# A GRÜSS TYPE INTEGRAL INEQUALITY ASSOCIATED WITH GAUSS HYPERGEOMETRIC FUNCTION FRACTIONAL INTEGRAL OPERATOR

• Choi, Junesang (Department of Mathematics Dongguk University) ;
• Purohit, Sunil Dutt (Department of Mathematics University College of Engineering Rajasthan Technical University)
• Received : 2014.10.04
• Published : 2015.04.30
• 122 34

#### Abstract

In this paper, we aim at establishing a generalized fractional integral version of Gr$\ddot{u}$ss type integral inequality by making use of the Gauss hypergeometric function fractional integral operator. Our main result, being of a very general character, is illustrated to specialize to yield numerous interesting fractional integral inequalities including some known results.

#### Acknowledgement

Supported by : Ministry of Education, Science and Technology of the Republic of Korea

#### References

1. G. A. Anastassiou, Fractional Differentiation Inequalities, Springer Publishing Company, New York, 2009.
2. G. A. Anastassiou, Advances on Fractional Inequalities, Springer Briefs in Mathematics, Springer, New York, 2011.
3. G. A. Anastassiou, Fractional representation formulae and right fractional inequalities, Math. Comput. Modelling 54 (2011), no. 11-12, 3098-3115. https://doi.org/10.1016/j.mcm.2011.07.040
4. A. Atangana and N. Bildik, The use of fractional order derivative to predict the groundwater flow, Math. Probl. Eng. 2013 (2013), Article ID 543026, 9 pages.
5. A. Atangana and A. Kilicman, A novel integral operator transform and its application to some FODE and FPDE with some kind of singularities, Math. Probl. Eng. 2014(2014), Article ID 531984, 7 pages.
6. A. Atangana and A. Secer, A note on fractional order derivatives and table of fractional derivatives of some special functions, Abstr. Appl. Anal. 2013 (2013), Article ID 279681, 8 pages.
7. D. Baleanu, S. D. Purohit, and P. Agarwal, On fractional integral inequalities involving hypergeometric operators, Chin. J. Math. 2014 (2014), Article ID 609476, 5 pages.
8. D. Baleanu, S. D. Purohit, and F. Ucar, On Gruss type integral inequality involving the Saigo's fractional integral operators, J. Comput. Anal. Appl. 19 (2015), no. 3, 480-489.
9. P. Cerone and S. S. Dragomir, New upper and lower bounds for the Chebyshev functional, J. Inequal. Pure App. Math. 3 (2002), no. 2, Article 77, 13 pages.
10. 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.
11. J. Choi and P. Agarwal, Some new Saigo type fractional integral inequalities and their q-analogues, Abstr. Appl. Anal. 2014 (2014), Article ID 579260, 11 pages.
12. J. Choi and P. Agarwal, Certain fractional integral inequalities involving hypergeometric operators, East Asian Math. J. 30 (2014), 283-291. https://doi.org/10.7858/eamj.2014.018
13. J. Choi and P. Agarwal, Certain new pathway type fractional integral inequalities, Honam Math. J. 36(2014), no. 2, 455-465. https://doi.org/10.5831/HMJ.2014.36.2.455
14. J. Choi and P. Agarwal, Certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions, Abstr. Appl. Anal. 2014 (2014), Article ID 735946, 7 pages.
15. L. Curiel and L. Galue, A generalization of the integral operators involving the Gauss' hypergeometric function, Rev. Tech. Ingr. Unlv. Zulla 19 (1996), no. 1, 17-22.
16. Z. Dahmani and A. Benzidane, New weighted Gruss type inequalities via (${\alpha},\;{\beta}$) fractional q-integral inequalities, IJIAS 1 (2012), no. 1, 76-83.
17. Z. Dahmani, L. Tabharit, and S. Taf, New generalizations of Gruss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl. 2 (2010), no. 2, 93-99.
18. A. Debbouche, D. Baleanu, and R. P. Agarwal, Nonlocal nonlinear integrodifferential equations of fractional orders, Bound. Value Probl. 2012 (2012), 78, 10 pages.
19. Z. Denton and A. S. Vatsala, Monotonic iterative technique for finite system of nonlinear Riemann-Liouville fractional differential equations, Opusc. Math. 31 (2011), no. 3, 327-339. https://doi.org/10.7494/OpMath.2011.31.3.327
20. S. S. Dragomir, A generalization of Gruss's inequality in inner product spaces and applications, J. Math. Anal. Appl. 237 (1999), no. 1, 74-82. https://doi.org/10.1006/jmaa.1999.6452
21. S. S. Dragomir, A Gruss type inequality for sequences of vectors in inner product spaces and applications, J. Inequal. Pure Appl. Math. 1 (2000), no. 2, Article 12, 11 pages.
22. S. S. Dragomir, Some integral inequalities of Gruss type, Indian J. Pure Appl. Math. 31 (2000), no. 4, 397-415.
23. S. S. Dragomir, Operator Inequalities of the Jensen, Cebysev and Gruss Type, Springer Briefs in Mathematics, Springer, New York, 2012.
24. S. S. Dragomir S. S. Dragomir and S. Wang, An inequality of Ostrowski-Gruss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl. 13 (1997), no. 11, 15-20.
25. H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl. 47 (2004), no. 2-3, 281-300. https://doi.org/10.1016/S0898-1221(04)90025-9
26. B. Gavrea, Improvement of some inequalities of Chebysev-Gruss type, Comput. Math. Appl. 64 (2012), no. 5, 2003-2010. https://doi.org/10.1016/j.camwa.2012.03.101
27. D. Gruss, Uber das maximum des absoluten Betrages von${\frac{1}{b-a}}{\int_{a}^{b}}f(x)g(x)dx-{\frac{1}{(b-a)^2}}{\int_{a}^{b}}f(x)dx{\int_{a}^{b}}g(x)dx$, Math. Z. 39 (1935), no. 1, 215-226. https://doi.org/10.1007/BF01201355
28. S. L. Kalla and A. Rao, On Gruss type inequality for a hypergeometric fractional integrals, Matematiche 66 (2011), no. 1, 57-64.
29. G. Kapoor, On some discrete Gruss type inequalities, Inter. J. Math. Sci. Appl. 2 (2012), no. 2, 729-734.
30. V. Kiryakova, Generalized Fractional Calculus and Applications, (Pitman Res. Notes Math. Ser. 301), Longman Scientific & Technical, Harlow, 1994.
31. V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. 11 (2007), no. 3-4, 395-402.
32. Z. Liu, Some Ostrowski-Gruss type inequalities and applications, Comput. Math. Appl. 53 (2007), no. 1, 73-79. https://doi.org/10.1016/j.camwa.2006.12.021
33. Y. Liu, J. J. Nieto, and S. Otero-Zarraquinos, Existence results for a coupled system of nonlinear singular fractional differential equations with impulse effects, Math. Probl. Eng. 2013 (2013), Article ID 498781, 21 pages.
34. M. Matic, Improvement of some inequalities of Euler-Gruss type, Comput. Math. Appl. 46 (2003), no. 8-9, 1325-1336. https://doi.org/10.1016/S0898-1221(03)90222-7
35. M. A. Mercer, An improvement of the Gruss inequality, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 93, 4 pages.
36. N. A.Mir and R. Ullah, Some inequalities of Ostrowski and Gruss type for triple integrals on time scales, Tamkang J. Math. 42 (2011), no. 4, 415-426.
37. D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, 1993.
38. S. K. Ntouyas, S. D. Purohit, and J. Tariboon, Certain Chebyshev type integral inequalities involving Hadamard's fractional operators, Abstr. Appl. Anal. 2014 (2014), Article ID 249091, 7 pages.
39. U. M. Ozkan and H. Yildirim, Gruss type inequalities for double integrals on time scales, Comput. Math. Appl. 57 (2009), no. 3, 436-444. https://doi.org/10.1016/j.camwa.2008.11.006
40. B. G. Pachpatte, On Gruss type integral inequalities, J. Inequal. Pure Appl. Math. 3 (2002), no. 1, Article 11, 5 pages.
41. B. G. Pachpatte, A note on Chebyshev-Gruss inequalities for differentiable functions, Tamsui Oxf. J. Math. Sci. 22 (2006), no. 1, 29-36.
42. S. D. Purohit and R. K. Raina, Chebyshev type inequalities for the Saigo fractional integrals and their q-analogues, J. Math. Inequl. 7 (2013), no. 2, 239-249.
43. S. D. Purohit and R. K. Raina, Certain fractional integral inequalities involving the Gauss hypergeometric function, Rev. Tec. Ing. Univ. Zulia 37 (2014), no. 2, 167-175.
44. J. D. Ramirez and A. A. Vatsala, Monotonic iterative technique for fractional differential equations with periodic boundary conditions, Opusc. Math. 29 (2009), no. 3, 289-304. https://doi.org/10.7494/OpMath.2009.29.3.289
45. M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11 (1978), no. 2, 135-143.
46. H.-R. Sun, Y.-N. Li, J. J. Nieto, and Q. Tang, Existence of solutions for Sturm-Liouville boundary value problem of impulsive differential equations, Abstr. Appl. Anal. 2012 (2012), Article ID 707163, 19 pages.
47. J. Tariboon, S. K. Ntouyas, and W. Sudsutad, Some new Riemann-Liouville fractional integral inequalities, Int. J. Math. Math. Sci. 2014 (2014), Article ID 869434, 6 pages.
48. G. Wang, P. Agarwal, and M. Chand, Certain Gruss type inequalities involving the generalized fractional integral operator, J. Inequal. Appl. 2014 (2014), 147, 8 pages. https://doi.org/10.1186/1029-242X-2014-8
49. W. Yang, On weighted q-Cebysev-Gruss type inequalities, Comput. Math. Appl. 61 (2011), no. 5, 1342-1347. https://doi.org/10.1016/j.camwa.2011.01.001
50. Z.-H. Zhao, Y.-K. Chang, and J. J. Nieto, Asymptotic behavior of solutions to abstract stochastic fractional partial integrodifferential equations, Abstr. Appl. Anal. 2013 (2013), Article ID 138068, 8 pages.
51. C. Zhu, W. Yang, and Q. Zhao, Some new fractional q-integral Gruss-type inequalities and other inequalities, J. Inequal. Appl. 2012 (2012), 299, 15 pages. https://doi.org/10.1186/1029-242X-2012-15