AIMS Mathematics, 2020, 5(6): 5495-5509. doi: 10.3934/math.2020352

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Perturbed trapezoid inequalities for n th order differentiable convex functions and their applications

Department of Mathematics, Manisa Celal Bayar University, Manisa, Turkey

In this study, we introduce a new general identity for n th order differentiable functions. Also, we establish some new inequalities regarding general perturbed trapezoid inequality for the functions whose the absolute values of n th derivatives are convex. Finally, some applications for special means are provided.
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1. M. A. Ardic, Inequalities via n-times differentiable convex functions, arXiv:1310.0947v1, 2013.

2. P. Agarwal, J. Tarioon, S. K. Ntouyas, Some generalized Riemann-Liouville k-fractional integral inequalities, J. Inequalities Appl., Article number: 122 (2016).

3. M. Bessenyei, Z. Páles, Characterizations of convexity via Hadamard's inequality, Math. Ineq. Appl., 9 (2006), 53-62.

4. P. Cerone, S. S. Dragomir, C. E. M. Pearce, A generalized trapezoid inequality for functions of bounded variation, Turk. J. Math., 24 (2000), 147-163.

5. P. Cerone, On perturbed trapezoidal and midpoint rules, J. Appl. Math. Comput., 9 (2002), 423-435.

6. S. S. Dragomir, P. Cerone, A. Sofo, Some remarks on the trapezoid rule in numerical integration, Indian J. Pure Appl. Math., 31 (2000), 475-494.

7. S. S. Dragomir, S. Wang, An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Computers Math. Applic., 33 (1997), 15-20.

8. S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91-95.

9. A. Fernandez, P. O. Mohammed, Hermite-Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels, Math. Methods Appl. Sci., 2020, 1-18.

10. S. Jain, K. Mehrez, D. Baleanu, et al. Certain Hermite-Hadamard inequalities for logarithmically convex functions with applications, Mathematics, 7 (2019), 163.

11. W. Liu, J. Park, Some perturbed versions of the generalized trapezoid inequality for functions of bounded variation, J. Comput. Anal. Appl., 22 (2017), 11-18.

12. X. F. Ma, L. C. Wang, Two mapping related to Minkowski's inequalities, JIPAM, 10 (2009), 1-8.

13. K. Mehrez, P. Agarwal, New Hermite-Hadamard type integral inequalities for convex functions and their applications, J. Comput. Appl. Ivlath., 350 (2019), 274-285.    

14. D. S. Mitrinović, J. Pečarić, A. M. Fink, Classical and new inequalities in analysis, Kluwer Academic, Dordrecht, 1993.

15. P. O. Mohamed, Inequalities of -type for Riemann-Liouville fractional integrals, Appl. Ivlath. ENotes, 17 (2017), 199-206.

16. P. O. Mohammed, Some new Hermite-Hadamard type inequalities for MT-convex functions on differentiable coordinates, J. King Saud Univ. Sci., 30 (2018), 258-262.    

17. P. O. Mohammed, New integral inequalities for preinvex functions via generalized beta function, J. Inter. Ivlath., 22 (2019), 539-549.

18. P. O. Mohammed, Hermite-Hadamard inequalities for Riemann-Liouville fractional integrals of a convex function with respect to a monotone function, Math. Meth. Appl. Sci., 2019, 1-11.

19. P. O. Mohammed, M. Z. Sarikaya, Hermite-Hadamard type inequalities for F-convex function involving fractional integrals, J. Jnequal. Appl., 2018, 359.

20. P. O. Mohammed, F. K. Hamasalh, New conformable fractional integral inequalities of HermiteHadamard type for convex functions, Symrretry, 11 (2019), 263.

21. P. O. Mohammed, M. Z. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Ivlath., 372 (2020), 112740.

22. P. O. Mohammed, T. Abdeljawad, Modification of certain fractional integral inequalities for convex functions, Adv. Differ. Equ., 2020, 69.

23. C. P. Niculescu, L. E. Persson, Convex functions and their applications: A Contemporary Approach, CMS Books in Mathematics, Vol. 23, Springer-Verlag, New York, 2006.

24. M. Z. Sarikaya, N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Modell., 54 (2011), 2175-2182.    

25. M. Tomar, P. Agarwal, J. Choi, Hermite-Hadamard type inequalities for generalized convex functions on fractal sets style, Bal. Soc. Paran. Ivlat., 38 (2020), 101-116.

26. M. Tunç, G. Şanal, Some perturbed trapezoid inequalities for convex, s-convex and tgs-convex functions and applications, Tbilisi Math, J., 8 (2015), 87-102.    

27. N. Ujević, Perturbed trapezoid and mid-point inequalities and applications, Soochow J. Math., 29 (2003), 249-257.

28. F. Qi, P. O. Mohammed, J. C. Yao, et al. Generalized fractional integral inequalities of HermiteHadamard type for (α, m)-convex functions, J. Inequal. Appl., DOI: 10.1186/s13660-019-2079-6, 2019, 135.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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