Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Turkish Journal of Analysis and Number Theory. 2014, 2(3), 65-69
DOI: 10.12691/tjant-2-3-2
Open AccessArticle

A Note on Saigo’s Fractional Integral Inequalities

Guotao Wang1, , Harshvardhan Harsh2, S.D. Purohit3 and Trilok Gupta4

1School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi, People’s Republic of China

2Department of Mathematics, Amity University, Jaipur, India

3Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur, India

4Department of Civil Engineering, College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur, India

Pub. Date: June 12, 2014

Cite this paper:
Guotao Wang, Harshvardhan Harsh, S.D. Purohit and Trilok Gupta. A Note on Saigo’s Fractional Integral Inequalities. Turkish Journal of Analysis and Number Theory. 2014; 2(3):65-69. doi: 10.12691/tjant-2-3-2


In this paper, some new integral inequalities related to the bounded functions, involving Saigo’s fractional integral operators, are eshtablished. Special cases of the main results are also pointed out.

Integral inequalities Gauss hypergeometric function Saigo’s fractional integral operators

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/


[1]  Anastassiou, G.A.: Advances on Fractional Inequalities, Springer Briefs in Mathematics, Springer, New York, 2011.
[2]  Ahmadmir, N. and Ullah, R.: Some inequalities of Ostrowski and Gr¨uss type for triple integrals on time scales, Tamkang J. Math., 42(4) (2011), 415-426.
[3]  Baleanu, D. and Purohit, S.D.: Chebyshev type integral inequalities involving the fractional hypergeometric operators, Abstract Appl. Anal., 2014, Article ID 609160, 10 (pp).
[4]  Baleanu, D., Purohit, S.D. and Agarwal, P.: On fractional integral inequalities involving hypergeometric operators, Chinese Journal of Mathematics, 2014, Article ID 609476, 5(pp.).
[5]  Belarbi, S. and Dahmani, Z.: On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10(3)(2009), Art. 86, 5 (pp).
[6]  Cerone, P. and Dragomir, S.S.: New upper and lower bounds for the Chebyshev functional, J. Inequal. Pure App. Math., 3 (2002), Article 77.
[7]  Chebyshev, P.L.: Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2(1882), 93-98.
[8]  Dahmani, Z. and Benzidane, A.: New weighted Gruss type inequalities via (α, β) fractional qintegral inequalities, International Journal of Innovation and Applied Studies, 1(1)(2012), 76-83.
[9]  Dahmani, Z., Tabharit, L. and Taf, S.: New generalisations of Gr¨uss inequality using Riemann- Liouville fractional integrals, Bull. Math. Anal. Appl., 2 (3)(2010), 93-99.
[10]  Dragomir, S.S.: A generalization of Grüss’s inequality in inner product spaces and applications, J. Math. Anal. Appl., 237 (1999), 74-82.
[11]  Dragomir, S.S.: A Grüss type inequality for sequences of vectors in inner product spaces and applications, J. Inequal. Pure Appl. Math., 1(2) (2000), 1-11.
[12]  Dragomir, S.S.: Some integral inequalities of Grüss type, Indian J. Pure Appl. Math., 31(4)(2000), 397-415.
[13]  Dragomir, S.S.: Operator Inequalities of the Jensen,Čebyšev and Grüss Type, Springer Briefs in Mathematics, Springer, New York, 2012.
[14]  Dragomir, S.S. and Wang, S.: An inequality of Ostrowski-Grüss’ type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl., 13(11) (1997), 15-20.
[15]  Gauchman, H.: Integral inequalities in q-calculus, Comput. Math. Appl., 47 (2004), 281-300.
[16]  Gavrea, B.: Improvement of some inequalities of Chebysev-Grüss type, Comput. Math. Appl., 64 (2012), 2003-2010.
[17]  Grüss, D.: Uber das maximum des absoluten Betrages von Math. Z., 39(1935), 215-226.
[18]  Kalla, S.L. and Rao, Alka: On Grüss type inequality for hypergeometric fractional integrals, Le Matematiche, 66 (1)(2011), 57-64.
[19]  Kapoor, G.: On some discrete Gruss type inequalities, Int Jr. of Mathematical Sciences & Applications, 2(2) (2012), 729-734.
[20]  Liu, Z.: Some Ostrowski-Grüss type inequalities and applications, Comput. Math. Appl., 53 (2007), 73-79.
[21]  Maticć, M.: Improvment of some inequalities of Euler-Grüss type, Comput. Math. Appl., 46 (2003), 1325-1336.
[22]  Mercer, McD A.: An improvement of the Grüss inequality, J. Inequa. Pure Appl. Math., 6(4) (2005), 1-4.
[23]  Mitrinović, D.S., Pečarić, J.E. and Fink, A.M.: Classical and New Inequalities in Analysis, Kluwer Academic, 1993.
[24]  Ntouyas, S.K., Purohit, S.D. and Tariboon, J.: Certain Chebyshev type integral inequalities involving the Hadamard’s fractional operators, Abstract Appl. Anal. 2014, Article ID 249091, 7(pp).
[25]  Öğünmez, H. and Özkan, U.M.: Fractional quantum integral inequalities, J. Inequal. Appl., Volume 2011, Article ID 787939, 7 (pp).
[26]  Özkan, U.M. and Yildirim, H.: Grüss type inequalities for double integrals on time scales, Comput. Math. Appl., 57 (2009), 436-444.
[27]  Pachpatte, B.G.: On Grüss type integral inequalities, J. Inequa. Pure Appl. Math., 3 (1) (2002), 1-5.
[28]  Pachpatte, B.G.: A note on Chebyshev-Grüss inequalities for differential equations, Tamsui Oxford Journal of Mathematical Sciences, 22(1), (2006), 29-36.
[29]  Purohit, S.D. and Raina, R.K.: Chebyshev type inequalities for the Saigo fractional integrals and their q-analogues, J. Math. Inequal., 7(2) (2013), 239-249.
[30]  Purohit, S.D. and Raina, R.K.: Certain fractional integral inequalities involving the Gauss hypergeometric function, Rev. Téc. Ing. Univ. Zulia, 37(2)(2014), In press.
[31]  Purohit, S.D., U¸car, F. and Yadav, R.K.: On fractional integral inequalities and their q-analogues, Revista Tecnocientifica URU, 6 (2014), In press.
[32]  Wang, G., Agarwal, P. and Chand, M.: Certain Grss type inequalities involving the generalized fractional integral operator. Journal of Inequalities and Applications 2014, 2014:147.
[33]  Tariboon, J., Ntouyas, S.K. and Sudsutad, W.: Some new Riemann-Liouville fractional integral inequalities, Int. J. Math. Math. Sci., 2014, Article ID 869434, 6 (pp).
[34]  Yang, W.: On weighted q-Čebyšev-Grüss type inequalities, Comput. Math. Appl., 61 (2011), 1342-1347.
[35]  Zhu, C., Yang, W. and Zhao, Q.: Some new fractional q-integral Gr¨uss-type inequalities and other inequalities, J. Inequal. Appl., 2012 (2012), 299.
[36]  Saigo, M.: A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ., 11 (1978) 135-143.
[37]  Kiryakova, V.: Generalized Fractional Calculus and Applications (Pitman Res. Notes Math. Ser. 301), Longman Scientific & Technical, Harlow, 1994.