Some general integral inequalities for Lipschitzian functions via conformable fractional integral

Abstract

In this paper, the author establishes some Hadamard-type and Bullen-type inequalities for Lipschitzian functions via Riemann Liouville fractional integral.

Keywords

• Lipschitzian functions,Hadamard inequality,Bullen inequality,conformable fractional integral

References

1. Ostrowski, A. Uber die Absolutabweichung einer dierentienbaren Funktionen von ihren Integralmittelwert. Comment. Math. Helv., 10 (1938), 226-227.
2. Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional via fractional integration. Ann. Funct. Anal., 1 (1) (2010), 51-58.
3. Dragomir, S.S., Agarwal R.P., Cerone, P., On Simpson's inequality and applications, J. of Inequal. Appl., 5 (2000), 533-579.
4. Dragomir, S.S., Cho, Y.J., Kim, S.S., Inequalities of Hadamard's Type for Lipschitzian Mappings and Their Applications, J. Math. Anal. Appl., 245 (2000), 489-501.
5. Dragomir, S.S., Pearce, C.E.M. , Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
6. Dragomir, S.S., Rassias, Th. M., Ostrowski type inequalities and applications in numerical integration, Kluwer Academic Publishers, Dorcdrecht, Boston, London, 2002.
7. Hadamard, J.S., Etude sur les propiètés des fonctions entieres et en particulier d'une fontion considerer per Riemann, J. Math. Pure and Appl., 58 (1893), 171-215.
8. Hwang, S.-R., Hsu, K.-C. and Tseng, K.-L., Hadamard-type inequalities for Lipschitzian functions in one and two variables with applications, J. Math. Anal. Appl., 405 (2013), 546--554.

9. İşcan, İ., New general integral inequalities for Lipschitzian functions via Hadamard fractional integrals. Int. J. Anal., 2014 (2014), Article ID 353924, 8 pages.
10. İşcan,İ. , Hadamard-type and Bullen-type inequalities for Lipschitzian functions via fractional integrals, Mathematical Sciences and Applications E-Notes, 4 (1) (2016), 77-87.
11. Sarıkaya, M.Z.,Aktan N. , On the generalization of some integral inequalities and their applications, Mathematical and Computer Modelling, 54 (2011), 2175-2182.
12. Set, E. , Akdemir, A.O. , Mumcu, I., The Hermite-Hadamard's inequality and its extentions for conformable fractioanal integrals of any order α>0. Available online at: https://www.researchgate.net/publication/303382221.
13. Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives Theory and Application, Gordan and Breach Science, New York, 1993.
14. Sarıkaya, M.Z., Ogunmez, H., On new inequalities via Riemann-Liouville fractional integration. Abstr. Appl. Anal. 2012 (2012), Article ID 428983, 10 pages.
15. Sarıkaya, M.Z., Set, E., Yaldız, H., Başak, N., Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities. Math. Comput. Modelling, 57 (2013), 2403-2407.
16. Tseng, K.-L., Hwang, S.-R., Dragomir, S.S., Fejér-type inequalities (1). J. Inequal. Appl., 2010 (2010), Article ID 531976, 7 pages.
17. Tseng, K.-L., Hwang, S.-R., Hsu, K.-C., Hadamard-type and Bullen-type inequalities for Lipschitzian functions and their applications, Comput. Math. Appl., 64 (4) (2012), 651-660.
18. Yang, G.-S., Tseng, K.-L., Inequalities of Hadamard's Type for Lipschitzian Mappings, J. Math. Anal. Appl., 260 (1) (2001), 230--238.
19. Zhu, C., Fečkan, M., Wang, J., Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula, J. Appl. Math. Stat. Inform. 8 (2) (2012), 21-28.