New estimates of Gauss-Jacobi and trapezium type inequalities for strongly $(h_{1},h_{2})$-preinvex mappings via general fractional integrals

Document Type : Research Paper


1 Department of Mathematics, Faculty of Technical Science, University “Ismail Qemali”, 9400, Vlora, Albania

2 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, 210023, China

3 Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce, Turkey


In this paper, authors discover two interesting identities regarding Gauss--Jacobi and trapezium type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss--Jacobi type integral inequalities for a new class of functions called strongly $(h_{1},h_{2})$--preinvex of order $\sigma>0$ with modulus $\mu>0$ via general fractional integrals are established. Also, using the second lemma, some new estimates with respect to trapezium type integral inequalities for strongly $(h_{1},h_{2})$--preinvex functions of order $\sigma>0$ with modulus $\mu>0$ via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different real numbers and new approximation error estimates for the trapezoidal are provided as well. These results give us the generalizations of some previous known results. The ideas and techniques of this paper may stimulate further research in the fascinating field of inequalities.


[1] S.M. Aslani, M.R. Delavar and S.M. Vaezpour, Inequalities of Fejer type related to generalized convex functions with applications, Int. J. Anal. Appl. 16(1) (2018) 38–49.
[2] M.R. Delavar and M. De La Sen, Some generalizations of Hermite–Hadamard type inequalities, Springer Plus 5 (1661) (2016).
[3] S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett. 11(5) (1998) 91–95.
[4] T.S. Du, J.G. Liao and Y.J. Li, Properties and integral inequalities of Hadamard–Simpson type for the generalized (s, m)–preinvex functions, J. Nonlinear Sci. Appl. 9 (2016) 3112–3126.
[5] R. Hussain, A. Ali, G. Gulshan, A. Latif and M. Muddassar, Generalized coordinated integral inequalities for convex functions by way of k–fractional derivatives, Miskolc Math. Notes Submitted.
[6] S. Karamardian, The nonlinear complementarity problem with applications, Part 2, J. Optim. Theory Appl. 4 (1969) 167–181.
[7] A. Kashuri and R. Liko, Some new Hermite–Hadamard type inequalities and their applications, Stud. Sci. Math. Hung. 56(1) (2019) 103–142.
[8] A. Kashuri and R. Liko, Hermite–Hadamard type fractional integral inequalities for generalized (r; s, m, ϕ)– preinvex functions, Eur. J. Pure Appl. Math. 10(3) (2017) 495–505.
[9] A. Kashuri and R. Liko, Hermite–Hadamard type inequalities for generalized (s, m, ϕ)–preinvex functions via k–fractional integrals, Tbil. Math. J. 10(4) (2017) 73–82.
[10] A. Kashuri, R. Liko and S.S. Dragomir, Some new Gauss–Jacobi and Hermite–Hadamard type inequalities concerning (n+ 1)–differentiable generalized ((hp1, hq2); (η1, η2))–convex mappings, Tamkang J. Math. 49(4) (2018) 317–337.
[11] M. A. Khan, Y.-M. Chu, A. Kashuri and R. Liko, Hermite–Hadamard type fractional integral inequalities for MT(r;g,m,φ)–preinvex functions, J. Comput. Anal. Appl. 26(8) (2019) 1487–1503.
[12] M. A. Khan, Y.-M. Chu, A. Kashuri, R. Liko and G. Ali, Conformable fractional integrals versions of Hermite– Hadamard inequalities and their generalizations, J. Funct. Spaces 2018 (2018), Article ID 6928130, pp. 9.
[13] G.H. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optim. Theory Appl. 118 (2003) 67–80.
[14] W. Liu, New integral inequalities involving beta function via P–convexity, Miskolc Math. Notes 15(2) (2014) 585–591.
[15] W.J. Liu, Some Simpson type inequalities for h–convex and (α, m)–convex functions, J. Comput. Anal. Appl. 16(5) (2014) 1005–1012.
[16] C. Luo, T.S. Du, M.A. Khan, A. Kashuri and Y. Shen, Some k–fractional integrals inequalities through generalized λφm–MT–preinvexity, J. Comput. Anal. Appl. 27(4) (2019) 690–705.
[17] S. Mubeen and G.M. Habibullah, k–Fractional integrals and applications, Int. J. Contemp. Math. Sci. 7 (2012) 89–94.
[18] M.A. Noor, K.I. Noor, M.U. Awan and S. Khan, Hermite–Hadamard inequalities for s–Godunova–Levin preinvex functions, J. Adv. Math. Stud. 7(2) (2014) 12–19.
[19] M.E. Ozdemir, S.S. Dragomir and C. Yildiz, ¨ The Hadamard’s inequality for convex function via fractional integrals, Acta Math. Sci., Ser. B, Engl. Ed. 33(5) (2013) 153–164.
[20] M.E. Ozdemir, E. Set and M. Alomari, Integral inequalities via several kinds of convexity, Creat. Math. Inf. 20(1) (2011) 62–73.
[21] C. Peng, C. Zhou and T. S. Du, Riemann–Liouville fractional Simpson’s inequalities through generalized (m, h1, h2) preinvexity, Ital. J. Pure Appl. Math. 38 (2017) 345–367.
[22] B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Sov. Math. Dokl. 7 (1966) 72–75.
[23] S. Rashid, M.A. Latif, Z. Hammouch and Y.-M. Chu, Fractional integral inequalities for strongly h–preinvex functions for a kth order differentiable functions, Symmetry 11(1448) (2019) pp. 18.
[24] M.Z. Sarikaya and F. Ertugral, On the generalized Hermite–Hadamard inequalities, 2017 (2017) https://www.
[25] M.Z. Sarikaya and H. Yildirim, On generalization of the Riesz potential, Indian J. Math. Math. Sci. 3(2) (2007) 231–235.
[26] E. Set, M.A. Noor, M.U. Awan and A. Gozpinar, Generalized Hermite–Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl. 169 (2017) 1–10.
[27] D.D. Stancu, G. Coman and P. Blaga, Analiza numerica si teoria aproximarii, Cluj-Napoca: Presa Universitara Clujeana. 2 (2002).
[28] H. Wang, T.S. Du and Y. Zhang, k–fractional integral trapezium–like inequalities through (h, m)–convex and (α, m)–convex mappings, J. Inequal. Appl. 2017(311) (2017) pp. 20.
[29] T. Weir and B. Mond, Preinvex functions in multiple objective optimizations, J. Math. Anal. Appl. 136 (1988) 29–38.
[30] X.M. Zhang, Y.-M. Chu and X. H. Zhang, The Hermite–Hadamard type inequality of GA–convex functions and its applications, J. Inequal. Appl. 2010 (2010), Article ID 507560, pp. 11.
[31] Y. Zhang, T.S. Du, H. Wang, Y.J. Shen and A. Kashuri, Extensions of different type parameterized inequalities for generalized (m, h)–preinvex mappings via k–fractional integrals, J. Inequal. Appl. 2018(49) (2018) pp. 30.
Volume 12, Issue 1
May 2021
Pages 979-996
  • Receive Date: 03 February 2020
  • Accept Date: 01 October 2020