Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Certain Inequalities Involving Pathway Fractional Integral Operators

  • Choi, Junesang (Department of Mathematics, Dongguk University) ;
  • Agarwal, Praveen (Department of Mathematics, Anand International College of Engineering)
  • Received : 2013.11.24
  • Accepted : 2014.04.04
  • Published : 2016.12.23
Download PDF
⟨ Previous Next ⟩

Abstract

Belarbi and Dahmani [3], recently, using the Riemann-Liouville fractional integral, presented some interesting integral inequalities for the Chebyshev functional in the case of two synchronous functions. Subsequently, Dahmani et al. [5] and Sulaiman [17], provided some fractional integral inequalities. Here, motivated essentially by Belarbi and Dahmani's work [3], we aim at establishing certain (presumably) new inequalities associated with pathway fractional integral operators by using synchronous functions which are involved in the Chebychev functional. Relevant connections of the results presented here with those involving Riemann-Liouville fractional integrals are also pointed out.

Keywords

References

  1. G. A. Anastassiou, Advances on Fractional Inequalities, Springer Briefs in Mathematics, Springer, New York, 2011.
  2. 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.
  3. S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10(3)(2009), Art. 86, 5 pp(electronic).
  4. P. L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre lesmemes limites, Proc. Math. Soc. Charkov, 2(1882), 93-98.
  5. 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.
  6. S. S. Dragomir, Some integral inequalities of Gruss type, Indian J. Pure Appl. Math., 31(4)(2000), 397-415.
  7. S. L. Kalla and A. Rao, On Gruss type inequality for hypergeometric fractional integrals, Matematiche(Catania) 66(1)(2011), 57-64.
  8. 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.
  9. V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11(2007), 395-402.
  10. 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
  11. 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
  12. A. M. Mathai and H. J. Haubold, Pathway model, superstatistics, Tsallis statistics and a generalize measure of entropy, Phys. A, 375(2007), 110-122. https://doi.org/10.1016/j.physa.2006.09.002
  13. S. S. Nair, Pathway fractional integration operator, Fract. Calc. Appl. Anal., 12(3)(2009), 237-252.
  14. H. Ogunmez and U.M. Ozkan, Fractional quantum integral inequalities, J. Inequal. Appl., 2011, Article ID 787939, 7 pp.
  15. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  16. 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.
  17. W. T. Sulaiman, Some new fractional integral inequalities, J. Math. Anal., 2(2)(2011), 23-28.