Some different type integral inequalities concerning twice differentiable generalized relative semi-(r; m; h)-preinvex mappings

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Abstract

In this article, we first present some integral inequalities for Gauss-Jacobi type quadrature formula involving generalized relative semi-(r; m; h)-preinvex mappings. And then, a new identity concerning twice differentiable mappings defined on m-invex set is derived. By using the notion of generalized relative semi-(r; m; h)-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard, Ostrowski and Simpson type inequalities via fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article.

References

  • [1] T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 60, (2005), 1473-1484.

  • [2] F. Chen, A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals, Ital. J. Pure Appl. Math., 33, (2014), 299-306.

  • [3] F. X. Chen and S. H. Wu, Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, J. Nonlinear Sci. Appl., 9(2), (2016), 705-716.

  • [4] Y. M. Chu, M. A. Khan, T. U. Khan and T. Ali, Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9(5), (2016), 4305-4316.

  • [5] Y. M. Chu, G. D. Wang and X. H. Zhang, Schur convexity and Hadamard's inequality, Math. Inequal. Appl., 13(4), (2010), 725-731.

  • [6] S. S. Dragomir, J. Pečarić and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21, (1995), 335-341.

  • [7] 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.

  • [8] C. Fulga and V. Preda, Nonlinear programming with '-preinvex and local '-preinvex functions, Eur. J. Oper. Res., 192, (2009), 737-743.

  • [9] A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s; m; ')- preinvex functions, Aust. J. Math. Anal. Appl., 13(1), (2016), Article 16, 1-11.

  • [10] A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (g; s; m; ')-preinvex functions, Extr. Math., 32(1), (2017), 105-123.

  • [11] A. Kashuri and R. Liko, Generalizations of Hermite-Hadamard and Ostrowski type inequalities for MTm-preinvex functions, Proyecciones, 36(1), (2017), 45-80.

  • [12] A. Kashuri and R. Liko, Hermite-Hadamard type inequalities for MTm-preinvex functions, Fasc. Math., 58(58), (2017), 77-96.

  • [13] A. Kashuri and R. Liko, Hermite-Hadamard type fractional integral inequalities for generalized (r; s; m; ')-preinvex functions, Eur. J. Pure Appl. Math., 10(3), (2017), 495-505.

  • [14] A. Kashuri and R. Liko, Hermite-Hadamard type fractional integral inequalities for twice differentiable generalized (s; m; ')-preinvex functions, Konuralp J. Math., 5(2), (2017), 228-238.

  • [15] A. Kashuri and R. Liko, Some new Ostrowski type fractional integral inequalities for generalized (r; g; s; m; ')-preinvex functions via Caputo k-fractional derivatives, Int. J. Nonlinear Anal. Appl., 8(2), (2017), 109-124.

  • [16] A. Kashuri and R. Liko, Hermite-Hadamard type inequalities for generalized (s; m; ')-preinvex functions via k-fractional integrals, Tbil. Math. J., 10(4), (2017), 73-82.

  • [17] A. Kashuri and R. Liko, Ostrowski type fractional integral operators for generalized (r; s; m; ')- preinvex functions, Appl. Appl. Math., 12(2), (2017), 1017-1035.

  • [18] A. Kashuri and R. Liko, Hermite-Hadamard type fractional integral inequalities for MT(m;')- preinvex functions, Stud. Univ. Babeş-Bolyai, Math., 62(4), (2017), 439-450.

  • [19] A. Kashuri and R. Liko, Hermite-Hadamard type fractional integral inequalities for twice differentiable generalized beta-preinvex functions, J. Fract. Calc. Appl., 9(1), (2018), 241-252.

  • [20] A. Kashuri, R. Liko and T. S. Du, Ostrowski type fractional integral operators for generalized beta (r; g)-preinvex functions, Khayyam J. Math., 4(1), (2018), 39-58.

  • [21] H. Kavurmaci, M. Avci and M. E. Özdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, arXiv:1006.1593v1 [math. CA], (2010), 1-10.

  • [22] H. Kavurmaci, M. Avci and M. E. Özdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, J. Inequal. Appl., 86, (2011), 1{11.

  • [23] M. Adil Khan, Y. Khurshid and T. Ali, Hermite-Hadamard inequality for fractional integrals via η-convex functions, Acta Math. Univ. Comenianae, 79(1), (2017), 153-164.

  • [24] M. Adil Khan, Y. Khurshid, T. Ali and N. Rehman, Inequalities for three times differentiable functions, J. Math., Punjab Univ., 48(2), (2016), 35-48.

  • [25] M. Adil Khan, Y. M. Chu, A. Kashuri, R. Liko and G. Ali, New Hermite-Hadamard inequalities for conformable fractional integrals, J. Funct. Spaces, (In press).

  • [26] W. Liu, New integral inequalities involving beta function via P-convexity, Miskolc Math. Notes, 15(2), (2014), 585-591.

  • [27] W. J. Liu, Some Simpson type inequalities for h-convex and (;m)-convex functions, J. Comput. Anal. Appl., 16(5), (2014), 1005-1012.

  • [28] W. Liu, W. Wen and J. Park, Ostrowski type fractional integral inequalities for MT-convex functions, Miskolc Math. Notes, 16(1), (2015), 249-256.

  • [29] W. Liu, W. Wen and J. Park, Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl., 9, (2016), 766-777.

  • [30] C. Luo, T. S. Du, M. A. Khan, A. Kashuri and Y. Shen, Some k-fractional integrals inequalities through generalized '-m-MT-preinvexity, J. Comput. Anal. Appl., 240, (2019), (Accepted paper).

  • [31] M. Mat loka, Inequalities for h-preinvex functions, Appl. Math. Comput., 234, (2014), 52-57.

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

  • [33] M. A. Noor, K. I. Noor, M. U. Awan and S. Khan, Hermite-Hadamard type inequalities for differentiable h'-preinvex functions, Arab. J. Math., 4, (2015), 63-76.

  • [34] O. Omotoyinbo and A. Mogbodemu, Some new Hermite-Hadamard integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1(1), (2014), 1-12.

  • [35] M. E. Özdemir, E. Set and M. Alomari, Integral inequalities via several kinds of convexity, Creat. Math. Inform., 20(1), (2011), 62-73.

  • [36] B. G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll., 6, (2003).

  • [37] 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.

  • [38] R. Pini, Invexity and generalized convexity, Optimization, 22, (1991), 513-525.

  • [39] F. Qi and B. Y. Xi, Some integral inequalities of Simpson type for GA ε-convex functions, Georgian Math. J., 20(5), (2013), 775-788.

  • [40] E. Set, M. A. Noor, M. U. Awan and A. Gözpinar, Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., 169, (2017), 1-10.

  • [41] H. N. Shi, Two Schur-convex functions related to Hadamard-type integral inequalities, Publ. Math. Debrecen, 78(2), (2011), 393-403.

  • [42] D. D. Stancu, G. Coman and P. Blaga, Analiză numerică şi teoria aproximării, Cluj-Napoca: Presa Universitară Clujeană., 2, (2002).

  • [43] M. Tunć, Ostrowski type inequalities for functions whose derivatives are MT-convex, J. Com- put. Anal. Appl., 17(4), (2014), 691-696.

  • [44] M. Tunć, E. Göv and Ü. Şanal, On tgs-convex function and their inequalities, Facta Univ. Ser. Math. Inform., 30(5), (2015), 679-691.

  • [45] S. Varoşanec, On h-convexity, J. Math. Anal. Appl., 326(1), (2007), 303-311.

  • [46] Y.Wang, M. M. Zheng and F. Qi, Integral inequalities of Hermite-Hadamard type for functions whose derivatives are -preinvex, J. Inequal. Appl., 97, (2014), 1-10.

  • [47] J. Wang, C. Zhu and Y. Zhou, New generalized Hermite-Hadamard type inequalities and applications to special means, J. Inequal. Appl., 325, (2013), 1-15.

  • [48] E. A. Youness, E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl., 102, (1999), 439-450.

  • [49] Y. Zhang and J. Wang, On some new Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, J. Inequal. Appl., 220, (2013), 1-27.

  • [50] X. M. Zhang, Y. M. Chu and X. H. Zhang, The Hermite-Hadamard type inequality of GAconvex functions and its applications, J. Inequal. Appl., (2010), Article ID 507560, 11 pages.