# On Opial-type inequalities via a new generalized integral operator

## Main Article Content

## Abstract

Opial inequality and its consequences are useful in establishing existence and uniqueness of solutions of initial and boundary value problems for differential and difference equations. In this paper we analyze Opial-type inequalities for convex functions. We have studied different versions of these inequalities for a generalized integral operator. Further difference of Opial-type inequalities are utilized to obtain generalized mean value theorems, which further produce various interesting derivations for fractional and conformable integral operators.

## Article Details

## References

[1] M. Andric , A. Barbir, G. Farid, J. PecVaric , More on certain Opial-type inequality for fractional derivatives and exponentially convex functions, Nonlinear Funct. Anal. Appl., 19 (4) (2014), 563-583. Google Scholar

[2] M. Andric , J. PecVaric , I. Peric , Improvement of composition rule for the Canavati fractional derivatives and applications to Opial-type inequalities, Dynam. Systems. Appl., 20 (2011), 383- 394. Google Scholar

[3] M. Andric , J. PecVaric , I. Peric , Composition identities for the Caputo fractional derivatives and application to Opial type inequalities, Math. Inequal. Appl., 16 (3)(2013), 657-670. Google Scholar

[4] M. Andric , A. Barbir, G. Farid and J. PecVaric , Opial-type inequality due to Agarwal-Pang and fractional differential inequalities, Integral Transforms Spec. Funct., 25 (4) (2013), 324-335. Google Scholar

[5] G. A. Anastassiou, General fractional Opial type inequalities, Acta Appl. Math., 54 (1998), 303–317. Google Scholar

[6] G. A. Anastassiou, Opial type inequalities involving fractional derivatives of functions, Nonlinear Stud., 6 (2)(1999), 207–230. Google Scholar

[7] D. W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math. 30 (1969), 367– 383. Google Scholar

[8] H. Chen, U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fej er type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274-1291. Google Scholar

[9] D. W. Boyd, J. S. W. Wong, An extension of Opial’s inequality, J. Math. Anal. Appl., 19 (1967), 100–102. Google Scholar

[10] G. Farid, J. PecVaric , Opial type integral inequalities for fractional derivatives II, Fractional Differ. Calc., 2 (2) (2012), 139-155. Google Scholar

[11] G. Farid, A. U. Rehman, S. Ullah, A. Nosheen, M. Waseem, Y. Mehboob, Opial-type inequalities for convex functions and associated results in fractional calculus, Adv. Difference Equ., 2019 (2019), 2019:152. Google Scholar

[12] G. Farid, J. PecVaric , Opial type integral inequalities for fractional derivatives, Fractional Differ. Calc., 2 (1) (2012), 31-54. Google Scholar

[13] A. Ur. Rehman, G. Farid, J. Pecaric Mean value theorem associated to the differences of recent Opial-type inequalities and their fractional versions, Fractional Differ. Calc., 10 (2) (2020), 213–224. Google Scholar

[14] G. Farid, Existence of an integral operator and its consequences in fractional and conformable integrals, Open J. Math. Sci. 3 (2019), 210–216. Google Scholar

[15] S. Habib, S. Mubeen, M. N. Naeem, Chebyshev type integral inequalities for generalized k- fractional conformable integrals, J. Inequal. Spec. Funct., 9 (4) (2018), 53–65. Google Scholar

[16] F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), 247pp. Google Scholar

[17] Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, S. M. Kang, Generalized Riemann-Liouville k- Fractional Integrals Associated With Ostrowski Type Inequalities and Error Bounds of Hadamard Inequalities, IEEE Access, 6 (2018), 64946–64953. Google Scholar

[18] T. U. Khan, M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389. Google Scholar

[19] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and application of fractional differential equations, North-Holland Mathematics studies, 204, Elsevier, New York-London, 2006. Google Scholar

[20] D. S. Mitrinovic, J. E. Pecaric, Generalization of two inequalities of Godunova and Levin, Bull. Polish Acad. Sci. Math., 36 (1988), 645–648. Google Scholar

[21] S. Mubeen, G. M. Habibullah, k-fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), 89–94. Google Scholar

[22] Z. Opial, Sur une ine galite , Ann. Polon. Math., 8 (1960), 29-32. Google Scholar

[23] B. G. Pachpatte, On Opial-type integral inequalities, J. Math. Anal. Appl., 120 (1986), 547–556. Google Scholar

[24] B. G. Pachpatte, A note on generalization Opial type inequalities, Tamkang J. Math., 24 (1993), 229–235. Google Scholar

[25] J. PecVaric , F. Proschan, Y. C. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Inc., (1992). Google Scholar

[26] M. Z. Sarikaya, M. Dahmani, M. E. Kiris, F. Ahmad, (k, s)-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45 (1) (2016), 77–89. doi:10.15672/HJMS.20164512484. Google Scholar