# Unified integral operator inequalities via convex composition of two functions

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## Abstract

In this paper we have established inequalities for a unified integral operator by using convexity of composition of two functions. The obtained results are directly connected to bounds of various fractional and conformable integral operators which are already known in literature. A generalized Hadamard integral inequality is obtained which further leads to its various versions for associated fractional integrals. Further, some implicated results are discussed.

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## References

[1] M. Andri c, G. Farid and J. PeVcari c, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal. 21 (5) (2018), 1377-1395. Google Scholar

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

[3] C. Cesarano, Generalized special functions in the description of fractional diffusive equations, CAIM 10 (2019), 31–40. Google Scholar

[4] S. S. Dragomir, Inequalities of Jensens type for generalized k-g-fractional integrals of functions for which the composite f ◦ g−1 is convex, Fract. Differ. Calc. 8 (1) (2018), 127–150. Google Scholar

[5] R. Dubey, N. L. Mishra and C. Cesarano, Multi objective fractional symmetric duality in math- ematical programming with (C,Gf)-invexity assumptions, AXIOMS 8 (2019), 1–11. Google Scholar

[6] G. Farid, Some Riemann-Liouville fractional integrals inequalities for convex function, J. Anal. 27 (4) (2019), 1095–1102. Google Scholar

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

[8] Z. He, G. Farid, A. U. Haq and K. Mahreen, Bounds of a unified integral operator for (s, m)- convex functions and their consequences, AIMS Mathematics 5 (6) (2020), 5510–5520. Google Scholar

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

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

[11] S. Kermausuor, Simpson’s type inequalities for strongly (s,m)-convex functions in the second sense and applications, Open J. Math. Sci. 3 (1) (2019), 74–83. Google Scholar

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

[13] A. A. Kilba ̧s, H. M. Srivastava and J. J Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006. Google Scholar

[14] Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah and 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

[15] Y. C. Kwun, G. Farid, S. Ullah, W. Nazeer, K. Mahreen and S. M. Kang, Inequalities for a unified integral operator and associated results in fractional calculus, IEEE Access 7 (2019), 126283–126292. Google Scholar

[16] S. Mubeen and A. Rehman, A note on k-Gamma function and Pochhammer k-symbol, J. Math. Sci. 6 (2) (2014), 93–107. Google Scholar

[17] S. Mubeen and G. M. Habibullah, k-fractional integrals and applications, Int. J. Contemp. Math. 7 (2) (2012), 89–94. Google Scholar

[18] T. R. Parbhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15. Google Scholar

[19] S. I. Butt, M. Nadeem and G. Farid,On Caputo fractional derivatives via exponential convex functions, Turkish Journal of Science 5 (2) (2020), 140–146. Google Scholar

[20] G. Rahman, D. Baleanu, M. A. Qurashi, S. D. Purohit, S. Mubeen and M. Arshad, The extended Mittag-Leffeler function via fractional calculus, J. Nonlinear Sci. Appl. 10 (2013), 4244–4253. Google Scholar

[21] G. Rahman, A. Khan, T. Abdeljwad and K. S. Nisar, The Minkowski inequalities via generalized proportional fractional integral operators, Adv. Difference equ. (2019) 2019:287. Google Scholar

[22] S. Rashid, T. Abdeljawad, F. Jarad and M. A. Noor, Some Estimates for Generalized Riemann- Liouville fractional integrals of exponentially convex functions and their applications, Mathematics 7 (9) (2019), 807. Google Scholar

[23] A. W. Roberts and D. E. Varberg, Convex functions, Acadamic press New York and London 1993. Google Scholar

[24] T. O. Salim, and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with integral calculus, J. Fract. Calc. Appl. 3 (5) (2012), 1–13. Google Scholar

[25] M. Z. Sarikaya and N. Alp, On Hermite-Hadamard-Fej er type integral inequalities for generalized convex functions via local fractional integrals, Open J. Math. Sci. 3 (1) (2019), 273-284. Google Scholar

[26] M. Z. Sarıkaya, M. Dahmani, M. E. Kiri ̧s and 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

[27] E. Set, M. A. Noor and M. U. Awan, A. G ̈ozpinar, Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., (2017), 2017:169. Google Scholar

[28] E. Set, J. Choi and B. C ̧ elik, Certain Hermite-Hadamard type inequalities involving generalized fractional integral operators, RACSAM 112 (4) (2018), 1539–1547. Google Scholar

[29] H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (1) (2009), 198– 210. Google Scholar

[30] T. Tunc ̧, H. Budak, F. Usta and M. Z. Sarıkaya, On new generalized fractional integral operators and related fractional inequalities, https://www.researchgate.net/publication/313650587. Google Scholar