On Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals

Authors

  • Ghulam FARID COMSATS Institute of Information Technology Attock Campus, Pakistan, e-mail: faridphdsms@hotmail.com ghlmfarid@ciit-attock.edu.pk https://orcid.org/0000-0003-0693-8414
  • Atiq Ur REHMAN COMSATS Institute of Information Technology Attock, Pakistan, e-mail: atiq@mathcity.org https://orcid.org/0000-0001-8742-8875
  • Bushra TARIQ Department of Mathematics Comsats Institute of Information Technology Attock, Pakistan, e-mail: bushratariq38@yahoo.com

DOI:

https://doi.org/10.24193/subbmath.2017.2.01

Keywords:

Convex functions, Hadamard inequalities, fractional integrals.

Abstract

In this paper we prove the Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals and the Hadamard-type in- equalities for convex functions via Riemann-Liouville fractional integral are deduced. Also we find connections with some well known results related to the Hadamard inequality.

Mathematics Subject Classification (2010): 26A51, 26A33, 26D10.

References

Azpeitia, A.G., Convex functions and the Hadamard inequality, Revista Colombiana Mat., 28(1994), 7-12.

Bakula, M.K., Ozdemir, M.E., Peˇcari´c, J., Hadamard type inequalities for m-convex and (α, m)-convex functions, J. Ineq. Pure Appl. Math., 9(2008), no. 4, art. 96.

Bakula, M.K., Peˇcari´c, J., Note on some Hadamard-type inequalities, J. Ineq. Pure Appl. Math., 5(2004), no. 3, art. 74.

David, S.A., Linares, J.L., Pallone, E.M.J.A., Fractional order calculus: historical apologia, basic concepts and some applications, Rev. Bras. Ensino Fs., 33(2011), no. 4, 4302- 4302.

Dragomir, S.S., On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 33(2002), no. 1, 45-56.

Farid, G., Rehman, A.U., Zahra, M., On Hadamard inequalities for relative convex functions via fractional integrals, Nonlinear Anal. Forum, 21(2016), no. 1, 77-86.

Farid, G., Marwan, M., Rehman, A.U., New mean value theorems and generalization of Hadamard inequality via coordinated m-convex functions, J. Inequal. Appl. 2015, Article ID 283, 2015, 11p.

Gorenflo, R., Mainardi, F., Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien, 1997, 223-276.

Hilfer, R., (ed.), Applications of fractional calculus in physics, World Scientific Publish- ing Co. 2000.

Kirmaci, U.S., Bakula, M.K., Ozdemir, M.E., Peˇcari´c, J., Hadamard-tpye inequalities for s-convex functions, Appl. Math. Comput., 193(2007), 26-35.

Bakula, M.K., Peˇcari´c, J., Ribicic, M., Companion inequalities to Jensen’s inequality for m-convex and (α,m)-convex functions, J. Inequal. Pure Appl. Math., 7(2006), art. 194, online:htpp://jipam.vu.edu.au.

Loverro, A., Fractional calculus: history, definitions and applications for the engineer, Department of Aerospace and Mechanical Engineering, University of Notre Dame, 2004.

Miller, S., Ross, B., An introduction to fractional calculus and fractional differential equations, John Wiley And Sons, USA, 1993.

Mocanu, P.T., Serb, I., Toader, G., Real star-convex functions, Studia Univ. Babe¸s- Bolyai Math., 42(1997), no. 3, 65-80.

Nishimoto, K., An essence of Nishimoto’s fractional calculus, Descartes Press Co., 1991. [16] O¨ zdemir, M.E., Avci, M., Set, E., On some inequalities of Hermite-Hadamard type via-convexity, Appl. Math. Lett., 23(2010), no. 9, 1065-1070.

Podlubny, I., Fractional differential equations, Mathematics in Science and Engineering V198, Academic Press 1999.

Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N., Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, J. Math. Comput. Model., 57(2013), 2403-2407.

Sarikaya, M.Z., Yildirim, H., On Hermite-Hadamard type inequalities for Riemann- Liouville fractional integrals, RGMIA Research Report Collection, 17(98)(2014), 10 pp. [20] Toader, G.H., Some generalizations of convexity, Proc. Colloq. Approx. Optim., 1984, 29-338.

Downloads

Published

2017-06-15

How to Cite

FARID, G., REHMAN, A. U., & TARIQ, B. (2017). On Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals. Studia Universitatis Babeș-Bolyai Mathematica, 62(2), 141–150. https://doi.org/10.24193/subbmath.2017.2.01

Issue

Section

Articles