SOME HERMITE-HADAMARD AND SIMPSON LIKE INEQUALITIES FOR s-GEOMETRICALLY CONVEX FUNCTION

Year 2015, Volume: 3 Issue: 2, 1 - 19, 30.10.2015

Mevlüt Tunç , Ebru Yüksel

https://doi.org/10.36753/mathenot.421321

Abstract

In the paper, the authors establish and generalize some new integral
inequalities of Hermite-Hadamard and Simpson type for functions the
power of the absolute of whose first derivative is s-geometrically convex.

Keywords

Convex function, Hermite-Hadamard integral inequality, Simpson integral inequality, s-geometrically convex function

References

• [1] Alomari, M., Darus, M., Dragomir, S. S., New inequalities of Simpson’s type for sconvex functions with applications. RGMIA Res. Rep. Coll. 12 (4) (2009) Article 9. Online http://ajmaa.org/RGMIA/v12n4.php.
• [2] Alomari, M., Darus, M., Kırmacı, U. S., Refinements of Hadamard-type inequalities for quasiconvex functions with applications to trapezoidal formula and to special means, Comp. and Math. with Appl. Vol.59 (2010), 225-232.
• [3] Bai, R.-F., Qi, F., Xi, B.-Y., Hermite-Hadamard type inequalities for the m- and (α, m)- logarithmically convex functions. Filomat, 27 (2013), 1-7.
• [4] Dragomir, S. S., Agarwal, R. P., Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998) no. 5, 91-95.
• [5] Dragomir, S. S., Agarwal, R. P., Cerone, P., On Simpson’s inequality and applications. J. of Ineq. and Appl., 5 (2000), 533-579.
• [6] Dragomir, S. S., Pearce, C. E. M., Selected topics on Hermite-Hadamard inequalities and applications, RGMIA monographs, Victoria University, 2000. [Online:http://www.staff.vu.edu.au/RGMIA/monographs/hermite-hadamard.html].
• [7] Hadamard, J., Etude sur les propri´et´es des fonctions enti`eres et en particulier d’une fonction ´ consider´ee par Riemann. J. Math Pures Appl., 58, (1893) 171-215.
• [8] Hudzik, H., Maligranda, L., Some remarks on s-convex functions. Aequationes Math., Vol. 48 (1994), 100-111.
• [9] Mitrinovic, D. S., Pecaric, J., Fink,A. M., Classical and new inequalities in analysis. KluwerAcademic, Dordrecht, 1993.
• [10] Pecari´c, J. E., Proschan, F. Tong, Y. L., Convex Functions, Partial Orderings, and Statistical Applications. Academic Press Inc., 1992.
• [11] Sarikaya, M. Z., Set, E., Ozdemir, M.E., On new inequalities of Simpson’s type for convex ¨ functions. RGMIA Res. Rep. Coll. 13 (2) (2010) Article2.
• [12] Sarikaya, M. Z., Set, E., Ozdemir, M.E., On new inequalities of Simpson’s type for ¨ s-convex functions. Comp. and Math. with Appl. 60 (2010) 2191-2199.
• [13] Tunç, M., On some new inequalities for convex functions. Turk. J. Math. 36 (2012), 245-251.
• [14] Xi, B.-Y., Qi, F., Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means. Journal of Function Spaces and Appl., Volume 2012, Article ID 980438, 14 p., doi:10.1155/2012/980438.
• [15] Zhang, T.-Y., Ji, A.-P., Qi, F., On integral inequalities of Hermite-Hadamard type for sgeometrically convex function. Abstract and Applied Analysis, doi:10.1155/2012/560586.
• [16] Zhang, T.-Y., Tunç, M., Ji, A.-P., Xi, B.-Y., Corrections to the paper ”On integral inequalities of Hermite-Hadamard type for s-geometrically convex function”. Abstract and Applied Analysis, (2014), Article ID 294739, http://dx.doi.org/10.1155/2014/294739 .