Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via fractional integrals

Abstract

In this paper, firstly, Hermite-Hadamard-Fejér type inequality for harmonically convex functions in fractional integral forms have been established. Secondly, an integral identity and some Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions in fractional integral forms have been obtained. The some results presented here would provide extensions of those given in earlier works.

Keywords

• Hermite-Hadamard inequality,Hermite-Hadamard-Fejér inequality,Riemann-Liouville fractional integral

Kaynakça

1. M. Bombardelli and S. Varošanec, Properties of h-convex functions related to the Hermite Hadamard Fejér inequalities, Computers and Mathematics with Applications 58 (2009), 1869 1877.
2. F. Chen and S. Wu, Fejér and Hermite-Hadamard type inqequalities for harmonically convex functions, Jurnal of applied Mathematics, volume 2014, article id:386806.
3. Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1) (2010), 51-58.
4. L. Fejér, Uberdie Fourierreihen, II, Math. Naturwise. Anz Ungar. Akad., Wiss, 24 (1906), 369-390, (in Hungarian).
5. J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl., 58 (1893), 171-215.
6. İ. İşcan, New estimates on generalization of some integral inequalities for s-convex functions and their applications, Int. J. Pure Appl. Math., 86(4) (2013), 727-746.
7. İ. İşcan, Some new general integral inequalities for h-convex and h-concave functions, Adv. Pure Appl. Math. 5(1) (2014), 21-29 . doi: 10.1515/apam-2013-0029.
8. İ. İşcan, Generalization of different type integral inequalitiesfor s-convex functions via fractional integrals, Applicable Analysis, 2013. doi: 10.1080/00036811.2013.851785.

9. İ. İşcan, On generalization of different type integral inequalities for s-convex functions via fractional integrals, Mathematical Sciences and Applications E-Notes, 2(1) (2014), 55-67.
10. İ. İşcan, S. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014) 237-244.
11. İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (6) (2014), 935-942
12. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006).
13. M. A. Latif, S. S. Dragomir and E. Momoniat, Some Fejér type inequalities for harmonically-convex functions with applications to special means, http://rgmia.org/papers/v18/v18a24.pdf.
14. A. P. Prudnikov, Y. A. Brychkov, O. J. Marichev, Integral and series, Elementary Functions, vol. 1, Nauka, Moscow, 1981.
15. M.Z. Sarıkaya, On new Hermite Hadamard Fejér type integral inequalities, Stud. Univ. Babeş-Bolyai Math. 57(3) (2012), 377–386.
16. M.Z. Sarıkaya, E. Set, H. Yaldız and N. Başak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57(9) (2013), 2403-2407.
17. K.-L. Tseng, G.-S. Yang and K.-C. Hsu, Some inequalities for differentiable mappings and applications to Fejér inequality and weighted trapezoidal formula, Taiwanese journal of Mathematics, 15(4) (2011), 1737-1747.
18. J. Wang, X. Li, M. Fečkan and Y. Zhou, Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92(11) (2012), 2241-2253. doi:10.1080/00036811.2012.727986
19. J. Wang, C. Zhu and Y. Zhou, New generalized Hermite-Hadamard type inequalities and applications to special means, J. Inequal. Appl., 2013(325) (2013), 15 pages.