Some Integral Inequalities of Hermite-Hadamard Type for Functions Whose n-times Derivatives are (α,m)-Convex

Feng Qi^1, , Muhammad Amer Latif², Wen-Hui Li³ and Sabir Hussain⁴

¹Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

²College of Science, Department of Mathematics, University of Hail, Hail, Saudi Arabia

³Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China

⁴Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan

Cite this paper:
Feng Qi, Muhammad Amer Latif, Wen-Hui Li and Sabir Hussain. Some Integral Inequalities of Hermite-Hadamard Type for Functions Whose n-times Derivatives are (α,m)-Convex. Turkish Journal of Analysis and Number Theory. 2014; 2(4):140-146. doi: 10.12691/tjant-2-4-7

Abstract

In the paper, the authors find some new integral inequalities of Hermite-Hadamard type for functions whose derivatives of the n-th order are (α,m)-convex and deduce some known results. As applications of the newly-established results, the authors also derive some inequalities involving special means of two positive real numbers.

Keywords:
Hermite-Hadamard integral inequality convex function (αm)-convex function differentiable function; application; mean

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	R.-F. Bai, F. Qi, and B.-Y. Xi, Hermite-Hadamard type inequalities for the m- and (α,m)-logarithmically convex functions, Filomat 27 (2013), no. 1, 1-7.
[2]	S.-P. Bai, S.-H. Wang, and F. Qi, Some Hermite-Hadamard type inequalities for n-time differentiable (α,m)-convex functions, J. Inequal. Appl. 2012, 2012:267, 11 pages.
[3]	L. Chun and F. Qi, Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex, J. Inequal. Appl. 2013, 2013:451, 10 pages.
[4]	S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2000; Available online at http://rgmia.org/monographs/hermite_hadamard.html
[5]	J. Hua, B.-Y. Xi, and F. Qi, Hermite-Hadamard type inequalities for geometric-arithmetically s-convex functions, Commun. Korean Math. Soc. 29 (2014), no. 1, 51-63.
[6]	J. Hua, B.-Y. Xi, and F. Qi, Some new inequalities of Simpson type for strongly s-convex functions, Afrika Mat. (2014), in press.
[7]	D.-Y. Hwang, Some inequalities for n-time differentiable mappings and applications, Kyugpook Math. J. 43 (2003), no. 3, 335-343.
[8]	M. A. Latif and S. Hussain, New inequalities of Hermite-Hadamard type for n-time differentiable (α,m)-convex functions with applications to special means, RGMIA Res. Rep. Coll. 16 (2013), Art. 17, 12 pages; Available online at http://rgmia.org/v16.php
[9]	W.-H. Li and F. Qi, Hermite-Hadamard type in-equalities of functions whose derivatives of n-th or-der are (α,m)-convex.
[10]	W.-H. Li and F. Qi, Some Hermite-Hadamard type inequalities for functions whose n-th derivatives are (α,m)-convex, Filomat 27 (2013), no. 8, 1575-1582.
[11]	V. G. Mihe_san, A generalization of the convexity, Seminar on Functional Equations, Approx. Convex, Cluj-Napoca, 1993. (Romania)
[12]	C. P. Niculescu and L.-E. Persson, Convex Functions and their Applications, CMS Books in Mathematics, Springer-Verlag, 2005.
[13]	M. E.  Ozdemir, M. Avci, and H. Kavurmaci, Hermite-Hadamard-type ineuqlities via (α,m)- convexity, Comput. Math. Appl. 61 (2011), no. 9, 2614-2620.
[14]	M. E.  Ozdemir, M. Avci, and E. Set, On some inequalities of Hermite-Hadamard type via m-convexity, Appl. Math. Lett. 23 (2010), no. 9, 1065-1070.
[15]	F. Qi, Z.-L. Wei, and Q. Yang, Generalizations and refinements of Hermite-Hadamard's inequality, Rocky Mountain J. Math. 35 (2005), no. 1, 235-251.
[16]	F. Qi and B.-Y. Xi, Some integral inequalities of Simpson type for GA-"-convex functions, Georgian Math. J. 20 (2013), no. 4, 775-78.
[17]	M. Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications.
[18]	D.-P. Shi, B.-Y. Xi, and F. Qi, Hermite-Hadamard type inequalities for (m,h1,h2)-convex functions via Riemann-Liouville fractional integrals, Turkish J. Anal. Number Theory 2 (2014), no. 1, 22-27.
[19]	Y. Shuang, Y. Wang, and F. Qi, Some inequalities of Hermite-Hadamard type for functions whose third derivatives are (α;m)-convex, J. Comput. Anal. Appl. 17 (2014), no. 2, 272-279.
[20]	G. Toader, Some generalizations of the convexity, Univ. Cluj-Napoca, Cluj-Napoc. 1985, 329-338.
[21]	S.-H. Wang and F. Qi, Hermite-Hadamard type in-equalities for n-times differentiable and preinvex functions, J. Inequal. Appl. 2014, 2014: 49, 9 pages.
[22]	S.-H. Wang and F. Qi, Inequalities of Hermite-Hadamard type for convex functions which are Turkish Journal of Analysis and Number Theory 8 n-times differentiable, Math. In equal. Appl. 16 (2013), no. 4, 1269-1278.
[23]	S.-H. Wang, B.-Y. Xi, F. Qi, Some new inequalities of Hermite-Hadamard type for n-time differentiable functions which are m-convex, Analysis (Munich) 32 (2012), no. 3, 247-262.
[24]	Y. Wang, B.-Y. Xi, and F. Qi, Hermite-Hadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex, Matematiche (Catania) 69 (2014), no. 1, 89-96.
[25]	Y. Wang, M.-M. Zheng, and F. Qi, Integral inequalities of Hermite-Hadamard type for functions whose derivatives are (α,m)-preinvex, J. Inequal. Appl. 2014, 2014:97, 10 pages.
[26]	B.-Y. Xi, R.-F. Bai, and F. Qi, Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions, Aequationes Math. 84 (2012), no. 3, 261-269.
[27]	B.-Y. Xi, J. Hua, and F. Qi, Hermite-Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle, J. Appl. Anal. 20 (2014), no. 1, 29-39.
[28]	B.-Y. Xi and F. Qi, Some Hermite-Hadamard type inequalities for geometrically quasi-convex functions, Proc. Indian Acad. Sci. Math. Sci. (2014), in press.
[29]	B.-Y. Xi and F. Qi, Some inequalities of Hermite-Hadamard type for h-convex functions, Adv. Inequal. Appl. 2 (2013), no. 1, 1-15.
[30]	B.-Y. Xi and F. Qi, Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means, J. Funct. Spaces Appl. 2012 (2012), Article ID 980438, 14 pages.
[31]	B.-Y. Xi, S.-H. Wang, and F. Qi, Properties and inequalities for the h- and (h; m)-logarithmically convex functions, Creat. Math. Inform. 23 (2014), no. 1, 123-130.
[32]	B.-Y. Xi, S.-H. Wang, and F. Qi, Some inequalities for (h,m)-convex functions, J. Inequal. Appl. 2014, 2014: 100, 12 pages.
[33]	T.-Y. Zhang and F. Qi, Integral inequalities of Hermite-Hadamard type for m-AH convex func-tions, Turkish J. Anal. Number Theory 2 (2014), no. 3, 60-64.