), pp. 171-178

[3] Dragomir, S. S. On the Hadamards inequlality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., Volume 5 (2001), pp. 775-788

[4] Dragomir, S. S.; Pearce, C. E. M. Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2002

[5] Gill, P. M.; Pearce, C. E. M.; Pečarić, J. Hadamard's inequality for r-convex functions, J. Math. Anal. Appl., Volume 215 (1997), pp. 461-470

[6] Guo, X.-Y.; Qi, F.; Xi, B.-Y. Some new inequalities of Hermite-Hadamard type for geometrically mean convex functions on the co-ordinates, J. Comput. Anal. Appl., Volume 21 (2016), pp. 144-155

[7] Hwang, D.-Y.; Tseng, K.-L.; Yang, G.-S. Some Hadamard's inequalities for co-ordinated convex functions in a rectangle from the plane, Taiwanese J. Math., Volume 11 (2007), pp. 63-73

[8] Bakula, M. Klaričić; Pečarić, J. On the Jensen's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., Volume 10 (2006), pp. 1271-1292

[9] Özdemir, M. E.; Akdemir, A. O.; Kavurmacı, H. On the Simpsons inequality for co-ordinated convex functions, Turkish J. Anal. Number Theory, Volume 2 (2014), pp. 165-169

[10] Özdemir, M. E.; Akdemir, A. O.; Yıldız, Ç . On co-ordinated quasi-convex functions, Czechoslovak Math. J., Volume 62 (2012), pp. 889-900

[11] Özdemir, M. E.; Set, E.; Sarikaya, M. Z. Some new Hadamard type inequalities for co-ordinated m-convex and (\(\alpha,m\))-convex functions, Hacet. J. Math. Stat., Volume 40 (2011), pp. 219-229

[12] Özdemir, M. E.; Yıldız, Ç .; Akdemir, A. O. On some new Hadamard-type inequalities for co-ordinated quasi-convex functions, Hacet. J. Math. Stat., Volume 41 (2012), pp. 697-707

[13] Qi, F.; Xi, B.-Y. Some integral inequalities of Simpson type for \(GA-\varepsilon\)-convex functions, Georgian Math. J., Volume 20 (2013), pp. 775-788

[14] Sarikaya, M. Z. On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integral Transforms Spec. Funct., Volume 25 (2014), pp. 134-147

[15] Sarikaya, M. Z. Some inequalities for differentiable coordinated convex mappings, Asian-Eur. J. Math., Volume 8 (2015), pp. 1-21

[16] Sarikaya, M. Z.; Budak, H.; Yaldiz, H. Čebyševtype inequalities for co-ordinated convex functions, Pure Appl. Math. Lett., Volume 2 (2014), pp. 36-40

[17] Sarikaya, M. Z.; Budak, H.; Yaldiz, H. Some new Ostrowski type inequalities for co-ordinated convex functions, Turkish J. Anal. Number Theory, Volume 2 (2014), pp. 176-182

[18] Sarikaya, M. Z.; Set, E.; Ozdemir, M. E.; Dragomir, S. S. New some Hadamard's type inequalities for co-ordinated convex functions, Tamsui Oxf. J. Inf. Math. Sci., Volume 28 (2012), pp. 137-152

[19] Set, E.; Sarikaya, M. Z.; Akdemir, A. O. Hadamard type inequalities for \(\varphi\)-convex functions on co-ordinates, Tbilisi Math. J., Volume 7 (2014), pp. 51-60

[20] Set, E.; Sarikaya, M. Z.; Ögülmüş, H. Some new inequalities of Hermite-Hadamard type for h-convex functions on the co-ordinates via fractional integrals, Facta Univ. Ser. Math. Inform., Volume 29 (2014), pp. 397-414

[21] Wang, S.-H.; Qi, F. Hermite-Hadamard type inequalities for s-convex functions via Riemann-Liouville fractional integrals, J. Comput. Anal. Appl., Volume 22 (2017), pp. 1124-1134

[22] Wang, Y.; Xi, B.-Y.; Qi, F. Integral inequalities of Hermite-Hadamard type for functions whose derivatives are strongly \(\alpha\)-preinvex, Acta Math. Acad. Paedagog. Nyházi. (N.S.), Volume 32 (2016), pp. 79-87

[23] Wu, Y.; Qi, F.; On some Hermite-Hadamard type inequalities for (s;QC)-convex functions, SpringerPlus, Volume 5 (2016), pp. 1-13

[24] Xi, B.-Y.; Qi, F. Integral inequalities of Simpson type for logarithmically convex functions, Adv. Stud. Contemp. Math. (Kyungshang), Volume 23 (2013), pp. 559-566

[25] Zhang, J.; Qi, F.; Xu, G.-C.; Pei, Z.-L. Hermite-Hadamard type inequalities for n-times differentiable and geometrically quasi-convex functions, SpringerPlus, Volume 5 (2016), pp. 1-6