Dual Simpson Type Inequalities for Functions Whose Absolute Value of the First Derivatives are Preinvex

Year 2022, Volume: 10 Issue: 1, 73 - 78, 15.04.2022

Tarek Chiheb Badreddine Meftah , Amel Dih

Abstract

In this paper, we prove a new integral identity. Basing on this identity, we establish some new dual Simpson-type inequalities for functions whose absolute value of the first derivatives are preinvex. Applications are also given.

Keywords

Dual Simpson inequality, Newton-Cotes quadrature, preinvex functions, Hölder inequality

References

• [1] H. Budak, F. Usta and M. Z. Sarikaya, New upper bounds of Ostrowski type integral inequalities utilizing Taylor expansion. Hacet. J. Math. Stat. 47 (2018), no. 3, 567–578.
• [2] H. Budak, F. Usta and M. Z. Sarikaya, Refinements of the Hermite-Hadamard inequality for co-ordinated convex mappings. J. Appl. Anal. 25 (2019), no. 1, 73–81.
• [3] H. Budak, F. Usta, M. Z. Sarikaya and M. E. Ozdemir, On generalization of midpoint type inequalities with generalized fractional integral operators. Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 113 (2019), no. 2, 769–790.
• [4] Lj. Dedi´c, M. Mati´c and J. Peˇcari´c, On dual Euler-Simpson formulae. Bull. Belg. Math. Soc. Simon Stevin 8 (2001), no. 3, 479–504.
• [5] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), no. 2, 545–550.
• [6] P. T. Harker, Generalized Nash games and quasi-variational inequalities. European journal of Operational research 54 (1991), no.1, 81-94.
• [7] B. Meftah, Two dimensional Cˇ ebysˇev type inequalities for functions whose second derivatives are co-ordinated (h1;h2)-preinvex. Konuralp J. Math. 6 (2018), no. 1, 76-83.
• [8] B. Meftah, Fractional Ostrowski type inequalities for functions whose certain power of modulus of the first derivatives are prequasiinvex via power mean inequality. J. Appl. Anal. 25 (2019), no 1, 83-90.
• [9] B. Meftah and C. Marrouche, Some new Hermite-Hadamard type inequalities for n-times log-convex functions. Jordan J. Math. Stat. 14(2021), no. 4, 651-669.
• [10] B. B. Mordukhovich and J. V. Outrata, Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18 (2007), no. 2, 389–412.
• [11] A. Nagurney, Finance and variational inequalities. Quant. Finance 1 (2001), no. 3, 309–317.
• [12] M. A. Noor, Variational-like inequalities. Optimization 30 (1994), no. 4, 323–330.
• [13] M. A. Noor, Invex equilibrium problems. J. Math. Anal. Appl. 302 (2005), no. 2, 463–475.
• [14] J. Peˇcari´c, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
• [15] R. Pini, Invexity and generalized convexity. Optimization 22 (1991), no. 4, 513–525.
• [16] F. Usta, H. Budak, M. Z. Sarikaya and E. Set, On a generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators. Filomat 32 (2018), no. 6, 2153–2171.
• [17] F. Usta and M. Z. Sarikaya, On generalization conformable fractional integral inequalities. Filomat 32 (2018), no. 16, 5519–5526.
• [18] F. Usta and M. Z. Sarikaya, On bivariate retarded integral inequalities and their applications. Facta Univ. Ser. Math. Inform. 34 (2019), no. 3, 553–561.
• [19] F. Usta, H. Budak and M. Z. Sarikaya, Montgomery identities and Ostrowski type inequalities for fractional integral operators. Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 113 (2019), no. 2, 1059–1080.
• [20] F. Usta, H. Budak and M. Z. Sarikaya, Some new Chebyshev type inequalities utilizing generalized fractional integral operators. AIMS Math. 5 (2020), no. 2, 1147–1161.
• [21] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136 (1988), no. 1, 29–38.
• [22] X. -M. Yang and D. Li, On properties of preinvex functions. J. Math. Anal. Appl. 256 (2001), no. 1, 229–241.
• [23] G. X. -Z.Yuan, G. Isac, K. -K. Tan and J. Yu, The study of minimax inequalities, abstract economics and applications to variational inequalities and Nash equilibria. Acta Appl. Math. 54 (1998), no. 2, 135–166.