Research article

The Ostrowski inequality for $ s $-convex functions in the third sense

  • Received: 21 October 2021 Revised: 28 December 2021 Accepted: 30 December 2021 Published: 10 January 2022
  • MSC : 26A51

  • In this paper, the Ostrowski inequality for $ s $-convex functions in the third sense is studied. By applying Hölder and power mean integral inequalities, the Ostrowski inequality is obtained for the functions, the absolute values of the powers of whose derivatives are $ s $-convex in the third sense. In addition, by means of these inequalities, an error estimate for a quadrature formula via Riemann sums and some relations involving means are given as applications.

    Citation: Gültekin Tınaztepe, Sevda Sezer, Zeynep Eken, Sinem Sezer Evcan. The Ostrowski inequality for $ s $-convex functions in the third sense[J]. AIMS Mathematics, 2022, 7(4): 5605-5615. doi: 10.3934/math.2022310

    Related Papers:

  • In this paper, the Ostrowski inequality for $ s $-convex functions in the third sense is studied. By applying Hölder and power mean integral inequalities, the Ostrowski inequality is obtained for the functions, the absolute values of the powers of whose derivatives are $ s $-convex in the third sense. In addition, by means of these inequalities, an error estimate for a quadrature formula via Riemann sums and some relations involving means are given as applications.



    加载中


    [1] G. Adilov, I. Yesilce, $B^{-1}$-convex functions, J. Convex Anal., 24 (2017), 505–517. http://dx.doi.org/10.81043/aperta.44759 doi: 10.81043/aperta.44759
    [2] G. Anastassiou, General Grüss and Ostrowski type inequalities involving s-convexity, Bull. Allahabad Math. Soc., 28 (2013), 101–129.
    [3] A. Bayoumi, Foundation of complex analysis in non locally convex spaces: function theory without convexity condition, Amsterdam: Elsevier Science, 2003.
    [4] W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter Funktionen in topologischen linearen Raumen, Publ. Inst. Math., 23 (1978), 13–20.
    [5] W. Briec, C. Horvath, $B$-convexity, Optimization, 53 (2004), 103–127. http://dx.doi.org/10.1080/02331930410001695283 doi: 10.1080/02331930410001695283
    [6] S. Dragomir, C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Science Direct Working Paper, 2003, S1574-0358(04)70845-X.
    [7] T. Du, C. Luo, Z. Cao, On the Bullen-type inequalities via generalized fractional integrals and their applications, Fractals, 29 (2021), 2150188. http://dx.doi.org/10.1142/S0218348X21501887 doi: 10.1142/S0218348X21501887
    [8] Z. Eken, S. Kemali, G. Tinaztepe, G. Adilov, The Hermite-Hadamard inequalities for $p$-convex functions, Hacet. J. Math. Stat., 50 (2021), 1268–1279. https://dx.doi.org/10.15672/hujms.775508 doi: 10.15672/hujms.775508
    [9] K. Gdawiec, Fractal patterns from the dynamics of combined polynomial root finding methods, Nonlinear Dyn., 90 (2017), 2457–2479. https://dx.doi.org/10.1007/s11071-017-3813-6 doi: 10.1007/s11071-017-3813-6
    [10] S. Kemali, I. Yesilce, G. Adilov, $B$-convexity, $B^{-1}$-convexity, and their comparison, Numer. Func. Anal. Opt., 36 (2015), 133–146. https://dx.doi.org/10.1080/01630563.2014.970641 doi: 10.1080/01630563.2014.970641
    [11] S. Kemali, S. Sezer, G. Tınaztepe, G. Adilov, $s$-Convex functions in the third sense, Korean J. Math., 29 (2021), 593–602. https://dx.doi.org/10.11568/kjm.2021.29.3.593 doi: 10.11568/kjm.2021.29.3.593
    [12] Y. Kwun, M. Tanveer, W. Nazeer, K. Gdawiec, S. Kang, Mandelbrot and Julia Sets via Jungck-CR iteration with $s$-convexity, IEEE Access, 7 (2019), 12167–12176. https://dx.doi.org/10.1109/ACCESS.2019.2892013 doi: 10.1109/ACCESS.2019.2892013
    [13] W. Orlicz, A note on modular spaces Ⅰ, Bull. Acad. Polon. Sci., 9 (1961), 157–162.
    [14] A. Ostrowski, Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1937), 226–227. doi: 10.1007/BF01214290
    [15] M. Özdemir, A. Ekinci, Some new integral inequalities for functions whose derivatives of absolute values are s-convex, Turkish Journal of Analysis and Number Theory, 7 (2019), 70–76. http://dx.doi.org/10.12691/tjant-7-3-3 doi: 10.12691/tjant-7-3-3
    [16] M. Sarikaya, F. Ertuğral, F. Yıldırım, On the Hermite-Hadamard-Fejér type integral inequality for s-convex function, Konuralp Journal of Mathematics, 6 (2018), 35–41.
    [17] S. Sezer, Z. Eken, G. Tınaztepe, G. Adilov, $p$-convex functions and some of their properties, Numer. Func. Anal. Opt., 42 (2021), 443–459. http://dx.doi.org/10.1080/01630563.2021.1884876 doi: 10.1080/01630563.2021.1884876
    [18] S. Sezer, The Hermite-Hadamard inequalities for $s$-convex functions in the third sense, AIMS Mathematics, 6 (2021), 7719–7732. https://dx.doi.org/10.3934/math.2021448 doi: 10.3934/math.2021448
    [19] I. Yesilce, G. Adilov, Some operations on $B^{-1}$-convex sets, Journal of Mathematical Sciences: Advances and Applications, 39 (2016), 99–104. http://dx.doi.org/10.18642/jmsaa_7100121669 doi: 10.18642/jmsaa_7100121669
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1971) PDF downloads(326) Cited by(0)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog