# CERTAIN NEW PATHWAY TYPE FRACTIONAL INTEGRAL INEQUALITIES

Choi, Junesang;Agarwal, Praveen

• Accepted : 2014.05.14
• Published : 2014.06.25
• 43 6

#### Abstract

In recent years, diverse inequalities involving a variety of fractional integral operators have been developed by many authors. In this sequel, here, we aim at establishing certain new inequalities involving pathway type fractional integral operator by following the same lines, recently, used by Choi and Agarwal [7]. Relevant connections of the results presented here with those earlier ones are also pointed out.

#### Keywords

integral inequalities;extended Chebyshev functional;Riemann-Liouville fractional integral operator;Erd$\acute{e}$lyi-Kober fractional integral operator;pathway fractional integral operator

#### References

1. H. Ogunmez and U.M. Ozkan, Fractional quantum integral inequalities, J. Inequal. Appl. 2011, Article ID 787939, 7 pp.
2. A. M. Mathai and H. J. Haubold, Pathway model, superstatistics, Tsallis statistics and a generalized measure of entropy, Phys. A 375 (2007), 110-122. https://doi.org/10.1016/j.physa.2006.09.002
3. D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, Berlin, 1970.
4. S. S. Nair, Pathway fractional integration operator, Fract. Calc. Appl. Anal. 12(3) (2009), 237-252.
5. A. M. Ostrowski, On an integral inequality, Aequations Math. 4 (1970), 358-373. https://doi.org/10.1007/BF01844168
6. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
7. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
8. W. T. Sulaiman, Some new fractional integral inequalities, J. Math. Anal. 2(2) (2011), 23-28.
9. J. Choi and P. Agarwal, Some new Saigo type fractional integral inequalities and their q-analogues, Abst. Appl. Anal. 2014 (2014), Article ID 579260, 11 pages.
10. D. Baleanu and P. Agarwal, Certain inequalities involving the fractional q-integral operators, Abst. Appl. Anal., 2014, in press.
11. S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math. 10(3) (2009), Art. 86, 5 pp (electronic).
12. P. L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov 2 (1882), 93-98.
13. J. Choi and P. Agarwal, Certain inequalities involving pathway fractional integral operators, submitted.
14. J. Choi and P. Agarwal, Certain fractional integral inequalities involving hypergeometric operators, East Asian Math. J. 30 (2014), in press.
15. Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 9 (2010), 493-497.
16. Z. Dahmani, O. Mechouar and S. Brahami, Certain inequalities related to the Chebyshev's functional involving a type Riemann-Liouville operator, Bull. Math. Anal. Appl. 3(4) (2011), 38-44.
17. S. S. Dragomir, Some integral inequalities of Gruss type, Indian J. Pure Appl. Math. 31(4) (2000), 397-415.
18. S. L. Kalla and A. Rao, On Gruss type inequality for hypergeometric fractional integrals, Matematiche (Catania) 66(1) (2011), 57-64.
19. J. C. Kuang, Applied Inequalities, Shandong Sciences and Technologie Press, 2004 (Chinese).
20. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland Mathematics Studies 204, Amsterdam, London, New York, and Tokyo, 2006.
21. V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. 11 (2007), 395-402.
22. A. M. Mathai, A pathway to matrix-variate gamma and normal densities, Linear Algebra Appl. 396 (2005), 317-328. https://doi.org/10.1016/j.laa.2004.09.022
23. A. M. Mathai and H. J. Haubold, On generalized distributions and path-ways, Phys. Lett. A 372 (2008), 2109-2113. https://doi.org/10.1016/j.physleta.2007.10.084
24. G. A. Anastassiou, Advances on Fractional Inequalities, Springer Briefs in Mathematics, Springer, New York, 2011.
25. D. Baleanu, S. D. Purohit and P. Agarwal, On fractional integral inequalities involving hypergeometric operators, Chinese J. Math. 2014 (2014), Article ID 609476, 5 pages.

#### Cited by

1. Certain recent fractional integral inequalities associated with the hypergeometric operators vol.28, pp.1, 2016, https://doi.org/10.1016/j.jksus.2015.04.002
2. A GRÜSS TYPE INTEGRAL INEQUALITY ASSOCIATED WITH GAUSS HYPERGEOMETRIC FUNCTION FRACTIONAL INTEGRAL OPERATOR vol.30, pp.2, 2015, https://doi.org/10.4134/CKMS.2015.30.2.081