HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS

Year 2016, Volume: 4 Issue: 1, 108 - 113, 01.04.2016

G. Farıd

Abstract

Fractional calculus is as important as calculus. This paper is due to presentation of Hadamard and Fejer-Hadamard inequalities for fractional calculus. We prove Hadamard and Fejer-Hadamard inequalities for general- ized fractional integral involving Mittag-Lefter function. Also, inequalities for special cases are obtained.

Keywords

Convex function, Hermite-Hadamard inequality, Fejer-Hadamard in- equality, fractional integrals, Mittag-Leer function

References

• [1] R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Dierential and Dierence Equations, Kluwer Academic Publishers, Dordrecht, Boston, London 1995.
• [2] G. A. Anastassiou, Advanced inequalities, 11, World Scientic, 2011.
• [3] A. G. Azpeitia, Convex functions and the Hadamard inequality, Revista Colombina Mat. 28 (1994)7-12.
• [4] M. K. Bakula, J. Pecaric, Note on some Hadamard type inequalities, J. ineq. Pure Appl. Math. 5 (3)(2004) Art. 74.
• [5] L. Curiel, L. Galue, A generalization of the integral operators involving the Gauss hypergeo- metric function, Revista Tecnica de la Facultad de Ingenieria Universidad del Zulia, 19 (1) (1996), 17{22.
• [6] S. S. Dragomir, R. P. Agarwal, Two inequalities for dierentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett. 11 (5)(1998), 91-95.
• [7] S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Math. Sic. Marh. Roum., 47 (2004), 3-14.
• [8] G. Farid, J. Pecaric and Z. Tomovski, Opial-type inequalities for fractional integral operator involving Mittag-Leer function, Fractional Dier. Calc., Vol. 5 , No. 1 (2015), 93-106.
• [9] L. Fejer, Uberdie Fourierreihen, II, Math. Naturwise. Anz Ungar. Akad., Wiss, 24 (1906), 369-390, (in Hungarian).
• [10] A. A. Kilbas, M. Saigo, R.K. Saxena, Generalized MittagLeer function and generalized fractional calculus operators, Integral Transform. Spec. Funct. 15 (2004) 3149.
• [11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of fractional derivatial Equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006.
• [12] K. Miller and B. Ross, An introduction to the fractional calculus and fractional dierential Equations, John Wiley and Sons Inc., New York, 1993.
• [13] K. Oldham and J. Spanier The fractional calculus, Academic Press, New York - London, 1974.
• [14] M. E. Ozdemir, M Avci, E. Set, On some inequalities of Harmite-Hadamard type via m- convexity, Appl. Math. Lett. 23 (9)(2010), 1065-1070.
• [15] E. Set, M. E. Ozdemir, S. S. Dragomir, On the Harmite-Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl. (2010) 9. Article ID 148102.
• [16] E. Set, M. E. Ozdemir, S. S. Dragomir, On Hadamard-type inequalities involving several kinds of convexity, J. Inequal. Appl. (2010) 12. Article ID 286845.
• [17] T. R. Prabhakar, A singular integral equation with a generalized Mittag{Leer function in the kernel, Yokohama Math. J. 19 (1971) 715.
• [18] I. Scan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional inte- grals, arXiv preprint arXiv: 1404. 7722 (2014).
• [19] T. O. Salim, and A. W. Faraj, A Generalization of Mittag{Leer function and integral operator associated with fractional calculus, J. Fract. Calc. Appl. Vol. 3 July 2012, No. 5, pp. 1 - 13.
• [20] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math., Comp., Modelling, 57 (2013), 2403-2407.
• [21] H. M. Srivastava, and Z. Tomovski, Fractional calculus with an integral operator con- taining generalized Mittag{Leer function in the kernal, Appl. Math. Comput. (2009), doi:10.1016/j.amc.2009.01.055
• [22] D. V. Widder, The Laplace transform, Princeton Uni. Press, New Jersey, 1941.