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AIMS Mathematics

2021, Volume 6, Issue 1: 362-377. doi: 10.3934/math.2021022
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Research article

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

  • 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: 26 May 2020 Accepted: 22 September 2020 Published: 13 October 2020

  • MSC : 26D15, 26A46, 26D10

  • 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.

    Citation: Samet Erden, Nuri Çelİk, Muhammad Adil Khan. Refined inequalities of perturbed Ostrowski type for higher-order absolutely continuous functions and applications[J]. AIMS Mathematics, 2021, 6(1): 362-377. doi: 10.3934/math.2021022

    Related Papers:

  • 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.


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    [1] M. A. Khan, S. Begum, Y. Khurshid, Y. Chu, Ostrowski type inequalities involving conformable fractional integrals, J. Inequal. Appl., 2018 (2018), 1-14. doi: 10.1186/s13660-017-1594-6
    [2] D. Bongiorno, Absolutely continuous functions with values in a Banach space, J. Math. Anal. Appl., 451 (2017), 1216-1223. doi: 10.1016/j.jmaa.2017.02.067
    [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. doi: 10.1016/j.jmaa.2004.08.002
    [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. doi: 10.1016/j.trmi.2017.03.004
    [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. doi: 10.2298/FIL1716305B
    [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. doi: 10.17776/csj.577459
    [18] S. Hencl, On the notion of absolute continuity for functions of several variables, Fund. Math., 173 (2002), 175-189. doi: 10.4064/fm173-2-5
    [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. doi: 10.1186/1029-242X-2013-9
    [21] J. Maly, Absolutely continuous functions of several variables, J. Math. Anal. Appl., 231 (1999), 492-508. doi: 10.1006/jmaa.1998.6246
    [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 Lp 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. doi: 10.22436/jnsa.009.02.19
    [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.
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