×
Kashuri, Artion; Liko, Rozana

Hermite-Hadamard type inequalities for generalized \((s,m,\varphi)\)-preinvex functions via \(k\)-fractional integrals. (English) Zbl 1375.26023

Tbil. Math. J. 10, No. 4, 73-82 (2017).
Summary: In the present paper, the notion of generalized (\(s\), \(m\), \(\varphi\))-preinvex function is applied to establish some new Hermite-Hadamard type inequalities via \(k\)-fractional Riemann-Liouville integrals. At the end, some applications to special means are given. These results not only extend the results appeared in the literature, but also provide new estimates on these types.

Cited in 19 Documents

MSC:

26A51 Convexity of real functions in one variable, generalizations
26A33 Fractional derivatives and integrals
26D07 Inequalities involving other types of functions
26D15 Inequalities for sums, series and integrals

Keywords:

Hermite-Hadamard-type inequality; Hölder inequality; power mean inequality; Riemann-Liouville fractional integral; \(s\)-convex function in the second sense; \(m\)-invex
Cite Review PDF
Full Text: DOI

References:

[1] T. Antczak, Mean value in invexity analysis, Nonlinear Anal., Vol. 60 (2005), 1473-1484.; · Zbl 1100.26005
[2] P. S. Bullen, Handbook of Means and Their Inequalities, Kluwer Academic Publishers, Dordrecht, (2003).; · Zbl 1035.26024
[3] 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., Vol. 9 (2016), No. 5, 4305-4316.; · Zbl 1345.26032
[4] Y. M. Chu, G. D. Wang and X. H. Zhang, Schur convexity and Hadamard’s inequality, Math. Inequal. Appl., Vol. 13 (2010), No. 4, 725-731.; · Zbl 1205.26030
[5] R. Díaz and E. Pariguan, On hypergeometric functions and pochhammer k-symbol, Divulg. Mat., Vol. 15 (2007), No. 2, 179-192.; · Zbl 1163.33300
[6] 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., Vol. 9 (2016), 3112-3126.; · Zbl 1345.26034
[7] G. Farid, A. Rehman and M. Zahra, On Hadamard-type inequalities for k-fractional integrals, Konuralp J. Math., Vol. 4 (2016), No. 2, 79-86.; · Zbl 1355.26019
[8] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., Vol. 48 (1994), 100-111.; · Zbl 0823.26004
[9] A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s; m; ’)- preinvex functions, Aust. J. Math. Anal. Appl., Vol. 13 (2016), No. 1, 1-11.; · Zbl 1350.26014
[10] 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.; · Zbl 1273.26031
[11] 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., Vol. 9 (2016), 766-777.; · Zbl 1329.26014
[12] S. Mubeen and G. M. Habibullah, k-Fractional integrals and applications, Int. J. Contemp. Math. Sci., Vol. 7 (2012), 89-94.; · Zbl 1248.33005
[13] R. Pini, Invexity and generalized convexity, Optimization, Vol. 22 (1991), 513-525.; · Zbl 0731.26009
[14] F. Qi and B. Y. Xi, Some integral inequalities of Simpson type for GA ε ”-convex functions, Georgian Math. J., Vol. 20 (2013), No. 5, 775-788.; · Zbl 1284.26012
[15] X. M. Yang, X. Q. Yang and K. L. Teo, Generalized invexity and generalized invariant mono- tonicity, J. Optim. Theory Appl., Vol. 117 (2003), 607-625.; · Zbl 1141.90504
[16] 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.; · Zbl 1187.26012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.