Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-Türkiye.

2 Department of Mathematics, Faculty of Science, Bartın-Türkiye.

Abstract

We first construct new Hermite-Hadamard type inequalities which include generalized fractional integrals for convex functions by using an operator which generates some significant fractional integrals such as Riemann-Liouville fractional and the Hadamard fractional integrals. Afterwards, Trapezoid and Midpoint type results involving generalized fractional integrals for functions whose the derivatives in modulus and their certain powers are convex are established. We also recapture the previous results in the particular situations of the inequalities which are given in the earlier works.

Keywords

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[1] G.A. Anastassiou, General fractional Hermite--Hadamard inequalities using $m$-convexity and $(s,m)$-convexity, Front. Time Scal. Ineq., 1 (2016), pp. 237-255.
[2] M.A. Ardıc, A.O. Akdemir and E. Set, On New Integral Inequalities via Geometric-Arithmetic Convex Functions with Applications,     Sahand Commun. Math. Anal. 19(2) (2022), pp. 1-14.

[3] H. Budak and M.Z. Sarikaya, Hermite-Hadamard type inequalities for s-convex mappings via fractional integrals of a function with respect to another function, Fasc. Math., 27 (2016) pp.25-36, 2016.
[4] H. Budak, On Fejer type inequalities for convex mappings utilizing fractional integrals of a function with respect to another function,     Result. Math., 74(1) (2019), 29.
[5] H. Budak, New Hermite-Hadamard type inequalities for convex mappings utilizing generalized fractional integrals, Filomat, 33(8) (2019), pp. 2329-2344.
[6] H. Budak, H. Kara, R Kapucu, New midpoint type inequalities for generalized fractional integral, Comput. Methods Differ. Equ., 10(1) (2022), pp. 93-108.
[7] H. Budak, C.C. Bilişik, M.Z. Sarikaya, On Some New Extensions of Inequalities of Hermite-Hadamard Type for Generalized Fractional Integrals, Sahand Commun. Math. Anal., 19(2) (2022), pp. 65-79.
[8] S.I. Butt, M. Umar, S. Rashid, A. O. Akdemir and Y. M. Chu, New Hermite--Jensen--Mercer-type inequalities via $k$ -fractional integrals, Adv. Difference Equ., 1 (2020), pp. 1-24.
[9] B. Celik, M.E. Ozdemir, A.O. Akdemir and E. Set, Integral Inequalities for Some Convexity Classes via Atangana-Baleanu Integral Operators, TJOI, 5(2) (2021), 82-92.
[10] H. Chen and U.N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integrals,J. Math. Anal. Appl. 446 (2017), pp. 1274-1291
[11] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1 (2010), pp. 51-58.
[12] B. Daraby, . Generalizations of Some Inequalities for Sugino Integrals, Sahand Commun. Math. Anal., 19(3) (2022), pp. 141-168.
[13] J. Deng and J. Wang, Fractional Hermite-Hadamard inequalities for ($\alpha ,m$)-logarithmically convex functions. J. Inequal. Appl. 2013 (2013), art. 364.
[14] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monog., Vic. Univ., 2000.
[15] S.S. Dragomir, 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), pp. 91-95.
[16] S.S. Dragomir, Some inequalities of Hermite-Hadamard type for symmetrized convex functions and Riemann-Liouville fractional integrals, RGMIA Res. Rep. Coll., 20 (2017), 15.

[17] S. Erden, Weighted inequalities involving conformable integrals and its applications for random variable and numerical integration, Filomat 34(8) 2020, pp. 2785-2796.
[18] A. Ekinci, M. Ozdemir, Some new integral inequalities via Riemann-Liouville integral operators, Appl. Comput. Math., 18(3) (2019), pp. 288-295.
[19] G. Farid, A. ur Rehman and M. Zahra, On Hadamard type inequalities for $k$-fractional integrals, Konuralp J. Math., 4(2) 2016, 79-86.
[20] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien (1997), 223-276.
[21] J. Hadamard, Etude sur les proprietes des fonctions entieres en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl. 58 (1893), pp. 171-215.
[22] M. Iqbal, M.I. Bhatti and K. Nazeer, Generalization of inequalities analogous to Hermite-Hadamard inequality via fractional integrals, Bull. Korean Math. Soc., 52(3) (2015), pp. 707-716.
[23] I. Iscan and S. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Compt., 238 (2014), pp. 237-244.
[24] I. Iscan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals, Stud. Univ. Babeș-Bolyai, Math., 60(3) (2015), pp. 355-366
[25] M. Jleli and B. Samet, On Hermite-Hadamard type inequalities via fractional integrals of a function with respect to another function, J. Nonlinear Sci. Appl., 9 (2016), pp. 1252-1260.
[26] U.N. Katugampola, New approach to a generalized fractional integrals, Appl. Math. Comput., 218(4) (2011), pp. 860-865.
[27] M.A. Khan, T.U. Khan, Parameterized Hermite-Hadamard Type Inequalities For Fractional Integrals, TJOI, 1(1), 2017, 26–37.
[28] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
[29] U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147(5) (2004), pp. 137-146.
[30] A.R. Khan, H. Nasir and S.S. Shirazi, Weighted Cebysev Type Inequalities for Double Integrals and Application, Sahand Commun. Math. Anal., 18(4) (2021), pp. 59-72.
[31] M. Kunt, İ. İşcan, Fractional Hermite–Hadamard–Fejer type inequalities for GA-convex functions, TJOI, 2(1), 1–20, 2018.
[32] S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993, pp.2.
[33] S. Mubeen, S. Iqbal and M. Tomar, On Hermite-Hadamard type inequalities via fractional integrals of a function with respect to another function and $k$-parameter, J. Inequal. Math. Appl., 1 (2016), pp. 1-9.

[34] M.A. Noor and M.U. Awan, Some integral inequalities for two kinds of convexities via fractional integrals, TJMM, 5(2) (2013), pp. 129-136.
[35] M.E. Ozdemir, New Refinements of Hadamard Integral inequlaity via k-Fractional Integrals for p-convex function, Turkish J. Science, 6(1) (2021), pp. 1-5.
[36] M.E. Ozdemir, M. Avci-Ardinc and H. Kavurmaci-Onalan, Hermite-Hadamard type inequalities fors-convex ands-concave functions via fractional integrals, Turkish J. Science, 1 (2016), pp. 28-40.
[37] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992.
[38] I. Podlubni, Fractional Differential Equations, Academic Press, San Diego, 1999.
[39] S. Peng, W. Wei and J-R. Wang, On the Hermite-Hadamard inequalities for convex functions via Hadamard fractional integrals, Facta Univ., Ser. Math. Inf., 29(1) (2014), pp. 55-75.
[40] M.Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 7(2) (2016), pp. 1049-1059.
[41] M.Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite -Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57 (2013), pp. 2403-2407.
[42] M.Z. Sarikaya and H. Budak, Generalized Hermite-Hadamard type integral inequalities for fractional integrals, Filomat, 30(5) (2016), pp. 1315-1326.
[43] M.Z. Sarikaya, A. Akkurt , H. Budak, M.E. Yildirim and H. Yildirim, Hermite-hadamard's inequalities for conformable fractional integrals. RGMIA Res. Rep. Col., 19(83), (2016).
[44] E. Set, M.Z. Sarikaya, M.E. Ozdemir and H. Yildirim, The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results, JAMSI, 10(2) (2014), pp. 69-83.
[45] E. Set, M.Z. Sarikaya, M.E. Ozdemir and H. Yildirim, The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results, JAMSI, 10(2) (2014).
[46] E. Set, J. Choi and B. Celik, New Hermite-Hadamard type inequalities for product of different convex functions involving certain fractional integral operators, J. Math. and Comp. Sci., 18(1) (2018), pp. 29-36
[47] E. Set, A. Gozpnar, A. Ekinci, Hermite-Hadamard type inequalities via confortable fractional integrals, Acta Math. Univ. Comen., 86 (2017), art. 309320.
[48] J. Wang, X. Li, M. Feckan, Y. Zhou, Hermite--Hadamard-type inequalities for Riemann--Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92(11) (2012), pp. 2241--2253.
[49] J.R. Wang, X. Li, C. Zhu, Refinements of Hermite-Hadamard type inequalities involving fractional integrals, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), pp. 655-666.
[50] J.R. Wang, C. Zhu, Y. Zhou, New generalized Hermite--Hadamard type inequalities and applications to special means, J. Inequal. Appl. 2013 (2013), art. 325.
[51] Y. Zhang and J. Wang, On some new Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, J. Inequal. Appl. 2013 (2013), art. 220.
[52] Z. Zhang, W. Wei, J. Wang, Generalization of Hermite-Hadamard inequalities involving Hadamard fractional integrals, Filomat, 29(7) (2015), pp. 1515-1524.
[53] Z. Zhang, J.R. Wang and J.H. Deng, Applying GG-convex function to Hermite-Hadamard inequalities involving Hadamard fractional integrals, Int. J. Math. Comput. Sci., 2014 (2014), art. 136035.