NEW REFINEMENTS AND INTEGRAL INEQUALITIES FOR CONCAVE FUNCTIONS

Abstract

In this paper, we establish new renements and integral inequalities including concave functions. The reason why we choose the concave functions in this study is that the methods we use are applicable to these functions. Also some applications are provided.

Keywords

• Concave function,Hermite-Hadamard inequalitiy,Holder inequality,Jensen inequality,Favard inequality.

References

1. [1] Mitrinovi´c, D.S., Pecari´c, J.E., Fink, A.M., (1993), Classical and New Inequalities in Analysis, Kluwar Academic Publishers, p.106, 10, 15 . [2] Kirmaci, U.S., (2004), Inequalities for differentiable mappings and applications to special means of real number and midpoint formula, Appl.Math. Comput. 147.
2. [3] Favard, J., (1933), Sur les valeurs moyennes, Bull. Sci. Math. (2) 57, 54-64.
3. [4] Kirmaci, U.S. and Ozdemir, M.E., (2004), On some inequalities for differentiable mappings and ap- ¨ plications to special means of real numbers and to midpoint formula. Appl. Math. Comput., 153, 361-368.
4. [5] Kirmaci, U.S., Bakula, M.K., Ozdemir, M.E. and Peˇcari´c, J., (2007), Hadamard-type inequalities of ¨ s−convex functions, Applied Mathematics and Computation 193, 26-35.
5. [6] Yıldız, C¸. and Ozdemir, M.E., (2017), On generalized Inequalities of Hermite- Hadamard Type for ¨ Convex Functions, International Journal of Analysis and Applications, ISSN 2291-8639 Volume 14, Number1, 52-63.
6. [7] Ozdemir, M.E., Latif, M.A. and Akdemir, A.O., (2016), On Some Hadamard-Type Inequalities for ¨ Product of Two Convex Functions on the Co-ordinates, Turkish Journal of Science, Volume I, Issue I, 41-58.
7. [8] Ozdemir, M.E., Akdemir, A.O. and Set, E., (2016), On ( ¨ h − m)-Convexity and Hadamard type ınequalities, TJMM 8, No. 1, 51-58.
8. [9] Ozdemir, M.E., Akdemir, A.O. and Avci-Ardi¸c, M., (2016), New hadamard Type Inequalities for ¨ m−Convex Functions, Academic Journal of Science, ISSN: 2165-6282: 06(01): 53-60.

9. [10] Ozdemir, M.E., Set, E. and Akdemir, A.O., (2015), On The ( ¨ r, s) −convexity and Some HadamardType Inequalities, J. Applied Functional Analysis, Vol. 10, NO. 1-2, 95-1
10. [11] Bakula, M.K., Ozdemir, M.E. and Peˇcari´c, J., (2008), Hadamard type Inequalities for ¨ m−convex and (α, m)-Convex Functions, Journal of inequalities in pure and applied mathematics, volume 9, Issue 4, Article 96, 12pp.
11. [12] Akdemir, A.O. and Ozdemir, M.E., (2010), Some Hadamard-type inequalities for co-ordinated ¨ P−convex functions and Godunova-Levin functions, American Institu of Phsyics (AIP) Conference Proceedings, 1309, 7-15 pp.
12. [13] Akdemir, A.O., Ozdemir, M.E. and Varosanec, S., (2012), On some inequalities for ¨ h−concave functions, Mathematical and Computer Modelling , 55-(3-4), 746-753 pp., DOI: 10.1016/j.mcm.2011.08.051.
13. [14] Ozdemir, M.E., Ekinci, A. and Akdemir, A.O., (2012), Generalizations of integral inequalities for ¨ functions whose second derivatives are convex and m−convex, Miskolc Mathematical Notes , 13-2, 441-457 pp.
14. [15] Tun¸c, M. and Akdemir, A.O., (2015), Ostrowski type inequalities for s−logarithmically convex functions in the second sense with applications, Georgian Mathematical Journal , Volume 22 Issue 1, 1-7.
15. [16] Ozdemir, M.E., Yıldız, C¸ ., Akdemir, A.O. and Set, E., (2013), On some inequalities for ¨ s−convex functions and applications, Journal of Inequalities and Applications, 2013, DOI: 10.1186/1029-242X2013-333.
16. [17] Ozdemir, M.E., Akdemir, A.O. and Ekinci, A., (2014), New Hadamard-type inequalities for functions ¨ whose derivatives are (α, m)-convex functions, Tbilisi Mathematical Journal, DOI:10.2478/tmj-2014- 0017, 7 (2), pp. 61-72.
17. [18] Akdemir, A.O., Ekinci A. and Set, E., (2017), Conformable Fractional Integrals and Related New Integral Inequalities, Journal of Nonlinear and Convex Analysis, Volume 18, Number 4, 661-674.
18. [19] Ozdemir, M.E., and Ekinci, A., (2016), Generalized integral inequalities for convex functions, Math- ¨ ematical Inequalities and Applications, Vol. 19, Issue 4, pp. 1429-1439.
19. [20] Set, E., G¨ozpınar, A. and Ekinci, A., (2017), Hermite-Hadamard Type Inequalities via Conformable Fractional Integrals, Acta Mathematica Universitatis Comenianae, Vol. 86, Issue 2, pp. 309-320.
20. [21] Yıldız, C¸ . and G¨urb¨uz, M., (2018), Some New Integral Inequalities Related to Convex Functions, I˘gdır Univ. J. Inst. Sci. & Tech. 8 (3): 279-286.
21. [22] Set, E., Akdemir, A.O. and G¨urb¨uz, M., (2017), Integral Inequalities for Different Kinds of Convex Functions Involving Riemann-Liouville Fractional Integrals, Karaelmas Fen ve M¨uhendislik Dergisi 7 (1), 140-144.
22. [23] Sarıkaya, M.Z., Set, E. and Ozdemir, M.E., (2010), On Some New Inequalities of Hadamard Type ¨ Involving h−Convex Functions, Acta Math. Univ.Comenianae, Vol. LXXIX, 2, pp.265-272.
23. [24] Set, E., Ozdemir, M.E. and Dragomir, S.S., (2010), On Hadamard-Type Inequalities Involving Several ¨ Kinds of Convexity, Journal of inequalities and applications, volume 2010, Article ID 286845, 12 pages doi: 10.1155/2010/286845.
24. [25] Ozdemir, M.E., Avcı, M. and Set, E., (2010), On Some Inequalities of Hermite- Hadamard type via ¨ m−convexity, Applied Math Letters 23, 1065-1070.
25. [26] Ozdemir, M.E., Akdemir, A.O., (2011), On Some Hadamard-Type Inequalities for Convex Functions ¨ on A Rectangular Box, Journal of Nonlinear analysis and Application, Volume 2011, Article ID jnaa00101, 10 Pages, doi: 10:5899/2011/jnaa-00101.
26. [27] Avci, M., Kavurmaci, H. and Ozdemir, M.E., (2011), New inequalities of Hermite- Hadamard type ¨ via s-convex functions in the second sense with applications, Applied Mathematics and Computation 217, 5171-5176.
27. [28] Kavurmaci, H., Avci, M. and Ozdemir, M.E., (2011), New inequalities of Hermite-Hadamard type for ¨ convex functions with applications, Journal of Inequalities and Applications, 2011:86.
28. [29] Ozdemir, M.E., Avci, M., Kavurmacı, H., (2011), Hermite- Hadamard type inequalities via (α, m) −convexity, Computers and Mathematics with Applications 61, 2614-2620.
29. [30] Set, E., Sardari, M., Ozdemir, M.E. and Rooin, J., (2012), On Generalizations of the Hadamard ¨ Inequality for (α, m) −convex functions. Kyungpook Math.j. 52, 307-312.
30. [31] Ozdemir, M.E., G¨urb¨uz, M. and Kavurmacı, M., (2013), Hermite- Hadamard type inequalities for ¨ (g, ϕα) −convex dominated functions, Journal of inequalities and applications, 2013:184.
31. [32] Ozdemir, M.E. and Yıldız, C¸ ., (2013), New inequalities for Hermite-Hadamard and Simpson ¨ type with Applications, Tamkang Journal of Mathematics, Volume 44, Number 2 , 209-216, doi: 10.5556/j.tkjm.44.2013.117
32. [33] Ozdemir, M.E., Set, E. and Akdemir, A.O., (2014), On Some Hadamard Type Inequalities for ¨ (r, m) −Convex Functions, Applications and Applied Mathematics, Volume 9, Issue 1, pp. 388-401.
33. [34] Kırmacı, U.S., (2017), On Some Hermite-Hadamard Type Inequalities for Twice Differentiable (α, m) − convex functions and Applications, RGMIA Research Report Collection, 20, Article 51, 11pp