Gr$\ddot{u}$ss type integral inequalities for a new class of $k$-fractional integrals

Document Type : Research Paper


1 Department of Mathematics‎, ‎G.C‎. ‎University Faisalabad‎, ‎Faisalabad‎, ‎Pakistan

2 Department of Mathematics‎, ‎COMSATS University Islamabad‎, ‎Attock Campus‎, ‎Pakistan

3 Department of Mathematics‎, ‎University of Sargodha‎, ‎Sargodha‎, ‎Pakistan‎


‎The main aim of this research article is to present the generalized $k$-fractional conformable integrals and an improved version of Gr$\ddot{u}$ss integral inequality via the fractional conformable integral in status of a new parameter $k>0$‎. ‎Here for establishing Gr$\ddot{u}$ss inequality in fractional calculus the classical method of proof has been adopted also related results with Gr$\ddot{u}$ss inequality have been discussed‎. ‎This work contributes in the current research by providing mathematical results along with their verifications‎.


[1] G. Abbas, K.A. Khan, G. Farid and A. Ur Rehman, Generalizations of some fractional integral inequalities via generalized Mittag-Leffler function, J. Inequal. Appl. 2017 (2017) 121.
[2] G. Abbas and G. Farid, Some integral inequalities for m-convex functions via generalized fractional integral operator containing generalized Mittag-Leffler function, Cogent Math. 3 (2016) 1269589.
[3] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015) 57-66.
[4] P. Agarwal, S. Salahshour, S.K. Ntouyas, and J. Tariboon, Certain inequalities involving generalized Erdelyi Kober fractional q-integral operators, Sci. World J. 2014 (2014) Article ID 174126, 11 pages.
[5] W. Ayub, G. Farid, and A. Ur Rehman, Generalization of the Fejer-Hadamard type inequalities for p-convex functions via k-fractional integrals, Commun. Math. Model. Appl. 2(3) (2017) 1-15.
[6] E. Akin, S. Asliyuce, A.F. Guvenilir, and B. Kaymakcalan, Discrete Gruss type inequality on fractional calculus, J. Inequal. Appl. 2015 (2015) 174.
[7] D. Baleanu and A. Agarwal, Certain inequalities involving the fractional q-integral operators, Abstr. Appl. Anal. 2014 (2014) 10 pages.
[8] J. Choi and P. Agarwal, Some new Saigo type fractional integral inequalities and their q-analogues. Abstr. Appl. Anal. 2014 (2014) 11 pages.
[9] J. Choi, D. Ritelli, and P.  Agarwal, Some new inequalities involving generalized Erdelyi-Kober fractional q-integral operator, Appl. Math. Sci. 9 (2015) 3577-3591.
[10] F. Chen, A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals, Ital. J. Pure Appl. Math. 2014 (2014) 299-306.
[11] S.S. Dragomir, A generalization of Gruss inequality in inner product spaces and applications, J. Math. Anal. Appl. 237 (1999) 74-82.
[12] R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat. 15 (2007) 179-192.
[13] Z. Dahmani, About some integral inequalities using Riemann-Liouville integrals, Gen. Math. 20 (2012) 63-69.
[14] N. Elezovic, L.J. Marangunic, and J. Pecaric, Some improvements of Gruss type inequality, J. Math. Ineq. 1(3) (2007) 425-436.
[15] G. Farid and A. Ur Rehman, On Chebyshev functional and Ostrowski-Gruss type inequalities for two coordinates. Int. J. Anal. App. 12(2) (2016) 180-187.
[16] G. Farid, S.  Rafique, and A.  Ur Rehman, More on Ostrowski and Ostrowski-Gruss type inequalities, Commun. Optim. Theory 2017 (2017) 9 pages.
[17] G. Gruss, Uber das maximum des absoluten Betrages 1 b−a R b af(x)g(x)dx −1 (b−a) 2 R b a f(x)dx 1 b−a R b a g(x)dx, Math. Z. 39 (1935) 215-226.
[18] F. Jarad, E.  Ugurlu, T. Abdeljawad, and D. Baleanu, On a new class of fractional operators, Adv. Difference Equ. 1 (2017) 247.
[19] E. Kacar and H. Yildirim, Gruss type integral inequalities for generalized Riemann-Liouville fractional integrals, Int. J. Pure Appl. Math. 101 (2015) 55-70.
[20] V. Krasniqi, Inequalities and monotonicity for the ration of k-gamma functions. Sci. Magna. 6 (2010 ) 40-45.
[21] C.G. Kokologiannaki and V.  Krasniqi, Some properties of the k-gamma function, Le Math. 68 (2013) 13-22.
[22] X. Li, R.N. Mohapatra, and R.S. Rodriguez, Gruss-type inequalities, J. Math. Anal. Appl. 267 (2002) 434-443.
[23] Y. Liao, J. Deng, and J. Wang, Riemann-Liouville fractional Hermite-Hadamard inequalities. Part I. for once differentiable geometric-arithmetically s-convex functions, J. Inequal. Appl. 2013 (2013) 443.
[24] X. Liu, L.  Zhang, P.  Agarwal, and G. Wang, On some new integral inequalities of Gronwall-Bellman-Bihari type with delay for discontinuous functions and their applications, Indag. Math. 27(1) (2016) 1-10.
[25] S. Mubeen and G.M. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci. 7 (2012) 89-94.
[26] J. Park, Some integral inequalities for convex functions via Riemann-Liouville integrals, Appl. Math. Sci. 9 (2015) 1341-1353.
[27] B.G. Pachpatte, On multidimensional Gruss type inequalities, J. Inequal. Pure Appl. Math. 3 (2002) 1-15.
[28] L.G. Romero, L.L. Luque, G.A. Dorrego, and R.A. Cerutti, On the k-Riemann-Liouville fractional derivative, Int. J. Contemp. Math. Sci. 8 (2013) 41-51.
[29] M.Z. Sarikaya, H. Filiz, and M.E. Kiris, On some generalized integral inequalities for Riemann-Liouville fractional integrals, Filomat 29(6) (2015) 1307-1314.
[30] M.Z. Sarikaya and A. Karaca, On the k-Riemann-Liouville fractional integral and applications. Int. J. Math. Stat. 1 (2014) 033-043.
[31] M.Z. Sarikaya, Z. Dahmani, M.E. Kiris, and F. Ahmad, (k, s)-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat. 45(1) (2016) 77–89.
[32] J. Tariboon, S.K. Ntouyas, and W. Sudsutad, Some new Riemann-Liouville fractional integral inequalities, Int. J. Math. Math. Sci. 2014 (2014), 6 pages.
[33] T. Tunc, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat 27(4) (2013) 559-565.
[34] T. Tunc, F. Usta, H. Budak, and M.Z. Sarikaya, On Gruss type inequalities utilizing generalized fractional integral operators, AIP Conf. Proc.1833 (2017) 020045.
[35] F. Usta and M.Z. Sarkaya, Explicit bounds on certain integral inequalities via conformable fractional calculus, Cogent Math. 4 (2017) 1277505.
[36] F. Usta, H. Budak, and M.Z. Sarkaya, On Chebychev type inequalities for fractional integral operators, AIP Conf. Proc. 1833 (2017) 020045.
[37] F. Usta and M.Z. Sarkaya, Some improvements of conformable fractional integral inequalities, Int. J. Anal. Appl. 14 (2017) 162-166.
[38] F. Usta and M.Z. Sarkaya, On generalization conformable fractional integral inequalities, Filomat 32(16) (2018) 5519-5526.
[39] G. Wang, P. Agarwal and M. Chand, Certain Gruss type inequalities involving the generalized fractional integral operator, J. Inequal. Appl. 2014 (2014) 147.
[40] A. Waheed, G. Farid, A. Ur Rehman and W. Ayub, k-fractional integral inequalities for harmonically convex functions via Caputo k-fractional derivatives, Bull. Math. Anal. Appl. 10 (2018) 55–67.
Volume 12, Issue 1
May 2021
Pages 541-554
  • Receive Date: 28 May 2018
  • Revise Date: 11 November 2019
  • Accept Date: 06 February 2020