HERMITE-HADAMARD TYPE FRACTIONAL INTEGRAL INEQUALITIES FOR TWICE DIFFERENTIABLE GENERALIZED (s, m, ϕ)-PREINVEX FUNCTIONS

Year 2017, Volume: 5 Issue: 2, 228 - 238, 15.10.2017

ARTION Kashurı ROZANA Lıko

Abstract

In the present paper, a new class of generalized $(s,m,\varphi)$-preinvex function is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving generalized $(s,m,\varphi)$-preinvex functions along with beta function are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for generalized $(s,m,\varphi)$-preinvex functions that are twice differentiable via Riemann-Liouville fractional integrals are established. At the end, some applications to special means are given.

Keywords

Hermite-Hadamard type inequality, H\"{o}lder's inequality, power mean inequality, Riemann-Liouville fractional integral, $s$-convex function in the second sense, $m$-invex, $P$-function

References

• [1] Antczak, T., Mean value in invexity analysis, Nonlinear Anal., 60(2005), 1473-1484.
• [2] Budak, H., Usta, F., Sarikaya, M. Z. and Özdemir, M. E., On generalization of midpoint type inequalities with generalized fractional integral operators, https://www.researchgate.net/publication/312596723.
• [3] Bullen, P. S., Handbook of Means and Their Inequalities, Kluwer Academic Publishers, Dordrecht, (2003).
• [4] Dragomir, S. S., Pecaric, J. and Persson, L. E., Some inequalities of Hadamard type, Soochow J. Math., 21(1995), 335-341.
• [5] Du, T. S., Liao, J. G. and Li, Y. J., Properties and integral inequalities of Hadamard-Simpson type for the generalized (s;m)-preinvex functions, J. Nonlinear Sci. Appl., 9(2016), 3112-3126.
• [6] Hudzik, H. and Maligranda, L., Some remarks on s-convex functions, Aequationes Math., 48(1994), 100-111.
• [7] Liu, W., New integral inequalities involving beta function via P-convexity, Miskolc Math. Notes, 15(2014), no. 2, 585-591.
• [8] Liu,W., Wen, W. and Park, J., Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl., 9(2016), 766-777.
• [9]  Ozdemir, M. E., Set, E. and Alomari, M., Integral inequalities via several kinds of convexity, Creat. Math. Inform., 20(2011), no. 1, 62-73.
• [10] Pini, R., Invexity and generalized convexity, Optimization, 22(1991), 513-525.
• [11] Qi, F. and Xi, B. Y., Some integral inequalities of Simpson type for GA-convex functions, Georgian Math. J., 20(2013), no. 4, 775-788.
• [12] Sarikaya, M. Z. and Budak, H., Generalized Hermite-Hadamard type integral inequalities for fractional integrals, Filomat, 30(2016), no. 5, 1315-1326.
• [13] Sarikaya, M. Z., Set, E., Yaldiz, H. and Basak, N., Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57(2013), 2403-2407.
• [14] Stancu, D. D., Coman, G. and Blaga, P., Analiza numericasi teoria aproximarii, Cluj-Napoca: Presa Universitara Clujeana., 2(2002).
• [15] Tunç, T., Budak, H., Usta, F. and Sarikaya, M. Z., On new generalized fractional integral operators and related fractional inequalities, ResearchGate Article, Available online at: https://www.researchgate.net/publication/313650587.
• [16] Usta, F., Budak, H., Sarkaya, M. Z. and Set, E., On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators. Filomat, (in press).
• [17] Yang, X. M., Yang, X. Q. and Teo, K. L., Generalized invexity and generalized invariant monotonicity, J. Optim. Theory Appl., 117(2003), 607-625.