Some fractional weighted trapezoid type inequalities for preinvex functions

[1] S. Abbaszadeh and A. Ebadian, Nonlinear integrals and Hadamard-type inequalities, Soft Comput. 22(9) (2018) 2843–2849.

[2] S. Abbaszadeh and M. Eshaghi, A Hadamard-type inequality for fuzzy integrals based on r-convex functions, Soft Comput. 20(8) (2016) 3117–3124.

[3] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015) 57–66.

[4] S.M. Aslani, M.R. Delavar and S.M. Vaezpour, Inequalities of Fejer type related to generalized convex functions, Int. J. Anal. Appl. 16(1) (2018) 38–49.

[5] D. Baleanu, A. Kashuri, P.O. Mohammed and B. Meftah, General Raina fractional integral inequalities on coordinates of convex functions, Adv. Differ. Equ. 2021 (2021) 82.

[6] A. Barani, A.G. Ghazanfari and S.S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, J. Inequal. Appl. 2012 (2012) 247.

[7] M. Bezziou, Z. Dahmani and M.E. Kiri¸s, Applications of the (k, s, h)-Riemann-Liouville and (k, h)-Hadamard fractional operators on inequalities, Konuralp J. Math. 8(1) (2020) 197–206.

[8] P.S. Bullen, Handbook of Means and Their Inequalities, Revised from the 1988 original [P. S. Bullen, D. S. Mitrinovic and P. M. Vasi´c, Means and their inequalities, Reidel, Dordrecht; MR0947142]. Mathematics and its Applications, 560. Kluwer Academic Publishers Group, Dordrecht, 2003.

[9] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1(2) (2015) 1–13.

[10] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1) (2010) 51–58.

[11] Z. Dahmani, The Riemann-Liouville operator to generate some new inequalities, Int. J. Nonlinear Sci. 12(4) (2011) 452–455.

[12] Z. Dahmani, New classes of integral inequalities of fractional order, Matematiche (Catania) 69(1) (2014) 237–247.

[13] L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 2003(54) (2003) 3413–3442.

[14] M.R. Delavar and M. De La Sen, Some Hermite-Hadamard-Fej´er type integral inequalities for differentiable-convex functions with applications, J. Math. 2017 (2017).

[15] M.R. Delavar, S.M. Aslani and M. De La Sen, Hermite-Hadamard-Fejer inequality related to generalized convex functions via fractional integrals, J. Math. 2018 (2018).

[16] M.R. Delavar, New bounds for Hermite-Hadamard’s trapezoid and mid-point type inequalities via fractional integrals, Miskolc Math. Notes 20(2) (2019) 849–861.

[17] M.R. Delavar and S.S. Dragomir, Trapezoidal type inequalities related to h-convex functions with applications, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. (Esp.) 113(2) (2019) 1487–1498.

[18] S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11(5) (1998) 91–95.

[19] A.M.A. El-Sayed, S.Z. Rida and A.A.M. Arafa, Exact solutions of fractional-order biological population model, Commun. Theor. Phys. (Beijing) 52(6) (2009) 992–996.

[20] H.A. Fallahgoul, S.M. Focardi and F.J. Fabozzi, Fractional Calculus and Fractional Processes with Applications to Financial Economics, Theory and Application, Elsevier/Academic Press, London, 2017.

[21] J. Hadamard, Etude sur les proprietes des fonctions enti`eres et en particulier d’une fonction consideree par Riemann, J. Math. Pures Appl. 58 (1893) 171–215.

[22] A. Hadjian and M. R.Delavar, Trapezoid and mid-point type inequalities related to η-convex functions, J. Inequal. Spec. Funct. 8(3) (2017) 25–31.

[23] M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80(2) (1981) 545–550.

[24] C. Hermite, Sur deux limites d’une int´egrale d´efinie, Mathesis 3(1) (1883) 1–82.

[25] R. Hilfer, Applications of Fractional Calculus in Physics, Edited by R. Hilfer. World Scientific Publishing Co., Inc., River Edge, NJ, 2000.

[26] F. Jarad, E. U˘gurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Difference Equ. 2017 (2017) 247.

[27] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6(4) (2014) 1–15.

[28] A. Kashuri, B. Meftah and P.O. Mohammed, Some weighted Simpson type inequalities for differentiable s–convex functions and their applications, J. Frac. Calc. Nonlinear Sys. 1(1) (2021) 75–94.

[29] H. Kavurmaci, M. Avci and M. E. Ozdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, J. Inequal. Appl. 2011 (2011) 86.

[30] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65–70.

[31] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[32] B. Meftah and A. Souahi, Fractional Hermite-Hadamard type inequalities for functions whose derivatives are s-preinvex, Math.Sci. Appl. E-Notes. 7(2) (2019) 128–138.

[33] B. Meftah, M. Merad, N. Ouanas and A. Souahi, Some new Hermite-Hadamard type inequalities for functions whose n
th derivatives are convex, Acta Comment. Univ. Tartu. Math. 23(2) (2019) 163–178.

[34] B. Meftah and K. Mekalfa, Some weighted trapezoidal inequalities for differentiable log-convex functions, J. Interdiscip. Math. 23 (2020) 1–13.

[35] B. Meftah, M. Benssaad, W. Kaidouchi and S. Ghomrani, Conformable fractional Hermite-Hadamard type inequalities for product of two harmonic s-convex functions, Proc. Amer. Math. Soc. 149(4) (2021) 1495–1506.

[36] D.S. Mitrinovi´c, J.E. Peˇcari´c and A.M. Fink, Classical and new Inequalities in Analysis, Mathematics and its Applications (East European Series), 61. Kluwer Academic Publishers Group, Dordrecht, 1993.

[37] S.R. Mohan and S.K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189(3) (1995) 901–908.

[38] S. Mubeen and G.M. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci. 7(1-4) (2012) 89–94.
[39] M.A. Noor, Variational-like inequalities, Optim. 30(4) (1994) 323–330.

[40] M.A. Noor, Invex equilibrium problems, J. Math. Anal. Appl. 302(2) (2005) 463–475.

[41] M.A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory 2(2) (2007) 126–131.

[42] C.E.M. Pearce and J. Pecaric, Inequalities for differentiable mappings with application to special means and quadrature formula, Appl. Math. Lett. 13(2) (2000) 51–55.

[43] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.

[44] R. Pini, Invexity and generalized convexity, Optim. 22(4) (1991) 513–525.

[45] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives, Theory and applications. Edited and with a foreword by S. M. Nikol’skiı. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993.

[46] M.Z. Sarikaya, E. Set, H. Yaldiz and N. Ba¸sak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modell. 57(9-10) (2013) 2403–2407.

[47] 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.

[48] H. Vosoughian, S. Abbaszadeh and M. Oraki, Hadamard Integral Inequality for the Class of Harmonically (γ, η)-Convex Functions, Frontiers in Functional Equations and Analytic Inequalities, Springer, Cham, 2019.

[49] J. Wang, C. Zhu and Y. Zhou, New generalized Hermite-Hadamard type inequalities and applications to special means, J. Inequal. Appl. 2013 (2013) 325.

[50] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl. 136(1) (1988) 29–38.

[51] X.-M. Yang and D. Li, On properties of preinvex functions, J. Math. Anal. Appl. 256(1) (2001) 229–241.