The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results

Erhan Set; Mehmet Zeki Sarikaya; M. Emin Özdemir; Hüseyin Yıldırım

doi:10.2478/jamsi-2014-0014

$The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results Cover$

.blurhash-client-img { display: none !important; }

The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results

Journal of Applied Mathematics, Statistics and Informatics

Volume 10 (2014): Issue 2 (December 2014)

By: Erhan Set, Mehmet Zeki Sarikaya, M. Emin Özdemir and Hüseyin Yıldırım

Open Access

|Jan 2015

Abstract

In this paper, we establish Hermite-Hadamard type inequalities for s - convex functions in the second sense and m - convex functions via fractional integrals. The analysis used in the proofs is fairly elementary.

References

[1] Alomari, M., Darus, M., On the Hadamard's inequality for log-convex functions on the coordinates, Journal of Inequalities and Applications, vol. 2009, Article ID 283147, 13 pages, 2009.
Search in Google Scholar
[2] Azpeitia, A.G., Convex functions and the Hadamard inequality, Rev. Colombiana Math., 28 (1994), 7-12.
Search in Google Scholar
[3] Bakula, M.K., Özdemir, M.E., Pečarić, J., Hadamard tpye inequalities for m - convex and (a,m)-convex functions, J. Ineq. Pure and Appl. Math., 9(4) (2008), Art. 96.
Search in Google Scholar
[4] Bakula, M. K., Pečarić, J., Note on some Hadamard-type inequalities, Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, article 74, 2004.
Search in Google Scholar
[5] Dragomir, S. S., Pearce, C. E. M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
Search in Google Scholar
[6] 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) (1998), 91-95.
Open DOI Search in Google Scholar
[7] Dragomir, S. S., On some new inequalities of Hermite-Hadamard type for m - convex functions, Tamkang J. Math., 3(1) (2002).
Open DOI Search in Google Scholar
[8] Dragomir, S. S., Fitzpatrik, S, .The Hadamard's inequality for s -convex functions in the second sense, Demonstratio Math. 32(4), (1999), 687-696.
Open DOI Search in Google Scholar
[9] Gill, P. M. , Pearce, C. E. M., Pečarić, J., Hadamard's inequality for r -convex functions, Journal of Mathematical Analysis and Applications, vol. 215, no. 2, pp. 461--470, 1997.
Open DOI Search in Google Scholar
[10] Hudzik, H., Maligranda, L., Some remarks on s - convex functions, Aequationes Math. 48 (1994), 100-111.
Open DOI Search in Google Scholar
[11] Kirmaci, U.S., Bakula, M.K., Özdemir, M.E., Pečarić, J., Hadamard-tpye inequalities for s - convex functions, Appl. Math. and Comp., 193 (2007), 26-35.
Open DOI Search in Google Scholar
[12] Toader, G, .Some generalizations of the convexity, Proceedings of The Colloquium On Approximation And Optimization, Univ. Cluj-Napoca, Cluj-Napoca,1985, 329-338.
Search in Google Scholar
[13] Özdemir, M. E., Avci, M., Set, E, .On some inequalities of Hermite-Hadamard type via m - convexity, Applied Mathematics Letters, vol. 23, no. 9, pp. 1065--1070, 2010.
Search in Google Scholar
[14] Pečarić, J.E., Proschan, F., Tong, Y.L., Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992.
Search in Google Scholar
[15] Set, E., Özdemir, M. E. , Dragomir, S. S., On the Hermite-Hadamard inequality and other integral inequalities involving two functions, Journal of Inequalities and Applications, Article ID 148102, 9 pages, 2010.
Open DOI Search in Google Scholar
[16] Set, E., Özdemir, M. E. , Dragomir, S. S., On Hadamard-Type inequalities involving several kinds of convexity, Journal of Inequalities and Applications, Article ID 286845, 12 pages, 2010.
Open DOI Search in Google Scholar
[17] Mitrinović, D.S., Pečarić, J.E., Fink, A.M., Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993, p. 106.
Open DOI Search in Google Scholar
[18] Anastassiou, G., Hooshmandasl, M.R., Ghasemi, A., Moftakharzadeh, F., Montogomery identities for fractional integrals and related fractional inequalities, J. Ineq. Pure and Appl. Math., 10(4) (2009), Art. 97.
Search in Google Scholar
[19] Belarbi, S., Dahmani, Z., On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3) (2009), Art. 86.
Search in Google Scholar
[20] Dahmani, Z., New inequalities in fractional integrals, International Journal of Nonlinear Scinece, 9(4) (2010), 493-497.
Search in Google Scholar
[21] Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1) (2010), 51-58.
Open DOI Search in Google Scholar
[22] Dahmani, Z., Tabharit, L. , Taf, S., Some fractional integral inequalities, Nonl. Sci. Lett. A, 1(2) (2010), 155-160.
Search in Google Scholar
[23] Dahmani, Z., Tabharit, L. , Taf, S., New generalizations of Gruss inequality usin Riemann- Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3) (2010), 93-99.
Search in Google Scholar
[24] Gorenflo, R., Mainardi, F., Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien (1997), 223-276.
Open DOI Search in Google Scholar
[25] Miller, S., Ross, B, .An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993, p.2.
Search in Google Scholar
[26] Podlubni, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
Search in Google Scholar
[27] 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.
Open DOI Search in Google Scholar
[28] Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N., Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57 (2013), 2403--2407.
Open DOI Search in Google Scholar
[29] Set, E., New inequalities of Ostrowski type for mapping whose derivatives are s-convex in the second sense via fractional integrals, Computers and Math. with Appl. 63 (2012), 1147-1154.
Open DOI Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/jamsi-2014-0014 | Journal eISSN: 1339-0015 (formerly 1336-9180) | Journal ISSN: 1336-9180

Journal RSS Feed

Language: English

Page range: 69 - 83

Published on: Jan 27, 2015

Published by: University of Ss. Cyril and Methodius in Trnava

In partnership with: Paradigm Publishing Services

Keywords:

s - convex functions,

Hermite-Hadamard type inequalities,

Riemann-Liouville fractional integral

Related subjects:

Mathematics,

Applied mathematics

© 2015 Erhan Set, Mehmet Zeki Sarikaya, M. Emin Özdemir, Hüseyin Yıldırım, published by University of Ss. Cyril and Methodius in Trnava
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 10 (2014): Issue 2 (December 2014)