On generalization of different type inequalities for (α,m)-convex functions via fractional integrals

Yıl 2016, Cilt: 4 Sayı: 3, 49 - 57, 30.09.2016

İmdat Iscan , Mustafa Aydin Kerim Bekar

Öz

 In this paper,
new identity for fractional integrals have been defined. By using of this
identity, the authors obtained new general inequalities containing all of
Hadamard, Ostrowski and Simpson type inequalities for functions whose
derivatives in absolute value at certain power are -convex via Riemann Liouville fractional integral.

Anahtar Kelimeler

Hermite–Hadamard inequality, Riemann–Liouville fractional integral, (α

Kaynakça

• M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965.
• M.K. Bakula, M.E. Ozdemir and J. Pecaric, Hadamard type inequalities for m-convex and (α,m)-convex functions, J. Inequal. Pure Appl. Math. 9(4) (2008) Article 96, p. 12. [Online: http://jipam.vu.edu.au/article.php?sid=1032].
• R. Gorenflo and F. Mainardi,Fractional calculus, integral and differential equations of fractional order, Springer Verlag, Wien, 1997, 223-276.
• I. Iscan, A new generalization of some integral inequalities for (α,m)-convex functions, Mathematical Sciences 7(22) (2013)1-8.
• I. Iscan, New estimates on generalization of some integral inequalities for (α,m)-convex functions, Contemp. Anal. Appl. Math. 1(2) (2013), 253-264 .
• I. Iscan, Hermite-Hadamard type inequalities for functions whose derivatives are (α,m)-convex, International Journal of Engineering and Applied sciences 2(3) (2013), 69-78.
• V.G. Miheşan, A generalization of the convexity. Seminer on Functional Equations, Approximation and Convexity, Cluj-Napoca, Romania, 1993.
• S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993.
• M.E. Ozdemir, M. Avcı and H. Kavurmacı, Hermite-Hadamard-type inequalities via (α,m)-convexity, Comput. Math. Appl. 61 (2011), 2614-2620.
• M.E. Ozdemir, H. Kavurmacı and E. Set, Ostrowski’s type inequalities for (α,m)-convex functions, Kyungpook Math. J. 50 (2010), 371-378.
• J. Park, Hermite-Hadamard and Simpson-Like Type Inequalities for Differentiable (α,m)-Convex Mappings, Int. J. Math. Math. Sci. 2012 (2012), Article ID 809689, 12 pages . doi:10.1155/2012/809689.
• I. Podlubni, Fractional Differential Equations, Academic Press, San Diego, 1999.
• M.Z. Sarıkaya and N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Modelling 54 (2011), 2175-2182 .
• M.Z. Sarıkaya and H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstr. Appl. Anal. 2012 (2012), Article ID 428983, 10 pages. doi:10.1155/2012/428983. Zbl 1253.26012.
• M.Z. Sarıkaya, E. Set, H. Yaldız and N. Başak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling 57(9-10) (2013), 2403-2407.
• G. Toader, Some generalizations of the convexity, Proceedings of The Colloquium On Approximation And Optimization, Univ. Cluj-Napoca, Cluj-Napoca,1985, 329-338.