Hermite-Hadamard type fractional integral inequalities for generalized (s,m,φ)-preinvex functions

Year 2017, Volume: 5 Issue: 3, 97 - 106, 01.07.2017

Artion Kashuri , Rozana Liko

Abstract

In the present paper, by using new identity for fractional integrals some new estimates on generalizations of Hermite-Hadamard type inequalities for the class of generalized (s,m,φ)-preinvex functions via Riemann-Liouville fractional integral are established. These results not only extend the results appeared in the literature (see [2]), but also provide new estimates on these types. At the end, some applications to special means are given.

Keywords

Hermite-Hadamard inequality, Hölder’s inequality, power mean inequality, Riemann-Liouville fractional integral, s-convex in the second sense, m-invex

References

• A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s,m,φ)-preinvex functions, Aust. J. Math. Anal. Appl., 13, (1) (2016), Article 16, 1-11.
• V. M. Mihai, Some Hermite-Hadamard type inequalities via Riemann-Liouville fractional calculus, Tamkang J. Math., 44, (4) (2013), 411-416.
• 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.
• S. S. Dragomir, J. Pečarić and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21, (1995), 335-341.
• H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48, (1994), 100-111.
• T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 60, (2005), 1473-1484.
• X. M. Yang, X. Q. Yang and K. L. Teo, Generalized invexity and generalized invariant monotonicity, J. Optim. Theory Appl., 117, (2003), 607-625.
• R. Pini, Invexity and generalized convexity, Optimization., 22, (1991), 513-525.
• 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.
• Y. M. Chu, G. D. Wang and X. H. Zhang, Schur convexity and Hadamard’s inequality, Math. Inequal. Appl., 13, (4) (2010), 725-731.
• 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), Article ID 507560, 11 pages.
• 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.
• 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.
• M. Adil Khan, Y. Khurshid and T. Ali, Hermite-Hadamard inequality for fractional integrals via α-convex functions, Acta Math. Univ. Comenianae, 79, (1) (2017), 153-164.
• Y. M. Chu, M. Adil Khan, T. Ullah Khan and T. Ali, Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9, (2016), 4305-4316.
• H. N. Shi, Two Schur-convex functions related to Hadamard-type integral inequalities, Publ. Math. Debrecen, 78, (2) (2011), 393-403.
• 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.
• 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.
• W. Liu, W. Wen and J. Park, Ostrowski type fractional integral inequalities for MT-convex functions, Miskolc Mathematical Notes., 16, (1) (2015), 249-256.
• Y. M. Chu, G. D. Wang and X. H. Zhang, Schur convexity and Hadamard’s inequality, Math. Inequal. Appl., 13, (4) (2010), 725-731.
• 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), Article ID 507560, 11 pages.
• 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.
• M. Tunç, Ostrowski type inequalities for functions whose derivatives are MT-convex, J. Comput. Anal. Appl., 17, (4) (2014), 691-696.
• W. J. Liu, Some Simpson type inequalities for h-convex and (α,m)-convex functions, J. Comput. Anal. Appl., 16, (5) (2014), 1005-1012.
• 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.
• P. S. Bullen, Handbook of Means and Their Inequalities, Kluwer Academic Publishers, Dordrecht, (2003).