AIMS Mathematics, 2021, 6(1): 362-377. doi: 10.3934/math.2021022 Research article

Export file:

Format

RIS(for EndNote,Reference Manager,ProCite)
BibTex
Text

Content

Citation Only
Citation and Abstract

Refined inequalities of perturbed Ostrowski type for higher-order absolutely continuous functions and applications

Samet Erden, Nuri Çelİk, Muhammad Adil Khan

1 Department of Mathematics, Faculty of Science, Bartin University, Bartin, Turkey
2 Department of Mathematics, Faculty of Science, Gebze Technical University, Kocaeli, Turkey
3 Department of Mathematics, Huzhou University, Huzhou 313000, China
4 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan

Received: , Accepted: , Published:

Abstract Full Text(HTML) Figure/Table Related pages

First of all, we establish an identity for higher-order differentiable functions. Then, we prove some integral inequalities for mappings that have continuous derivatives up to the order $n-1$ with $n\geq 1$ and whose n-th derivatives are the element of $L_{1},~L_{r}$, and $L_{\infty }.$ In addition, estimates of new composite quadrature rules are examined. Finally, natural applications for exponential and logarithmic functions are given.

Figure/Table

Supplementary

Article Metrics

References

1. M. A. Khan, S. Begum, Y. Khurshid, Y. Chu, Ostrowski type inequalities involving conformable fractional integrals, J. Inequal. Appl., 2018 (2018), 1-14.

2. D. Bongiorno, Absolutely continuous functions with values in a Banach space, J. Math. Anal. Appl., 451 (2017), 1216-1223.

3. D. Bongiorno, On the Hencl's notion of absolute continuity, J. Math. Anal. Appl., 350 (2008), 562-567.

4. D. Bongiorno, Absolutely continuous functions in $\mathbb{R}^n$, J. Math. Anal. Appl., 303 (2005), 119-134.

5. H. Budak, M. Z. Sarikaya, A companion of Ostrowski type inequalities for mappings of bounded variation and some applications, Trans. A. Razmadze Math. Inst., 171 (2017), 136-143.

6. H. Budak, M. Z. Sarikaya, A. Qayyum, Improvement in companion of Ostrowski type inequalities for mappings whose first derivatives are of bounded variation and application, Filomat, 31 (2017), 5305-5314.

7. H. Budak, M. Z. Sarikaya, S. S. Dragomir, Some perturbed Ostrowski type inequality for twice differentiable functions, Adv. Math. Inequal. Appl., (2018), 279-294.

8. P. Cerone, S. S. Dragomir, J. Roumeliotis, An inequality of Ostrowski type for mappings whose second derivatives are bounded and applications, RGMIA Res. Rep. Coll., 1 (1998), 7-9.

9. P. Cerone, S. S. Dragomir, J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32 (1999), 697-712.

10. S. S. Dragomir, N. S. Barnett, An Ostrowski type inequality for mappings whose second derivatives are bounded and applications, RGMIA Res. Rep. Coll., 1 (1998), 67-75.

11. S. S. Dragomir, P. Cerone, J. Roumeliotis, A new generalization of Ostrowski's integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Appl. Math. Lett., 13 (2000), 19-25.

12. S. S. Dragomir, Some perturbed Ostrowski type inequalities for absolutely continuous functions (I), Acta Univ. M. Belii Ser. Math., 23 (2015), 71-86.

13. S. S. Dragomir, Some perturbed Ostrowski type inequalities for absolutely continuous functions (II), Acta Univ. Apulensis Math. Inform., 43 (2015), 209-228.

14. S. S. Dragomir, Perturbed companions of Ostrowski's inequality for absolutely continuous functions (I), An. Univ. Timis., Ser. mat.-inform., LIV (2016), 119-138.

15. S. Erden, H. Budak, M. Z. Sarıkaya, Some perturbed inequalities of Ostrowski type for twice differentiable functions, Math. Clu., 62 (2020), In press.

16. S. Erden, Perturbed Companions of Ostrowski type inequalities for N-times differentiable functions and applications, Probl. Anal. Issues Anal., 9 (2020), 45-57.

17. S. Erden, Companions of Perturbed type inequalities for higher order differentiable functions, Cumhuriyet Sci. J., 40 (2019), 819-829.

18. S. Hencl, On the notion of absolute continuity for functions of several variables, Fund. Math., 173 (2002), 175-189.

19. Y. Khurshid, M. A. Khan, Y. M. Chu, Ostrowski type inequalities involving conformable integrals via preinvex functions, AIP Adv., 10 (2020), 1-9.

20. W. Liu, Y. Zhu, J. Park, Some compenians of perturbed Ostrowski-type inequalities based on the quasratic kernel function with three sections and applications, J. Inequal. Appl., 2013 (2013), 9-14.

21. J. Maly, Absolutely continuous functions of several variables, J. Math. Anal. Appl., 231 (1999), 492-508.

22. A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv., 10 (1938), 226-227.

23. M. Z. Sarikaya, E. Set, On new Ostrowski type Integral inequalities, Thai J. Math., 12 (2014), 145-154.

24. M. Z. Sarikaya, H. Budak, T. Tunc, S. Erden, H. Yaldız, Perturbed companion of Ostrowski type inequality for twice differentiable functions, Facta Univ. Ser. Math. Inform., 31 (2016), 595-608.

25. A. Sofo, Integral inequalities for n-times differentiable mappings with multiple branches on the L_p norm, Soochow J. Math., 28 (2002), 179-222.

26. A. Qayyum, M. Shoaib, I. Faye, Companion of Ostrowski-type inequality based on 5-step quadratic kernel and applications, J. Nonlinear Sci. Appl., 9 (2016), 537-552.

27. A. Qayyum, M. Shoaib, I. Faye, On new refinements and applications of efficient quadrature rules using n-times differentiable mappings, J. Comput. Anal. Appl., 23 (2017), 723-739.

28. M. Wang, X. Zhao, Ostrowski type inequalities for higher-order derivatives, J. Inequal. Appl., 2009 (2009), 1-8.

Download full text in PDF

Export Citation

Article outline

Show full outline