Some Perturbed Trapezoid Inequalities for n-times Differentiable Strongly log-Convex Functions

Abstract

The aim of this study is to introduce some inequalities for n-times differentiable strongly log-convex functions. The perturbed trapezoid inequality is used to establish the new inequalities. It is seen that these inequalities have a better upper bound than the inequalities obtained for log-convex functions. Besides, the mentioned inequalities for strongly log-convex functions are reduced to the ones given for log-convex functions with a suitable choice of the arbitrary constant.

Keywords

• Convex function
• log-convex function
• strongly log-convex function
• perturbed trapezoid inequalities.

References

1. Polyak, BT. 1966. Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math. Dokl; 7: 2-75.
2. Necoara, I, Nesterov, Y, Glineur, F. 2019. Linear convergence of first order methods for non-strongly convex optimization. Mathematical Programming; 175: 69-107.
3. Karamardian, S. 1969. The nonlinear complementarity problems with applications, Part II. Journal of Optimization Theory and Applications; 4 (3): 167-181.
4. Nikodem, K, Pales, ZS. 2011. Characterizations of inner product spaces by strongly convex function. Banach J. Math. Anal.: 1(2): 83-87.
5. Zu, DL, Marcotte, P. 1996. Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM Journal on Optimization; 6(3): 714- 726.
6. Qu, G, Li, N. 2019. On the exponentially stability of primal-dual gradient dynamics. IEEE Control Syst. Letters; 3(1): 46-48.
7. Noor, MA, Noor, KI. 2019. On generalized strongly convex functions involving bifunction. Appl. Math. Inform. Sci.; 13(3): 411-416.
8. Mohsen, BB, Noor, MA, Noor, KI, Postolache, M. 2019. Strongly convex functions of higher order involving bifunction. Mathematics; 7(1028): 1-12.

9. Cristescu, G, Lupsa, L. Non-Connected Convexities and Applications. Kluwer Academic Publisher, Dordrechet, 2002.
10. Niculescu, CF, Persson, LE. Convex Functions and Their Applications. Springer-Verlag, New York, 2018.
11. Pecaric, J, Proschan, F, Tong, YL. Convex Functions, Partial Orderings and Statistical Applications. Academic Press: New York, 1992.
12. Noor, MA. 2004. Some developments in general variational inequalities. Appl. Math. Comput.; 152: 199-277.
13. Noor, MA, Noor, KI. 2021. Higher order general convex functions and general variational inequalities. Canadian J. Appl. Math.; 3(1): 1-17.
14. Merentes, N, Nikodem, K. 2010. Remarks on strongly convex functions. Aequationes Math.; 80: 193-199.
15. Azcar, A, Nikodem, K, Roa, G. 2012. Fejér-type inequalities for strongly convex functions. Annal. Math. Siles.; 26: 43-54.
16. Noor, MA, Noor, KI, Iftikhar, S. 2020. Inequalities via strongly (p,h)-harmonic convex functions. TWS J. App. Eng. Math.; 10(1): 81-94.
17. Noor, MA, Noor, KI, K. Iftikhar, S. 2016. Hermit-Hadamard inequalities for strongly harmonic convex functions. Journal of Inequalities and Special Functions; 7(3): 99-113.
18. Beckenbach, EF. 1948. Convex functions. Bull. Amer. Math. Soc.; 54: 439-460.
19. Dragomir, SS, Agarwal, RP. 1998. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett., 11(5): 91-95.
20. Şanal, G, Dönmez Demir, D. Some inequalities for n-times differentiable log-convex functions, Manisa Celal Bayar University, II. International University Industry Cooperation, R&D and Innovation Congress, Manisa, Türkiye, 2018, pp.89.
21. Paul, M. 2018. Sur les fonctions convexes et les fonctions sousharmoniques. Journal de Mathématiques Pures et Appliquées; 7: 29-60.
22. Hermite, C. 1883. Sur deux limites d’une integrale definie. Mathesis; 3: 82-83.
23. Dragomir, SS, Cerone, P, Sofo, A. 2000. Some remarks on the trapezoid rule in numerical integration. Indian J. Pure Appl. Math.; 31(5): 475-494.
24. Sykora, S. Mathematical Means and Averages: Basic Properties. 3. Stan’s Library: Castano Primo, Italy, Vol. III, 2009.