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Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions

Year 2024, Volume: 12 Issue: 1, 62 - 73, 30.04.2024

Samet Erden Burçin Gökkurt Özdemir Sevgi Kılıçer Canmert Demır

Abstract

The main purpose of this study is to establish new inequalities including Riemann-Liouville fractional integrals for various classes of functions with two variables. We first establish two identities involving Riemann-Liouville fractional integrals for higher-order partial differential functions. Then, some fractional Ostrowski type inequalities for functions of bounded variation of two variables are attained. Moreover, we present fractional integral inequalities for functions whose higher-order partial derivatives are elements of $L_{\infty }$ and $L_{1},$ respectively. Some special cases and midpoint versions of our main results are also examined.

Keywords

Bounded functions fuctions of bounded variation fractional integrals Ostrowski type inequalities.

References

  • [1] Agli´c Aljinovi´c, A. (2014). Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral. Journal of Mathematics, Article ID 503195, 6 pages.
  • [2] Anastassiou, G. (1995). Ostrowski type inequalities. Proc. of the American Math. Soc., 123 (12), 3775-378.
  • [3] Barnett, N.S., & Dragomir, S.S. (2001). An Ostrowski type inequality for double integrals and applications for cubature formulae. Soochow J. Math., 27(1), 1-10.
  • [4] Budak H. and Sarikaya, M.Z.(2016). On Ostrowski type inequalities for functions of two variables with bounded variation, International Journal of Analysis and Applications, 12 (2), 142-156.
  • [5] Budak, H.; Sarikaya, M.Z. and Erden, S. (2016). New weighted Ostrowski type inequalities for mappings whose nth derivatives are of bounded variation, International J. of Analysis and App., 12 (1), 71-79.
  • [6] Budak, H. and Sarikaya, M.Z. (2016). A new Ostrowski type inequality for functions whose first derivatives are of bounded variation, Moroccan J. Pure Appl. Anal., 2(1); 1–11.
  • [7] Cerone, P.; Dragomir, S.S. and Roumeliotis, (1999). Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32 (4), 697-712.
  • [8] Changjian, Z. and Cheung, W.S. (2010). On Ostrowski-type inequalities for heigher-order partial derivatives. Journal of Ineqaulities and Applications, 1-8.
  • [9] Clarkson, J.A., Adams, C.R. (1933). On definitions of bounded variation for functions of two variables, Bull. Amer. Math. Soc. V.35 pp.824-854.
  • [10] Dragomir, S.S. (2001). On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Mathematical Inequalities & Applications, 4 (1), 59–66.
  • [11] Dragomir, S.S. and Wang, S. (1997). A new inequality of Ostrowski’s type in L1-norm and applications to some special means and to some numerical quadrature rules. Tamkang J. of Math., 28(3), 239-244.
  • [12] Dragomir, S. S. and Wang, A. (1998). A new inequality of Ostrowski’s type in Lp-norm and applications to some special means and to some numerical quadrature rules. Indian Journal of Mathematics, 40(3), 299-304.
  • [13] Dragomir, S. S., Barnett, N. S. and Cerone, P. (2003). An Ostrowski type inequality for double integrals in terms of Lp-norms and applications in numerical integration. Anal. Num. Theor. Approx., 32(2), 161-169.
  • [14] Dragomir, S. S. (2017). Ostrowski type inequalities for generalized Riemann Liouville fractional integrals of bounded variation. H¨older and Lipschitzian functions. RGMIA Research Report Collection, 20, Article 48, 1-14.
  • [15] Dragomir, S. S. (2017).Ostrowski Type inequalities for riemann-Liouville fractional integrals of absolutely continuous functions in terms of ¥􀀀norms. RGMIA Research Report Collection, 20, Article 49, 1-14.
  • [16] Dragomir, S. S. (2017). Ostrowski Type inequalities for riemann-Liouville fractional integrals of absolutely continuous functions in terms of p􀀀norms. RGMIA Research Report Collection, 20, Article 50.
  • [17] Dragomir, S. S. (2020). Ostrowski and trapezoid type inequalities for Riemann-Liouville fractional integrals of absolutely continuous functions with bounded derivatives. Fractional Differential Calculus, 10(2), 307-320.
  • [18] Erden, S., Sarikaya, M. Z., and Budak, H. (2018). New weighted inequalities for higher order derivatives and applications. Filomat, 32(12), 4419-4433.
  • [19] Erden, S., Budak, H., Sarikaya, M. Z., Iftikhar, S. and Kumam, P. (2020). Fractional Ostrowski type inequalities for bounded functions. Journal of Inequalities and Applications, 123, 1-11.
  • [20] Erden, S., Budak, H., and Sarikaya, M. Z. (2020). Fractional Ostrowski type inequalities for functions of bounded variaton with two variables. Miskolc Mathematical Notes, 21(1), 171-188.
  • [21] Erden, S. and Baskir, B. M. (2021). Improved results of perturbed inequalities for higher-order differentiable functions and their various applications. Filomat, 35(10), 3475-3490.
  • [22] Farid, G. (2017). Some new Ostrowski type inequalities via fractional integrals. International Journal of Analysis and Applications, 14(1), 64-68.
  • [23] Fink, M. A. (1992). Bounds on the deviation of a function from its averages, Czechoslovak Mathematical Journal, 42 (117): 289-310.
  • [24] Gorenflo, R. and Mainardi, F. (1997). Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien, 223-276.
  • [25] Hanna, G., Dragomir, S. S. and Cerone, P. (2002). A general Ostrowski type inequality for double integrals, Tamkang Journal of Mathematics, 33 (4), 319-333
  • [26] Kashif, A. R., Shoaib, M. and Latif, M. A. (2016). Improved version of perturbed Ostrowski type inequalities for n-times differentiable mappings with three-step kernel and its application. J. Nonlinear Sci. Appl, 9, 3319-3332.
  • [27] Lakoud, A. G. and Aissaoui, F. (2013). New fractional inequalities of Ostrowski type, Transylv. J. Math. Mech., 5(2), 103-106
  • [28] Latif M.A. and Hussain, S. (2012). New inequalities of Ostrowski type for co-ordinated convex functions via fractional integrals, J Fractional Calc Appl. 2(9):1–15.
  • [29] Ostrowski, A.M. (1938). U¨ ber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10, 226-227.
  • [30] Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego.
  • [31] Qayyum, A., Shoaib, M. and Erden, S. (2019). Generalized fractional Ostrowski type inequality for higher order derivatives, New Trends in Mathematical Sciences (NTMSCI), 4 (2), 111-124.
  • [32] Qayyum, A.; Shoaib, M. and Faye, I. (2016). On new refinements and applications of efficient quadrature rules using n-times differentiable mappings, RGMIA Research Report Collection, 19, Article 9, 22 pp.
  • [33] Sarikaya, M. Z., Set, E., Yaldiz, H., and Bas¸ak, N. (2013). Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Mathematical and Computer Modelling, 57(9-10), 2403-2407.
  • [34] Sarıkaya, M. Z. (2014). On the Hermite–Hadamard-type inequalities for co-ordinated convex function via fractional integrals. Integral Transforms and Special Functions, 25(2), 134-147.
  • [35] Sarikaya, M. Z. and Filiz, H. (2014). Note on the Ostrowski type inequalities for fractional integrals. Vietnam Journal of Mathematics, 42, 187-190.
  • [36] Sarıkaya, M. Z. (2023). On the generalized Ostrowski type inequalities for co-ordinated convex functions. Filomat, 37(22), 7351-7366.
  • [37] Sofo, A. (2002). Integral inequalities for n- times differentiable mappings, with multiple branches,on the Lp norm, Soochow Journal of Mathematics, 28 (2), 179-221.
  • [38] Ujevi´c, N. (2003). Ostrowski-Gr¨uss type inequalities in two dimensional, J. of Ineq. in Pure and Appl. Math., 4 (5), article 101.
  • [39] Wang M. and Zhao, X. (2009). Ostrowski type inequalities for higher-order derivatives, J. of Inequalities and App., Vol. 2009, Article ID 162689, 8 pages

Year 2024, Volume: 12 Issue: 1, 62 - 73, 30.04.2024

Samet Erden Burçin Gökkurt Özdemir Sevgi Kılıçer Canmert Demır

Abstract

References

  • [1] Agli´c Aljinovi´c, A. (2014). Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral. Journal of Mathematics, Article ID 503195, 6 pages.
  • [2] Anastassiou, G. (1995). Ostrowski type inequalities. Proc. of the American Math. Soc., 123 (12), 3775-378.
  • [3] Barnett, N.S., & Dragomir, S.S. (2001). An Ostrowski type inequality for double integrals and applications for cubature formulae. Soochow J. Math., 27(1), 1-10.
  • [4] Budak H. and Sarikaya, M.Z.(2016). On Ostrowski type inequalities for functions of two variables with bounded variation, International Journal of Analysis and Applications, 12 (2), 142-156.
  • [5] Budak, H.; Sarikaya, M.Z. and Erden, S. (2016). New weighted Ostrowski type inequalities for mappings whose nth derivatives are of bounded variation, International J. of Analysis and App., 12 (1), 71-79.
  • [6] Budak, H. and Sarikaya, M.Z. (2016). A new Ostrowski type inequality for functions whose first derivatives are of bounded variation, Moroccan J. Pure Appl. Anal., 2(1); 1–11.
  • [7] Cerone, P.; Dragomir, S.S. and Roumeliotis, (1999). Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32 (4), 697-712.
  • [8] Changjian, Z. and Cheung, W.S. (2010). On Ostrowski-type inequalities for heigher-order partial derivatives. Journal of Ineqaulities and Applications, 1-8.
  • [9] Clarkson, J.A., Adams, C.R. (1933). On definitions of bounded variation for functions of two variables, Bull. Amer. Math. Soc. V.35 pp.824-854.
  • [10] Dragomir, S.S. (2001). On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Mathematical Inequalities & Applications, 4 (1), 59–66.
  • [11] Dragomir, S.S. and Wang, S. (1997). A new inequality of Ostrowski’s type in L1-norm and applications to some special means and to some numerical quadrature rules. Tamkang J. of Math., 28(3), 239-244.
  • [12] Dragomir, S. S. and Wang, A. (1998). A new inequality of Ostrowski’s type in Lp-norm and applications to some special means and to some numerical quadrature rules. Indian Journal of Mathematics, 40(3), 299-304.
  • [13] Dragomir, S. S., Barnett, N. S. and Cerone, P. (2003). An Ostrowski type inequality for double integrals in terms of Lp-norms and applications in numerical integration. Anal. Num. Theor. Approx., 32(2), 161-169.
  • [14] Dragomir, S. S. (2017). Ostrowski type inequalities for generalized Riemann Liouville fractional integrals of bounded variation. H¨older and Lipschitzian functions. RGMIA Research Report Collection, 20, Article 48, 1-14.
  • [15] Dragomir, S. S. (2017).Ostrowski Type inequalities for riemann-Liouville fractional integrals of absolutely continuous functions in terms of ¥􀀀norms. RGMIA Research Report Collection, 20, Article 49, 1-14.
  • [16] Dragomir, S. S. (2017). Ostrowski Type inequalities for riemann-Liouville fractional integrals of absolutely continuous functions in terms of p􀀀norms. RGMIA Research Report Collection, 20, Article 50.
  • [17] Dragomir, S. S. (2020). Ostrowski and trapezoid type inequalities for Riemann-Liouville fractional integrals of absolutely continuous functions with bounded derivatives. Fractional Differential Calculus, 10(2), 307-320.
  • [18] Erden, S., Sarikaya, M. Z., and Budak, H. (2018). New weighted inequalities for higher order derivatives and applications. Filomat, 32(12), 4419-4433.
  • [19] Erden, S., Budak, H., Sarikaya, M. Z., Iftikhar, S. and Kumam, P. (2020). Fractional Ostrowski type inequalities for bounded functions. Journal of Inequalities and Applications, 123, 1-11.
  • [20] Erden, S., Budak, H., and Sarikaya, M. Z. (2020). Fractional Ostrowski type inequalities for functions of bounded variaton with two variables. Miskolc Mathematical Notes, 21(1), 171-188.
  • [21] Erden, S. and Baskir, B. M. (2021). Improved results of perturbed inequalities for higher-order differentiable functions and their various applications. Filomat, 35(10), 3475-3490.
  • [22] Farid, G. (2017). Some new Ostrowski type inequalities via fractional integrals. International Journal of Analysis and Applications, 14(1), 64-68.
  • [23] Fink, M. A. (1992). Bounds on the deviation of a function from its averages, Czechoslovak Mathematical Journal, 42 (117): 289-310.
  • [24] Gorenflo, R. and Mainardi, F. (1997). Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien, 223-276.
  • [25] Hanna, G., Dragomir, S. S. and Cerone, P. (2002). A general Ostrowski type inequality for double integrals, Tamkang Journal of Mathematics, 33 (4), 319-333
  • [26] Kashif, A. R., Shoaib, M. and Latif, M. A. (2016). Improved version of perturbed Ostrowski type inequalities for n-times differentiable mappings with three-step kernel and its application. J. Nonlinear Sci. Appl, 9, 3319-3332.
  • [27] Lakoud, A. G. and Aissaoui, F. (2013). New fractional inequalities of Ostrowski type, Transylv. J. Math. Mech., 5(2), 103-106
  • [28] Latif M.A. and Hussain, S. (2012). New inequalities of Ostrowski type for co-ordinated convex functions via fractional integrals, J Fractional Calc Appl. 2(9):1–15.
  • [29] Ostrowski, A.M. (1938). U¨ ber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10, 226-227.
  • [30] Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego.
  • [31] Qayyum, A., Shoaib, M. and Erden, S. (2019). Generalized fractional Ostrowski type inequality for higher order derivatives, New Trends in Mathematical Sciences (NTMSCI), 4 (2), 111-124.
  • [32] Qayyum, A.; Shoaib, M. and Faye, I. (2016). On new refinements and applications of efficient quadrature rules using n-times differentiable mappings, RGMIA Research Report Collection, 19, Article 9, 22 pp.
  • [33] Sarikaya, M. Z., Set, E., Yaldiz, H., and Bas¸ak, N. (2013). Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Mathematical and Computer Modelling, 57(9-10), 2403-2407.
  • [34] Sarıkaya, M. Z. (2014). On the Hermite–Hadamard-type inequalities for co-ordinated convex function via fractional integrals. Integral Transforms and Special Functions, 25(2), 134-147.
  • [35] Sarikaya, M. Z. and Filiz, H. (2014). Note on the Ostrowski type inequalities for fractional integrals. Vietnam Journal of Mathematics, 42, 187-190.
  • [36] Sarıkaya, M. Z. (2023). On the generalized Ostrowski type inequalities for co-ordinated convex functions. Filomat, 37(22), 7351-7366.
  • [37] Sofo, A. (2002). Integral inequalities for n- times differentiable mappings, with multiple branches,on the Lp norm, Soochow Journal of Mathematics, 28 (2), 179-221.
  • [38] Ujevi´c, N. (2003). Ostrowski-Gr¨uss type inequalities in two dimensional, J. of Ineq. in Pure and Appl. Math., 4 (5), article 101.
  • [39] Wang M. and Zhao, X. (2009). Ostrowski type inequalities for higher-order derivatives, J. of Inequalities and App., Vol. 2009, Article ID 162689, 8 pages
There are 39 citations in total.

Details

Primary Language English
Subjects Approximation Theory and Asymptotic Methods, Applied Mathematics (Other)
Journal Section Articles
Authors

Samet Erden

Burçin Gökkurt Özdemir

Sevgi Kılıçer 0009-0009-4013-9115

Canmert Demır

Early Pub Date April 29, 2024
Publication Date April 30, 2024
Submission Date December 4, 2023
Acceptance Date December 14, 2023
Published in Issue Year 2024 Volume: 12 Issue: 1

Cite

APA Erden, S., Gökkurt Özdemir, B., Kılıçer, S., Demır, C. (2024). Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions. Konuralp Journal of Mathematics, 12(1), 62-73.
AMA Erden S, Gökkurt Özdemir B, Kılıçer S, Demır C. Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions. Konuralp J. Math. April 2024;12(1):62-73.
Chicago Erden, Samet, Burçin Gökkurt Özdemir, Sevgi Kılıçer, and Canmert Demır. “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”. Konuralp Journal of Mathematics 12, no. 1 (April 2024): 62-73.
EndNote Erden S, Gökkurt Özdemir B, Kılıçer S, Demır C (April 1, 2024) Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions. Konuralp Journal of Mathematics 12 1 62–73.
IEEE S. Erden, B. Gökkurt Özdemir, S. Kılıçer, and C. Demır, “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”, Konuralp J. Math., vol. 12, no. 1, pp. 62–73, 2024.
ISNAD Erden, Samet et al. “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”. Konuralp Journal of Mathematics 12/1 (April 2024), 62-73.
JAMA Erden S, Gökkurt Özdemir B, Kılıçer S, Demır C. Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions. Konuralp J. Math. 2024;12:62–73.
MLA Erden, Samet et al. “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”. Konuralp Journal of Mathematics, vol. 12, no. 1, 2024, pp. 62-73.
Vancouver Erden S, Gökkurt Özdemir B, Kılıçer S, Demır C. Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions. Konuralp J. Math. 2024;12(1):62-73.
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