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

Year 2019, Volume: 68 Issue: 2, 2128 - 2142, 01.08.2019

Huriye Kadakal

https://doi.org/10.31801/cfsuasmas.511184

Cited By: 9

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

• 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.
• 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.
• Dragomir, S.S. and Pearce, CEM. Selected Topics on Hermite-Hadamard Inequalities and Its Applications, RGMIA Monograph (2002).
• Kadakal, H., (m₁,m₂)-convexity and some new Hermite-Hadamard type inequalities, Journal of Applied and Engineering Mathematics, (accepted for publications), 2019.
• 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.
• 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.
• 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.
• 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.
• Mihaly, B. Hermite-Hadamard-type inequalities for generalized convex functions. J. Inequal, Pure Appl. Math., 9(3), Article ID 63 (2008) (PhD thesis).
• 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.
• 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.
• 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.
• Pečarić, J. E., Proschan, F and Tong, Y. L. Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, INC, 1992.
• 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.
• Toader, G. The hierarchy of convexity and some classic inequalities, J. Math. Ineq.,3(3)(2009), 305--313.