HERMITE-HADAMARD TYPE INEQUALITIES FOR THE PRODUCT TWO MAPPINGS WHOSE DERIVATIVES ABSOLUTE VALUES ARE $s$-CONVEX

Year 2017, Volume: 5 Issue: 1, 123 - 131, 01.04.2017

M. Shafıeı A. G. Ghazanfarı

https://izlik.org/JA75AP24PN

Abstract

In this paper, we extend some estimates of the right hand side of a Hermite-Hadamard type inequality for the product two differentiable functions whose derivatives absolute values are $s$-convex. Some natural applications to special weighted means of real numbers are given. Finally, an error estimate for the Simpson's formula is also addressed.

Keywords

Hermite-Hadamard inequality , differentiable functions , $s$-convex functions , convex functions

References

• [1] A. Barani and F. Malmir, New Hermit-Hadamard type inequalities for convex functions on a rectangular box, Konuralp Journal of Mathematics, Vol:4, No.1, (2016), 1-22.
• [2] G. Cristescu, Improved Integral Inequalities for Products of Convex Functions, J. Inequal. Pure and Appl. Math., Vol:6, No.2 (2005), Art. 35.
• [3] S. S. Dragomir, Inequality of Hermit-Hadamard type for $\phi$-convex functions , Konuralp Journal of Mathematics, Vol:4, No.1, (2016), 54-67.
• [4] S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. Vol:11, No.5 (1998), 91-95.
• [5] Dragomir, S.S., and Pearce, C.E.M., Selected Topics on Hermite-Hadamard Inequalities and applications, (RGMIA Monographs http:// rgmia.vu.edu.au/ monographs/ hermite hadamard.html), Victoria University, 2000.
• [6] S.S. Dragomir, S. Fitzpatrick, The Hadamards inequality for s-convex functions in the second sense, Demonstratio Math. Vol:32, No.4, 1999, 687-696.
• [7] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math. Vol:48, (1994), 100-111.
• [8] Kikianty, E., Hermite-Hadamard inequality in the geometry of banach spaces, PhD thesis, Victoria University, 2010.
• [9] M. Latif, Generalization of integral inequalities for product of convex functions, International Journal of Analysis and Applications, Vol:5, No.2, (2014), 185-190.
• [10] M. Klaricic Bakula and J. Pecaric, Note on some Hadamard-type inequalities, J. Inequal. Pure Appl. Math. Vol:5, No.3, (2004), Article 74.
• [11] M. Tunc, On some new inequalities for convex functions, Turk. J. Math. Vol:36, (2012), 245-251.
• [12] S. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, Rocky Mountain J. Math. Vol:39, No.5, (2009), 1741-1749.