Generalizations of some integral inequalities for fractional integrals
Abstract
In this paper we give generalizations of the Hadamard-type inequalities for fractional integrals. As special cases we derive several Hadamard type inequalities.
Keywords
convex functions; Hadamard inequality; Fejér–Hadamard inequality; fractional integrals; Hölder inequality
References
2. Azócar A., Nikodem K., Roa G., Fejér-type inequalities for strongly convex functions, Ann. Math. Sil. 26 (2012), 43–53.
3. Azpeitia A.G., Convex functions and the Hadamard inequality, Rev. Colombiana Mat. 28 (1994), 7–12.
4. Díaz R., Pariguan E., On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat. 15 (2007), no. 2, 179–192.
5. Dragomir S.S., Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl. 167 (1992), no. 1, 49–56.
6. Dragomir S.S., On some new inequalities of Hermite–Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002), no. 1, 55–65.
7. 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 (1998), no. 5, 91–95.
8. Dragomir S.S., Pearce C.E.M., Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, Victoria University, Melbourne, 2000.
9. Farid G., Rehman A.U., Zahra M., On Hadamard inequalities for k-fractional integrals, Nonlinear Funct. Anal. Appl. 21 (2016), no. 3, 463–478.
10. Fejér L., Über die Fourierreihen, II, Math. Naturwiss. Anz. Ungar. Akad. Wiss. 24 (1906), 369–390 (in Hungarian).
11. Gill P.M., Pearce C.E.M., Pečarić J., Hadamard’s inequality for r-convex functions, J. Math. Anal. Appl. 215 (1997), no. 2, 461–470.
12. İşcan İ., Hermite–Hadamard–Fejér type inequalities for convex functions via fractional integrals, Stud. Univ. Babeş-Bolyai Math. 60 (2015), no. 3, 355–366.
13. Kirmaci U.S., Klaričić Bakula M., Özdemir M.E., Pečarić J., Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (2007), no. 1, 26–35.
14. Klaričić Bakula M., Özdemir M.E., Pečarić J., Hadamard type inequalities for m-convex and (α,m)-convex functions, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), no. 4, Article 96, 12 pp.
15. Klaričić Bakula M., Pečarić J., Note on some Hadamard-type inequalities, JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), no. 3, Article 74, 9 pp.
16. Lesnic D., Characterizations of the functions with bounded variation, Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics (ICTAMI 2003), Part A, Acta Univ. Apulensis Math. Inform. 6 (2003), 47–54.
17. Mubeen S., Habibullah G.M., k-fractional integrals and application, Int. J. Contemp. Math. Sci. 7 (2012), no. 1–4, 89–94.
18. Sarikaya M.Z., Erden S., On the Hermite–Hadamard–Féjer type integral inequality for convex function, Turkish J. Anal. Number Theory 2 (2014), no. 3, 85–89.
19. Sarikaya M.Z., Set E., Yaldiz H., Başak N., Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling 57 (2013), no. 9–10, 2403–2407.
20. Wang J., Zhu C., Zhou Y., New generalized Hermite–Hadamard type inequalities and applications to special means, J. Inequal. Appl. 2013, Article ID 325, 15 pp.
21. Xiang R., Refinements of Hermite–Hadamard type inequalities for convex functions via fractional integrals, J. Appl. Math. Inform. 33 (2015), no. 1–2, 119–125.
COMSATS Institute of Information Technology, Pakistan Pakistan
COMSATS Institute of Information Technology, Pakistan Pakistan
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
