Hermite–Hadamard Type Integral Inequalities Involving k–Riemann–Liouville Fractional Integrals and Their Applications

Authors: Artion Kashuri, Rozana Liko

Abstract:

In this paper, some generalization integral inequalities of Hermite–Hadamard type for functions whose derivatives are s–convex in modulus are given by using k–fractional integrals. Some applications to special means are obtained as well. Some known versions are recovered as special cases from our results. We note that our inequalities can be viewed as new refinements of the previous results. Finally, our results have a deep connection with various fractional integral operators and interested readers can find new interesting results using our idea and technique as well.

Keywords: Hermite–Hadamard’s inequalities, k–Riemann–Liouville fractional integral, H¨older’s inequality, Special means.

References:

[1] H. Hudzik and L. Maligranda, Some remarks on s–convex functions, Aequationes Math., 48 (1994), 100-111.
[2] S. S. Dragomir and S. Fitzpatrik, The Hadamard’s inequality for s–convex functions in the second sense, Demonstratio Math., 32(4) (1999), 687-696.
[3] S. ´’Ozcan and ˙I. ˙Is¸can, Some new Hermite–Hadamard type inequalities for s–convex functions and their applications, J. Inequal. Appl., 2019(201) (2019), 1–11.
[4] M. Muddassar, M. I. Bhatti and M. Iqbal, Some new s–Hermite-Hadamard type inequalities for differentiable functions and their applications, Proc. Pak. Acad. Sci., 49(1) (2012), 9–17.
[5] S. Rashid, M. A. Noor, K. I. Noor and A. O. Akdemir, Some New Generalizations for Exponentially s-Convex Functions and Inequalities via Fractional Operators, Int. J. Sci. Innovation Tech., 1(1) (2014), 1–12.
[6] M. J. Cloud, B. C. Drachman and L. Lebedev, Inequalities, Springer, Cham, Second edition, 2014.
[7] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies; Elsevier Sci. B.V.: Amsterdam, The Netherlands, 2006; Vol. 204.
[8] D. Baleanu, P. O. Mohammed, M. J. Vivas-Cortez, and Y.-R. Oliveros, Some modifications in conformable fractional integral inequalities, Adv. Differ. Equ., 2020(374), (2020).
[9] T. Abdeljawad, P. O. Mohammed and A. Kashuri, New Modified Conformable Fractional Integral Inequalities of Hermite–Hadamard Type with Applications, J. Funct. Spaces, 2020 Article 4352357, (2020).
[10] P. O. Mohammed, Some integral inequalities of fractional quantum type, Malaya J. Mat., 4(1) (2016), 93–99.
[11] P. O. Mohammed and T. Abdeljawad, Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel, Adv. Differ. Equ., 2020(363) (2020).
[12] P. O. Mohammed, New integral inequalities for preinvex functions via generalized beta function, J. Interdiscip. Math., 22(4) (2019), 539–549.
[13] M. Chudziak and M. ´Iołdak, Hermite–Hadamard and Fej´er Inequalities for Co-Ordinated (F,G)–Convex Functions on a Rectangle, Symmetry, 12(13) (2020).
[14] P. O. Mohammed and M. Z. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math., 372 Article 112740, (2020).
[15] P. O. Mohammed and I. Brevik, A New Version of the Hermite–Hadamard Inequality for Riemann–Liouville Fractional Integrals, Symmetry, 12(610) (2020).
[16] T.-Y. Zhang, A.-P. Ji and F. Qi, On Integral Inequalities of Hermite–Hadamard Type for s–Geometrically Convex Functions, Abst. Appl. Anal., 2012 Article 560586, (2012).
[17] T.-Y. Zhang, A.-P. Ji and F. Qi, Some inequalities of Hermite–Hadamard type for GA–convex functions with applications to means, Le Mat., 68 (2013), 229–239.
[18] P. O. Mohammed, Some new Hermite–Hadamard type inequalities for MT–convex functions on differentiable coordinates, J. King Saud Univ. Sci., 30 (2018), 258–262.
[19] J. Han, P. O. Mohammed and H. Zeng, Generalized fractional integral inequalities of Hermite–Hadamard-type for a convex function, Open Math., 18 (2020), 794–806.
[20] D.-P. Shi, B.-Y. Xi and F. Qi, Hermite-Hadamard type inequalities for Riemann–Liouville fractional integrals of (α,m)–convex functions, Fract. Differ. Calc., 4 (2014), 31–43.
[21] F. Qi, P. O. Mohammed, J. C. Yao and Y. H. Yao, Generalized fractional integral inequalities of Hermite–Hadamard type for (α,m)–convex functions, J. Inequal. Appl., 2019(135) (2019).
[22] P. O. Mohammed and M. Z. Sarikaya, Hermite–Hadamard type inequalities for F–convex function involving fractional integrals, J. Inequal. Appl., 2018(359) (2018).
[23] D. Baleanu, P. O. Mohammed and S. Zeng, Inequalities of trapezoidal type involving generalized fractional integrals, Alex. Eng. J., (2020).
[24] P. O. Mohammed, T. Abdeljawad, S. Zeng and A. Kashuri, Fractional Hermite–Hadamard Integral Inequalities for a New Class of Convex Functions, Symmetry, 12(1485) (2020).
[25] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs; Victoria University: Footscray, Australia, 2000.
[26] G. Farid, X. Qiang and S. B. Akbar, Generalized fractional integrals inequalities for exponentially (s,m)–convex functions, J. Inequal. Appl., (2020).
[27] A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115(2) (2002), 260–288.
[28] A. Iqbal, M. A. Khan, S. Ullah and Y.-M. Chu, Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Spaces, 2020, Article ID 9845407 (2020).
[29] Y. Khurshid, M. A. Khan and Y.-M. Chu, Conformable integral inequalities of the Hermite–Hadamard type in terms of GG– and GA–convexities, J. Funct. Spaces, 2019, Article ID 6926107 (2019).
[30] S. Mubeen and G. M. Habibullah, k–Fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), 89–94.
[31] M. Z. Sarikaya and H. Yaldiz, On generalized Hermite–Hadamard type integral inequalities involving Riemann–Liouville fractional integrals, Nihonkai Math. J., 25 (2014), 93–104.