Opial-type Inequalities for Generalized Integral Operators With Special Kernels in Fractional Calculus

Muhammad Samraiz, Muhammad Arslan Afzal, Sajid Iqbal, Artion Kashuri

Abstract


In this article, we originate some new Opial-type inequalities on fractional calculus involving generalized Riemann-Liouville fractional integral, the Riemann-Liouville \(k\)-fractional integral, the \((k,r)\) fractional integral of the Riemann-type and the generalized fractional integral operator involving Hypergeometric function in its kernel. As special case of our general results we obtain the results of Farid et al. [7].


Keywords


Opial-type inequalities; Generalized Riemann-Liouville fractional integral operator; Riemann-Liouville \(k\)-fractional integral; \((k,r)\) fractional integral of the Riemann-type

Full Text:

PDF

References


R.P. Agarwal and P.Y.H. Pang, Opial inequalities with applications in differential and difference equations, Kluwer Acad. Publ., Dordrecht (1995).

R.P. Agarwal and P.Y.H. Pang, Sharp Opial-type inequalities involving higher order derivatives of two functions, Math. Nachr. 174 (1995), 5 – 20.

H. Alzer, An Opial-type inequality involving higher-order derivatives ot two functions, Appl. Math. Lett. 10(4) (1997), 123 – 128.

W.S. Cheung, Some new Opial-type inequalities, Mathematika 37(1) (1990), 136 – 142.

L. Curiel and L. Galué, A generalization of the integral operators involving the Gauss hypergeometric function, Revista Técnica de la Facultad de Ingenieria Universidad del Zulia 19(1) (1996), 17 – 22.

K.M. Das, An inequality similar to Opial’s inequality, Proc. Amer. Math. Sot. 22 (1969), 258 – 261.

G. Farid, J. Peˇcari´c and Ž. Tomovski, Generalized Opial-type inequalities for differential and integral operator with special kernel in fractional calculus, J. Math. Ineq. 10(4) (2016), 1019 – 1040.

S. Iqbal, J. Peˇcari´c and M. Samraiz, Opial type inequalities for two functions with general kernels and applications, J. Math. Ineq. 8(4) (2014), 757 – 775.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, New York — London (2006).

D.S. Mitrinovic, J. Pecaric and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publisher (1993).

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 inequality for generalized Riemann-liouville k-fractionl integral, J. Inequal. Appl. Springer 109 (2016).

Z. Opial, it Sur une inegalite, Ann. Polon. Math. 8 (1960), 29 – 32.

M. Sarikaya, Z. Dahmani, Z. Kiris and M.E. Ahmad, (k, s)-Riemann-Liouville fractional Integral and applications, Hact. J. Math. Stat. 45(1) (2016), 77 – 89.




DOI: http://dx.doi.org/10.26713%2Fcma.v9i3.831

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905