On refinements of Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral operators

On refinements of Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral operators

In this paper, we first establish weighted versions of Hermite-Hadamardtype inequalities for Riemann-Liouville fractional integral operators utilizingweighted function. Then we obtain some refinements of these inequalities.The results obtained in this study would provide generalization of inequalitiesproved in earlier works.

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  • Azpeitia, A.G. (1994). Convex functions and the Hadamard inequality. Rev. Colombiana Math., 28 , 7-12.
  • Dragomir, S.S. and Pearce, C.E.M. (2000). Selected topics on Hermite-Hadamard in- equalities and applications. RGMIA Mono- graphs, Victoria University.
  • Dragomir, S.S. (1992). Two mappings in con- nection to Hadamard’s inequalities. J. Math. Anal. Appl., 167, 49-56.
  • Ertuğral, F., Sarıkaya, M. Z. and Bu- dak, H. (2018). On refinements of Hermite- Hadamard-Fejer type inequalities for frac- tional integral operators. Applications and Applied Mathematics, 13(1), 426-442.
  • Farissi, A.E. (2010). Simple proof and re nement of Hermite-Hadamard inequality. J. Math. Inequal., 4, 365-369.
  • Fejér, L. (1906). Uberdie Fourierreihen, II, Math., Naturwise. Anz Ungar. Akad. Wiss, 24, 369-390.
  • Gorenflo, R., Mainardi, F. (1997). Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien, 223- 276.
  • Hwang, S.R., Yeh S.Y. and Tseng, K.L. (2014). Refinements and similar extensions of Hermite–Hadamard inequality for frac- tional integrals and their applications. Ap- plied Mathematics and Computation, 24, 103- 113.
  • Hwang, S.R., Tseng, K.L., Hsu, K.C. (2013). Hermite–Hadamard type and Fejer type in- equalities for general weights (I). J. Inequal. Appl. 170.
  • Iqbal, M., Qaisar S. and Muddassar, M. (2016). A short note on integral inequality of type Hermite-Hadamard through convex- ity. J. Computational analaysis and applica- tions, 21(5), 946-953.
  • İşcan, I. (2015). Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals. Stud. Univ. Babeş-Bolyai Math. 60(3), 355-366.
  • Kilbas, A.A., Srivastava H.M. and Trujillo, J.J. (2006). Theory and applications of frac- tional differential equations. North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam.
  • Ahmad, B., Alsaedi, A., Kirane, M. and Torebek, B.T. (2019). Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals. Journal of Computational and Applied Mathematics, 353, 120-129.
  • Latif, M.A. (2012). On some refinements of companions of Fejér’s inequality via superquadratic functions. Proyecciones J. Math., 31(4), 309-332.
  • Miller S. and Ross, B. (1993). An introduc- tion to the fractional calculus and fractional differential equations. John Wiley and Sons, USA.
  • Noor, M.A., Noor K.I. and Awan, M.U. (2016). New fractional estimates of Hermite- Hadamard inequalities and applications to means, Stud. Univ. Babe ̧s-Bolyai Math. 61(1), 3-15.
  • Pečarić, J.E., Proschan F. and Tong, Y.L. (1992). Convex functions, partial orderings and statistical applications. Academic Press, Boston.
  • Podlubny, I. (1999). Fractional differential equations. Academic Press, San Diego.
  • Sarikaya, M.Z. and Yildirim, H. (2016). On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals. Miskolc Mathematical Notes, 17(2), 1049- 1059.
  • Sarikaya, M.Z., Set, E., Yaldiz H. and Basak, N. (2013). Hermite-Hadamard’s inequalities for fractional integrals and related frac- tional inequalities. Mathematical and Com- puter Modelling, 57, 2403-2407.
  • Sarikaya, M.Z. and Budak, H. (2016). Gen- eralized Hermite-Hadamard type integral in- equalities for fractional integral, Filomat, 30(5), 1315-1326 (2016).
  • Xiang, R. (2015). Refinements of Hermite- Hadamard type inequalities for convex func- tions via fractional integrals. J. Appl. Math. and Informatics, 33, No. 1-2, 119-125.
  • Tseng, K.L., Hwang, S.R. and Dragomir, S.S. (2012). Refinements of Fejér’s inequality for convex functions. Period. Math. Hung., 65, 17-28.
  • Yaldiz, H. and Sarikaya, M.Z. On Hermite- Hadamard type inequalities for fractional in- tegral operators, ResearchGate Article, Avail- able online at: https://www.researchgate. net/publication/309824275.
  • Yang, G.S. and Tseng, K.L. (1999). On cer- tain integral inequalities related to Hermite- Hadamard inequalities. J. Math. Anal. Appl., 239, 180-187.
  • Yang, G.S. and Hong, M.C. (1997). A note on Hadamard’s inequality, Tamkang J. Math., 28, 33-37.