Hatice YALDIZ, Zoubir DAHMANİ, Mehmet Zeki SARIKAYA

On the Hermite-Hadamard-Fejer-type inequalities for co-ordinated convex functions via fractional integrals

In this paper, using Riemann-Liouville integral operators, we establishnew fractional integral inequalities of Hermite-Hadamard-Fejer type for coordinatedconvex functions on a rectangle of R². The results presented herewould provide extensions of those given in earlier works.

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[1] Alomari, M., Darus, M., Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities, Int. J. Contemp. Math. Sciences, 3 (32),1557-1567 (2008).
[2] Alomari, M., Darus, M., On the Hadamard's inequality for log-convex functions on the coordinates, J. of Inequal. and Appl, Article ID 283147, 13 pages,(2009).
[3] Alomari, M., Darus, M., Fejer inequality for double integrals, Facta Universitatis (Nis) Ser. Math. Inform., 24 , 15-28(2009).
[4] Kilbas, A.A., Srivastava H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, (2006).
[5] Belarbi, S., Dahmani, Z., On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3) , Art. 86, (2009).
[6] Dahmani, Z., New inequalities in fractional integrals, International Journal of Nonlinear Scinece, 9(4) , 493- 497, (2010).
[7] Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1), 51-58, (2010).
[8] Dahmani, Z., Tabharit, L., Taf, S., Some fractional integral inequalities, Nonl. Sci. Lett. A, 1(2) , 155-160, (2010).
[9] Dahmani, Z., Tabharit, L., Taf, S., New generalizations of Gr¨uss inequality usin Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3) , 93-99, (2010).
[10] Dragomir, S.S., On Hadamard's inequality for convex functions on the co-ordinates in arectangle from the plane, Taiwanese Journal of Mathematics, 4, 775-788, (2001).
[11] Fej´er, L., Uberdie Fourierreihen, II, Math. Naturwise. Anz Ungar. Akad., Wiss, 24, 369-390(1906), (in Hungarian). [12] Gorenflo, R., Mainardi, F., Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien, 223-276, (1997).
[13] Hadamard, J., Etude sur les proprietes des fonctions entieres et en particulier d'une fonction considree par, Riemann, J. Math. Pures. et Appl. 58, 171-215,(1893).
[14] scan, I., Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals, 2014, arXiv:1404.7722v1.
[15] Latif, M.A., Alomari, M., Hadamard-type inequalities for product two convex functions on the co-ordinetes, Int. Math. Forum, 4(47), 2327-2338,(2009).
[16] Latif, M.A., Alomari, M., On the Hadamard-type inequalities for h-convex functions on the co-ordinetes, Int. J. of Math. Analysis, 3(33), 1645-1656,(2009).
[17] Latif, M.A., Hussain, S., New inequalities of Ostrowski type for co-ordineted convex functions via fractional integrals, Journal of Fractional Calculus and Applications,Vol. 2, No. 9, pp. 1-15, (2012).
[18] Miller, S., Ross, B., An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, USA, p.2, (1993).
[19] Minculete, N., Mitroi, F-C.,Fejer-type inequalities, Aust. J. Math. Anal. Appl. 9, no. 1, Art. 12, 8pp,(2012).
[20] Ozdemir, M.E., Set, E., Sarikaya, M.Z., New some ¨ Hadamard's type inequalities for co-ordinated mconvex and (?, m)-convex functions, Hacettepe Journal of Mathematics and Statistics , 40(2), 219-229, (2011).
[21] Sarikaya, M.Z., On new Hermite Hadamard Fej´er type integral inequalities, Stud. Univ. Babe¸s-Bolyai Math. 57 (3), 377-386, (2012).
[22] Sarikaya, M.Z., Set, E., Ozdemir M.E., Dragomir, S.S., New some Hadamard's type inequalities for coordinated convex functions, Tamsui Oxford Journal of Information and Mathematical Sciences, 28(2),137- 152, (2012) .
[23] Sarikaya M.Z., Ogunmez, H., On new inequalities via Riemann-Liouville fractional integration, Abstract and Applied Analysis, Volume 2012, Article ID 428983, 10 pages, doi:10.1155/2012/428983.
[24] Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N., Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57, 2403-2407,(2013).
[25] Sarikaya, M.Z., Ostrowski type inequalities involving the right Caputo fractional derivatives belong to Lp, Facta Universitatis, Series Mathematics and Informatics , Vol. 27 No 2, 191-197, (2012).
[26] Sarikaya, M.Z., Yaldiz, H., On weighted Montogomery identities for Riemann-Liouville fractional integrals, Konuralp Journal of Mathematics, Volume 1 No. 1 pp. 48-53, (2013).
[27] Sarikaya, M.Z., Yaldiz, H., On the Hadamard's type inequalities for L-Lipschitzian mapping, Konuralp Journal of Mathematics, Volume 1 No. 2 pp. 33-40 (2013).
[28] Sarikaya, M.Z., On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integral Transforms and Special Functions, Vol. 25, No. 2, 134-147, (2014).