INEQUALITIES GENERATED WITH RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OPERATOR

Abstract

The primary objective of this study is to handle new generalized midpoint, trapezoid and Simpson's type inequalities with the help of Riemann-Liouville fractional integral operator. In order to do this, a new fractional integral identity is obtained. Then by using this identity, some inequalities for the class of functions whose derivatives in absolute values at certain powers are convex are derived. It is observed that the obtained inequalities are generalizations of some results exist in the literature.

Keywords

• Riemann-Liouville fractional integrals,Hadamard type inequalities,Simpson's inequality,convex functions.

References

1. Erd´elyi A, Magnus W, Oberhettinger F, Tricomi F., (1981), Higher Transcendental Functions Vol. I-III. Melbourne, Florida, Krieger Pub.
2. Iscan, ˙I., (2012), A new generalization of some integral inequalities and their applications. arXiv preprint arXiv:1207.1828.
3. Katugampola UN., (2011), New approach to a generalized fractional integral. Appl Math Comput., 218 (3); 860-865.
4. Katugampola UN., (2014), New approach to generalized fractional derivatives. Bull Math Anal Appl., 6 (4); 1-15.
5. Kilbas AA, Srivastava HM, Trujillo JJ., (2014), Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies; 204 pp.
6. Kilbas AA., (2001), Hadamard-type fractional calculus. J. Korean Mathematical Society, 38 (6); 1191– 1204.
7. Peng, C., Zhou, C., & Du, T. S., (2017), Riemann-Liouville fractional Simpson’s inequalities through generalized (m, h1, h2)-preinvexity. Ital. J. Pure Appl. Math, 38, 345-367.
8. Qiu K, Wang JR., (2018), A fractional integral identity and its application to fractional Hermite- Hadamard type inequalities. Jour Interdisciplinary Math, 21 (1); 1-16.

9. Akdemir, A.O., Ekinci A. and Set, E., (2017), Conformable Fractıonal Integrals and Related New Integral Inequalıtıes, Journal of Nonlinear and Convex Analysis, Volume 18, Number 4, 661-674.
10. Set, E., G¨ozpınar, A. and Ekinci, A., (2017), Hermite-Hadamard Type Inequalities via Conformable
11. Fractional Integrals, Acta Mathematica Universitatis Comenianae, Vol. 86, Issue 2, pp. 309-320.