Differentiable Preinvex and Prequasiinvex Functions

Year 2020, Volume: 3 Issue: 2, 69 - 77, 22.06.2020

Huriye Kadakal

https://doi.org/10.32323/ujma.649117

Abstract

In this paper, a new identity for functions defined on an open invex subset of set of real numbers is established, and by using this identity and the Hölder and Power mean integral inequalities we present new integral inequalities for functions whose powers of derivatives in absolute value are preinvex and prequasiinvex. We should especially mention that the results obtained in special cases coincide with the well-known results in the literature.

Keywords

Invex set, Preinvex function, Prequasiinvex function, Integral inequalities

References

• [1] T. Antczak, Mean value in invexity analysis, Nonl. Anal. 60(2005), 1473-1484.
• [2] A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality through prequasiinvex functions, RGMIA Research Report Collection, 14(2011), Article 48, 7 pp.
• [3] A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, J. Inequal. Appl. 2012, 2012:247.
• [4] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
• [5] J. Hadamard, Etude sur les proprie´te´s des fonctions entie`res et en particulier d’une fonction considere´e par Riemann, J. Math Pures Appl., 58 (1893), 171–215.
• [6] D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Comp. Sci. Ser. Volume 34, 2007, Pages 82–87.
• [7] A. B. Israel and B. Mond, What is invexity? J. Aust. Math. Soc. Ser. B 28(1), 1-9 (1986). [8] İ. İşcan, H. Kadakal and M. Kadakal, Some New Integral Inequalities for n-Times Differentiable Quasi-Convex Functions, Sigma Journal of Engineering and Natural Sciences, 35 (3), 363-368, 2017.
• [9] M. Kadakal, Hermite-Hadamard and Simpson type inequalities for multiplicatively harmonically P-functions, Sigma Journal of Engineering and Natural Sciences, 37(4), (2019), 1311-1320.
• [10] H. Kadakal, M. Kadakal and İ. İşcan, Some New Integral Inequalities for n-Times Differentiable r-Convex and r-Concave Functions, Miskolc Mathematical Notes, 20(2) (2019), 997-1011.
• [11] M. A. Latif and S. S. Dragomir, Some Hermite-Hadamard type inequalities for functions whose partial derivatives in absloute value are preinvex on the co-oordinates, Facta Universitatis (NIˇ S) Ser. Math. Inform. Vol. 28, No 3 (2013), 257-270.
• [12] S. Maden, H. Kadakal, M. Kadakal and İ. İşcan, Some new integral inequalities for n-times differentiable convex functions, Journal of Nonlinear Sciences and Applications, 10 (12), (2017), 6141-6148.
• [13] M. Matloka, On some new inequalities for differentiable (h1; h2)-preinvex functions on the co-ordinates, Mathematics and Statistics 2(1), (2014), 6-14.
• [14] S.R. Mohan, S.K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901-908.
• [15] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory 2 (2007) 126-131.
• [16] M.A. Noor, Invex equilibrium problems. J. Math. Anal. Appl. 302, 463-475 (2005).
• [17] M.A. Noor, Variational-like inequalities. Optimization 30, 323-330 (1994).
• [18] S. Özcan, Some Integral Inequalities for Harmonically (a; s)-Convex Functions, Journal of Function Spaces, 2019, (2019) Article ID 2394021, 8 pages.
• [19] S. Özcan, On Refinements of Some Integral Inequalities for Differentiable Prequasiinvex Functions, Filomat, 33(14), (2019), 4377-4385.
• [20] S. Özcan, Some Integral Inequalities of Hermite-Hadamard Type for Multiplicatively Preinvex Functions, AIMS Mathematics, 5(2), (2020), 1505-1518.
• [21] S. Özcan and İ. İşcan, Some new Hermite-Hadamard type inequalities for s-convex functions and their applications, Journal of Inequalities and Applications 2019(201), (2019).
• [22] J.E. Pecaric, F. Porschan and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992.
• [23] R. Pini, Invexity and generalized convexity. Optimization 22, 513-525 (1991).
• [24] T. Toplu, İ. İşcan and M. Kadakal, On n-polinomial convexity and some related inequalities, Aims Mathematics, 5(2) (2020), 1304-1318.
• [25] T. Weir and B. Mond, Preinvex functions in multiple objective optimization, Journal of Mathematical Analysis and Applications, 136 (1998) 29-38.
• [26] X. M. Yang, X. Q. Yang, K. L. Teo, Generalized invexity and generalized invariant monotonicity, J. Optim. Theory. Appl., 117(2003), 607-625.
• [27] X.M.Yang and D. Li, On properties of preinvex functions. J. Math. Anal. Appl. 256, 229-241 (2001).