Abstract
In this work, we first establish Hermite–Hadamard–Fejer type inequalities for convex function involving fractional integrals with respect to another function which are generalization of some important fractional integrals such as the Riemann–Liouville fractional integrals and the Hadamard fractional integrals. Moreover, we obtain some trapezoid type inequalities for these kind of fractional integrals. The results given in this paper provide generalization of several inequalities obtained in earlier studies.
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Azpeitia, A.G.: Convex functions and the Hadamard inequality. Rev. Colomb. Math. 28, 7–12 (1994)
Ali, A., Gulshan, G., Hussain, R., Latif, A., Muddassar, M.: Generalized inequalities of the type of Hermite–Hadamard–Fejer with quasi-convex functions by way of \(\mathit{k}\)-fractional derivatives. J. Comput. Anal. Appl. 22(7), 1208–1219 (2017)
Bakula, M.K., Pečarić, J.: Note on some Hadamard-type inequalities. J. Inequal. Pure Appl. Math. 5(3), 74 (2004)
Bombardelli, M., Varosanec, S.: Properties of h-convex functions related to the Hermite–Hadamard–Fejer inequalities. Comput. Math. Appl. 58, 1869–1877 (2009)
Budak, H., Sarikaya, M.Z.: Hermite–Hadamard type inequalities for s-convex mappings via fractional integrals of a function with respect to another function. Fasc. Math. 27, 25–36 (2016)
Chen, H., Katugampola, U.N.: Hermite–Hadamard and Hermite–Hadamard–Fejer type inequalities for generalized fractional integrals. J. Math. Anal. Appl. 446, 1274–1291 (2017)
Chen, F., Wu, S.: Fejer and Hermite–Hadamard type inqequalities for harmonically convex functions. J. Appl. Math. 2014, 6 (2014)
Dahmani, Z.: On Minkowski and Hermite–Hadamard integral inequalities via fractional integration. Ann. Funct. Anal. 1(1), 51–58 (2010)
Deng, J., Wang, J.: Fractional Hermite–Hadamard inequalities for (\(\alpha, m\))-logarithmically convex functions. J. Inequal. Appl. 2013, 364 (2013)
Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs. Victoria University, Melbourne (2000)
Dragomir, S.S., Agarwal, R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11(5), 91–95 (1998)
Fejer, L.: Über die Fourierreihen, II. Math. Naturwiss. Anz Ungar. Akad. Wiss. 24, 369–390 (1906). (Hungarian)
Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Order, pp. 223–276. Springer, Wien (1997)
Iscan, I.: Hermite–Hadamard–Fejer type inequalities for convex functions via fractional integrals. Stud. Univ. Babe ş-Bolyai Math. 60(3), 355–366 (2015)
Jleli, M., Samet, B.: On Hermite–Hadamard type inequalities via fractional integrals of a function with respect to another function. J. Nonlinear Sci. Appl. 9, 1252–1260 (2016)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Sci. B.V, Amsterdam (2006)
Miller, S., Ross, B.: An introduction to the Fractional Calculus and Fractional Differential Equations, p. 2. John Wiley & Sons, Hoboken (1993)
Mubeen, S., Iqbal, S., Tomar, M.: On Hermite–Hadamard type inequalities via fractional integrals of a function with respect to another function and \(k\)-parameter. J. Inequal. Math. Appl. 1, 1–9 (2016)
Pečarić, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Academic Press, Boston (1992)
Podlubni, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Peng, S., Wei, W., Wang, J.-R.: On the Hermite–Hadamard inequalities for convex functions via Hadamard fractional integrals. Facta Univ. Ser. Math. Inform. 29(1), 55–75 (2014)
Sarikaya, M.Z., Yaldiz, H., Erden, S.: Some inequalities associated with the Hermite–Hadamard–Fejer type for convex function. Math. Sci. 8, 117–124 (2014)
Sarikaya, M.Z., Budak, H.: On Fejer type inequalities via local fractional integrals. J. Fract. Calc. Appl. 8(1), 59–77 (2017)
Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013)
Sarikaya, M.Z., Budak, H.: Generalized Hermite–Hadamard type integral inequalities for fractional integrals. Filomat 30(5), 1315–1326 (2016)
Sarikaya, M.Z.: On new Hermite Hadamard Fejer type integral inequalities. Stud. Univ. Babes-Bolyai Math. 57(3), 377–386 (2012)
Sarikaya, M.Z., Budak, H.: Some Hermite–Hadamard type integral inequalities for twice differentiable mappings via fractional integrals. Facta Univ. Ser. Math. Inform. 29(4), 371–384 (2014)
Set, E., Sarikaya, M.Z., Ozdemir, M.E., Yildirim, H.: The Hermite–Hadamard’s inequality for some convex functions via fractional integrals and related results. JAMSI 10(2), 69–83 (2014)
Set, E., Işcan, I., Sarikaya, M.Z., Ozdemir, M.E.: On new inequalities of Hermite–Hadmard–Fejer type for convex functions via fractional integrals. Appl. Math. Comput. 259, 875–881 (2015)
Tseng, K.-L., Yang, G.-S., Hsu, K.-C.: Some inequalities for differentiable mappings and applications to Fejer inequality and weighted trapezoidal formula. Taiwan. J. Math. 15(4), 1737–1747 (2011)
Wang, J.R., Li, X., Zhu, C.: Refinements of Hermite–Hadamard type inequalities involving fractional integrals. Bull. Belg. Math. Soc. Simon Stevin 20, 655–666 (2013)
Wang, J.R., Zhu, C., Zhou, Y.: New generalized Hermite–Hadamard type inequalities and applications to special means. J. Inequal. Appl. 2013, 325 (2013)
Zhang, Y., Wang, J.: On some new Hermite–Hadamard inequalities involving Riemann–Liouville fractional integrals. J. Inequal. Appl. 2013, 220 (2013)
Zhang, Z., Wei, W., Wang, J.: Generalization of Hermite–Hadamard inequalities involving Hadamard fractional integrals. Filomat 29(7), 1515–1524 (2015)
Zhang, Z., Wang, J.R., Deng, J.H.: Applying GG-convex function to Hermite–Hadamard inequalities involving Hadamard fractional integrals. Int. J. Math. Math. Sci. 2014, 20 (2014)
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Budak, H. On Fejer Type Inequalities for Convex Mappings Utilizing Fractional Integrals of a Function with Respect to Another Function. Results Math 74, 29 (2019). https://doi.org/10.1007/s00025-019-0960-8
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DOI: https://doi.org/10.1007/s00025-019-0960-8