(alpha,m1,m2)-convexity and some inequalities of Hermite-Hadamard type

Abstract

In this paper, we introduce a new class of extended (alpha;m1;m2)-convex functions. Some algebraic properties of these class functions have been investigated. Some new Hermite-Hadamard type inequalities are derived. Results represent signicant refinement and improvement of the previous results. Also, the author establish a new integral identity and, by this identity, Hölder's and power mean inequality, discover some new Hermite-Hadamard type inequalities for functions whose first derivatives are (alpha;m1;m2)-convex. Our results are new and coincide with the previous results in special cases.

Keywords

• Convex function
• m-Convex function
• (alpha;m)-Convex function
• (alpha;m1;m2)-Convex function
• Hermite-Hadamard type inequality

References

1. Bakula, M. K., Özdemir, M. E. and Pečarić, J. Hadamard tpye inequalities for m-convex and (α,m) -convex functions, J. Inequal. Pure and Appl. Math., 9(4)(2008), 1-25.
2. Dragomir, S.S. On Some New Inequalities of Hermite-Hadamard Type for m-Convex Functions, Tamkang Journal of Mathematics, Volume 33, Number 1, Spring 2002, 45--55.
3. Dragomir, S.S. and Pearce, CEM. Selected Topics on Hermite-Hadamard Inequalities and Its Applications, RGMIA Monograph (2002).
4. Kadakal, H., (m₁,m₂)-convexity and some new Hermite-Hadamard type inequalities, Journal of Applied and Engineering Mathematics, (accepted for publications), 2019.
5. Kavurmaci, H., Avci M. and Özdemir, M.E. New inequalities of Hermite-Hadamard type for convex functions with applications, Journal of Inequalities and Applications, 2011, 2011:86.
6. Kirmaci, U. S., Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. and Comput., 147 (2004) 137--146.
7. Kırmacı, U. S., Bakula, M. K., Özdemir, M. E. and Pe čarić, J. Hadamard-tpye inequalities for s-convex functions, Appl. Math. and Comp., 193(2007), 26--35.
8. Lara, T., Rosales, E. and Sánchez, J.L. New Properties of m-Convex Functions, International Journal of Mathematical Analysis, Vol. 9, 2015, No. 15, 735--742.

9. Mihaly, B. Hermite-Hadamard-type inequalities for generalized convex functions. J. Inequal, Pure Appl. Math., 9(3), Article ID 63 (2008) (PhD thesis).
10. Özdemir, M. E., Latif M. A. and Akdemir, A. O. On some Hadamard-type inequalities for product of two s-convex functions on the co-ordinates, Journal of Inequalities and Applications, 2012, 2012:21.
11. Özdemir, M. E., Set E. and Sarıkaya, M. Z. Some new Hadamard's type inequalities for co-ordinated m-convex and (α,m)-convex functions, Hacettepe J. of. Math. and Statistics, 40(2011), 219--229.
12. Park, J., New Ostrowski-Like type inequalities for differentiable (s,m)-convex mappings, International Journal of Pure and Applied Mathematics, Volume 78 No. 8 2012, 1077--1089.
13. Pečarić, J. E., Proschan, F and Tong, Y. L. Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, INC, 1992.
14. Sarikaya, M.Z., Sağlam A. and Yıldırım, H. On some Hadamard-type inequalities for h-convex functions, J. Math. Ineq., 2(3)(2008), 335--341.
15. Toader, G. The hierarchy of convexity and some classic inequalities, J. Math. Ineq.,3(3)(2009), 305--313.