New Perturbed Inequalities for Functions Whose Higher Degree Derivatives are Absolutely Continuous

Abstract

We firstly derive inequalities for high order differentiable functions with the property (S) and mappings whose higher derivatives are convex by using the same equality. Also, it is obtained Hermite Hadamard type and Bullen type inequalities for higher order differentiable functions. Then, we establish inequalities for high degree Lipschitzian derivatives via an equality which was presented previous by Erden in [12]. We also examine connection in between inequalities obtained in earlier works and our results. 

Keywords

• Absolutely continuous functions,convex functions,Lipschitzian derivatives,Perturbed inequalities

References

1. [1] M. A. Ardic, Inequalities via n􀀀times differentiable convex functions, arXiv:1310.0947v1, (2013).
2. [2] H. Budak, S. Erden and M. Z. Sarikaya, New weighted ostrowski type inequalities for mappings whose nth derivatıves are of bounded variation, International Journal of Analysis and Applications, 12 (2016), no. 1, 71-79.
3. [3] H. Budak, M. Z. Sarikaya and S. S. Dragomir, Some perturbed Ostrowski type inequality for twice differentiable functions, In: Agarwal P., Dragomir S., Jleli M., Sa¨oet B. (eds) Advances in Mathematical Inequalities and Applications. Trends in MAthematics, 2018.
4. [4] P. S. Bullen, Error estimates for some elementary quadrature rules, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., (1978) 602-633, (1979) 97-103 .
5. [5] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32 (1999), No. 4, 697-712.
6. [6] S. S. Dragomir and N. S. Barnett, An Ostrowski type inequality for mappings whose second derivatives are bounded and applications, RGMIA Research Report Collection, 1 (1998), no. 2.
7. [7] S. S. Dragomir, and A. Sofo, An integral inequality for twice differentiable mappings and application, Tamkang J. Math., 31 (2000) No. 4.
8. [8] S. S. Dragomir, Some perturbed Ostrowski type inequalities for absolutely continuous functions (I), Acta Universitatis Matthiae Belii, series Mathematics 23 (2015), 71–86.

9. [9] S. S. Dragomir, Some perturbed Ostrowski type inequalities for absolutely continuous functions (II), RGMIA Research Report Collection, 16 (2013), Article 93, 16 pp.
10. [10] S. S. Dragomir, Some perturbed Ostrowski type inequalities for absolutely continuous functions (III), TJMM, 7 (2015), no. 1, 31-43.
11. [11] S. Erden, H. Budak and M. Z. Sarıkaya, Some perturbed inequalities of Ostrowski type for twice differentiable functions, RGMIA Research Report Collection, 19 (2016), Article 70, 11 pp.
12. [12] S. Erden, Some perturbed inequalities of Ostrowski type for funtions whose nth derivatives are of bounded, Iranian Journal of Mathematical Sciences and Informatics, in press, (2019).
13. [13] S. Erden and M. Z. Sarikaya, Some perturbed inequalities of Ostrowski type for high order differentiable functions and applications, submited, (2018).
14. [14] A. El Farissi, Z. Latreuch and B. Belaidi, Hadamard-Type inequalities for twice diffrentiable functions, RGMIA Reseaech Report collection, 12 (2009), no. 1, art. 6 .
15. [15] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction consideree par Riemann, J. Math. Pures Appl., 58 (1893), 171-215.
16. [16] M. A. Latif and S. S. Dragomir, On Hermite-Hadamard type integral inequalities for n-times differentiable Log-Preinvex functions, Filomat, 29 (2015), no.7, 1651–1661.
17. [17] M. A. Latif and S. S. Dragomir, Generalization of Hermite-Hadamard type inequalities for n-times differentiable functions which are s-preinvex in the second sense with applications, Hacettepe J. of Math. and Stat., 44 (2015), no.4, 389-853.
18. [18] M. A. Latif and S. S. Dragomir, On Hermite-Hadamard type integral inequalities for n-times differentiable (a;m)-logarithmically convex functions, RGMIA Research Report Collection, 17 (2014), Article 14, 16 pages.
19. [19] A. M. Ostrowski, U¨ ber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv., 10 (1938), 226-227.
20. [20] M. E. Ozdemir and C¸ . Yıldız, A new generalization of the midpoint formula for n-time differentiable mappings which are convex, arXiv:1404.5128v1, (2014).
21. [21] B. G. Pachpatte, New inequalities of Ostrowski and Trapezoid type for n-time differentiable functions, Bull. Korean Math. Soc. 41 (2004), no. 4, pp. 633-639.
22. [22] M. Z. Sarikaya and E. Set, On new Ostrowski type Integral inequalities, Thai Journal of Mathematics, 12 (2014); no.4, 145-154.
23. [23] A. Sofo, Integral inequalities for n- times differentiable mappings, with multiple branches,on the Lp norm, Soochow Journal of Mathematics, 28 (2002), No. 2,179-221.
24. [24] M. Wang and X. Zhao, Ostrowski type inequalities for higher-order derivatives, J. of Inequalities and App., (2009), Article ID 162689, 8 pages.