ON GENERALIZATION OF DIFFERENT TYPE INTEGRAL INEQUALITIES FOR s-CONVEX FUNCTIONS VIAFRACTIONAL INTEGRALS

Year 2014, , 55 - 67, 01.06.2014

İMDAT İşcan

https://doi.org/10.36753/mathenot.207633

Cited By: 15

Abstract

In this paper, a new general identity for differentiable mappingsvia Riemann-Liouville fractional integrals has been defined. By using of thisidentity, author has obtained new estimates on generalization of Hadamard,Ostrowski and Simpson type inequalities for functions whose derivatives inabsolutely value at certain powers are s-convex in the second sense

Keywords

Hermite-Hadamard inequality, Riemann-Liouville fractional integral, Ostrowski inequality, Simpson type inequality, s-convex function

References

• M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1965.
• M. Alomari, M. Darus and S.S. Dragomir, P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense. Appl. Math. Lett. 23 (2010), 1071–1076. [3] M. Alomari, M. Darus and U.S. Kirmaci, Some inequalities of Hermite-Hadamard type for s-convex functions. Acta Math. Sci. 31B(4) (2011), 1643–1652.
• M. Avci, H. Kavurmaci and M.E. Ozdemir, New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications. Appl. Math. Comput. 217 (2011), 5171–5176.
• Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional via fractional integration. Ann. Funct. Anal. 1(1) (2010), 51-58.
• S.S. Dragomir and S. Fitzpatrick, The Hadamard’s inequality for s-convex functions in the second sense. Demonstratio Math. 32(4) (1999), 687–696.
• R. Gorenflo and F. Mainardi, Fractional calculus; integral and differential equations of frac- tional order. Springer Verlag, Wien, 1997, 223-276.
• H. Hudzik and L. Maligranda, Some remarks on s-convex functions. Aequationes Math. 48 (1994), 100-111.
• ˙I. ˙I¸scan, 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.
• ˙I. ˙I¸scan, Generalization of different type integral inequalitiesfor s-convex functions via frac- tional integrals. Appl. Anal. (2013), 1-17. doi:10.1080/00036811.2013.851785.
• S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, USA, 1993.
• M.E. Ozdemir, M. Avci and H. Kavurmaci, Hermite-Hadamard type inequalities for s-convex and s-concave functions via fractional integrals, arXiv preprint arXiv:1202.0380. (2012).
• J. Park, Generalization of some Simpson-like type inequalities via differentiable s-convex mappings in the second sense, Int. J. Math. Math. Sci. 2011 (2011), 13 pages, Article ID 493531. doi:10.1155/493531.
• I. Podlubni, Fractional Differential Equations. Academic Press, San Diego, 1999.
• J. Peˇcari´c, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications. Academic Press Inc.,1992.
• 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, E. Set and M.E. Ozdemir, On new inequalities of Simpson’s type for s-convex functions. Computers and Math. with Appl. 60 (2010), 2191-2199.
• M.Z. Sarıkaya, E. Set, H. Yaldız and N. Ba¸sak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Modelling. 57(9) (2013), 2403- 2407. doi:10.1016/j.mcm.2011.12.048.
• Department of Mathematics, Faculty of Arts and Sciences, Giresun University, Gire- sun, Turkey
• E-mail address: [e-mail]