Chebyshev-Grüss Type Inequalities for Hadamard \(k\)-Fractional Integrals

Sana Iqbal, Shahid Mubeen, Siddra Habib, Muharrem Tomar


Integral inequalities are taken up to be important as they are useful in the study of different classes of differential and integral equations. During the past several years, many researchers have obtained various fractional integral inequalities comprising the different fractional differential and integral operators. A considerable work is done associated with classical and variants of Grüss type inequality, which actually connects the integral of the product of two functions with the product of their integrals. In this paper, we present the Chebyshev-Grüss type inequalities for Hadamard fractional integrals in the framework of parameter \(k > 0\).


Fractional integral inequalities; Chebyshev functional; \((k, s)\)-fractional integrals

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S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math. 10 (2009), 86.

P. Cerone and S.S. Dragomir, New upper and lower bounds for the Chebyshev functional, J. Inequal. Pure App. Math. 3 (2002), 77.

P.L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov 2 (1882), 93 – 98.

Z. Dahmani and L. Tabharit, On Weighted Grüss type Inequalities via Fractional Integration, J. Adv. Res. Pure Math. 2 (2010), 31 – 38.

Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 9 (2010), 493 – 497.

Z. Dahmani, O. Mechouar and S. Brahami, Certain inequalities related to the Chebyshev’s functional involving a Riemann-Liouville operator, Bull. Math. Anal. Appl. 3 (4) (2011), 38 – 44.

R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulgaciones Matemticas 15 (2007), 179 – 192.

S.S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pure Appl. Math. 31 (4) (2000), 397 – 415.

J. Hadamard, Essai sur l’étude des fonctions données par leur développement de Taylor, Journal of Pure and Applied mathematics 4 (8) (1892), 101 – 186.

V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. Ser. 301, Longman, New York (1994).

C.G. Kokologiannaki and V. Karasniqi, Some properties of k-gamma function, Le Mathematics LXVIII (2013), 13 – 22.

C.G. Kokologiannaki and V.D. Sourla, Bounds for k-gamma and k-beta functions, Journal of Inequalities and Special Functions 4 (3) (2013), 1 – 5.

C.G. Kokologiannaki, Properties and inequalities of generalized k¡gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences 5 (14) (2010), 653 – 660.

V. Krasniqi, A limit for the k-gamma and k-beta function, Int. Math. Forum 5 (2010), 1613 – 1617.

V. Krasniqi, Inequalities and monotonicity for the ration of k-gamma function, Scientia Magna 6 (2010), 40 – 45.

S. Liu and J.Wang, Hermite-Hadamard type fractional integral inequalities for geometric-geometric convex functions, Le Matematiche LXX (Fasc. I) (2015), 3 – 20.

S. Mubeen and G.M. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci. 7 (2012), 89 – 94.

S. Mubeen and S. Iqbal, Grüss type integral inequalities for generalized Riemann-Liouville kfractional integrals, Journal of Inequalities and Applications 1 (2016), 1 – 13.

S.K. Ntouyas, S.D. Purohit and J. Tariboon, Certain Chebyshev type integral inequalities involving the Hadamard’s fractional operators, Abstr. Appl. Anal. volume number missing, 249091 (2014).

J. Pecaric and I. Peric, Identities for the Chebyshev functional involving derivatives of arbitrary order and applications, J. Math. Anal. Appl. 313 (2006), 475 – 483.

J.E. Pecaric, F. Prochan and Y. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, San Diego (1992).

I. Podlubny, Fractional Differential Equations, Academic Press, London (1999).

S.D. Purohit and R.K. Raina, Certain fractional integral inequalities involving the Gauss hypergeometric function, Rev. Téc. Ing. Univ. Zulia 37 (2) (2014), 167 – 175.

S.D. Purohit and S.L. Kalla, Certain inequalities related to the Chebyshev’s functional involving Erdélyi-Kober operators, Sci. Ser. A Math. Sci. 25 (2014), 55 – 63.

L.G. Romero, L.L. Luque, G.A. Dorrego and R.A. Cerutti, On the k-Riemann–Liouville fractional derivative, Int. J. Contemp. Math. Sciences 8 (1) (2013), 41 – 51.

H.A. Salem and A. El-Sayed, Weak solution for fractional order integral equations in reflexive Banach spaces, Mathematica Slovaca 55 (2) (2005), 169 – 181.

M.Z. Sarıkaya, E. Set, H. Yaldız and N. Basak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling 57 (9) (2013), 2403 – 2407.

E. Set, M. Tomar and M.Z. Sarıkaya, On generalized Grüss type inequalities for k-fractional integrals, Applied Mathematics and Computation 269 (2015), 29 – 34.

R. Suryanarayana and C. Rao, Some new inequalities for the generalized (epsilon)-gamma, beta and zeta functions, International Journal of Engineering Research and Technology 1 (9) (2012), 1 – 4.

J. Tariboon, S.K. Ntouyas and M. Tomar, Some new integral inequalities for k-fractional integrals, Malaya J. Mat. 4 (1) (2016), 100 – 110.

J. Wang, C. Zhu and Y. Zhou, New generalized Hermite–Hadamard type inequalities and applications to special means, Journal of Inequalities and Applications 1 (2013), 1 – 15.



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