Some Inequalities for n-Time Differentiable Mappings Using a Multi-Step Kernel with Applications

Obeidat, Sofian

doi:10.28924/2291-8639-18-2020-50

Research Article

Some Inequalities for n-Time Differentiable Mappings Using a Multi-Step Kernel with Applications

Sofian Obeidat¹

University of Hail

Abstract

In this paper, we develop a new multi-step kernel and use it to establish new Ostrowski's type inequalities for n-time differentiable mappings, whose n-th derivatives satisfy convexity and quasi-convexity conditions. Applications of our findings to random variables and approximation of integrals are given.

How to Cite

S. Obeidat, Some Inequalities for n-Time Differentiable Mappings Using a Multi-Step Kernel with Applications, Int. J. Anal. Appl. 18 (2020), 50–62. https://doi.org/10.28924/2291-8639-18-2020-50

BibTeX

@article{Obeidat2020,
  author  = {Obeidat, Sofian},
  title   = {Some Inequalities for n-Time Differentiable Mappings Using a Multi-Step Kernel with Applications},
  journal = {Int. J. Anal. Appl.},
  volume  = {18},
  year    = {2020},
  pages   = {50--62},
  doi     = {10.28924/2291-8639-18-2020-50}
}

References

[1] P. Cerone, S.S. Dragomir, J. Roumeliotis and J. Sunde, A new generalization of the trapezoid formula for n-time differentiable mappings and applications, Demonstr. Math. 33 (4) (2000), 719-736.
[2] X. L. Cheng, Improvement of some Ostrowski-Gr ¨uss type inequalities, Computers Math. Appl. 42 (2001), 109-114.
[3] L. Chun and F. Qi, Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex, J. Inequal. Appl. 2013 (2013), Art. ID 451.
[4] S. S. Dragomir and S. Wang, An inequality of Ostrowski-Gr ¨uss type and its applications to the estimation of error bounds for some special means and for home numerical quadrature rules, Computers Math. Appl. 33 (11) (1997), 15-20.
[5] D. Y. Hwang, Some Inequalities for n-time Differentiable Mappings and Applications, Kyung. Math. J. 43 (2003), 335-343.
[6] J. L. W. V. Jensen, On konvexe funktioner og uligheder mel lem middlvaerdier, Nyt. Tidsskr. Math. B. 16 (1905), 49-69.
[7] A. I. Kechriniotis and Y. A. Theodorou, New integral inequalities for n-time differentiable functions with applications for pdfs, Appl. Math. Sci. 2 (8) (2008), 353-362.
[8] A. Ostrowski, Uber die Absolutabweichung einer differentierbaren Funktion von ihren Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227.
[9] M. E. Ozdemir and C. Yildiz, A new generalization of the midpoint formula for n-time differentiable mappings which are convex, arXiv:1404.5128v1, 2014.
[10] B.G. Pachpatte, New inequalities of Ostrowski and Trapezoid type for n-time differentiable functions, Bull. Korean Math. Soc. 41 (4) (2004), 633-639.
[11] J. E. Pecaric, F. Proschan, Y.L. Tong, Convex Function, Partial Ordering and Statistical Applications, Academic Press, New York, 1991.
[12] S. H. Wang and F. Qi, Inequalities of Hermite-Hadamard type for convex functions which are n-times differentiable, Math. Inequal. Appl. 16 (4) (2013), 1269-1278.