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

Yıl 2024, Cilt: 12 Sayı: 1, 62 - 73, 30.04.2024

Samet Erden Burçin Gökkurt Özdemir Sevgi Kılıçer Canmert Demır

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • [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

Yıl 2024, Cilt: 12 Sayı: 1, 62 - 73, 30.04.2024

Samet Erden Burçin Gökkurt Özdemir Sevgi Kılıçer Canmert Demır

Öz

Kaynakça

  • [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
Toplam 39 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Yaklaşım Teorisi ve Asimptotik Yöntemler, Uygulamalı Matematik (Diğer)
Bölüm Articles
Yazarlar

Samet Erden

Burçin Gökkurt Özdemir

Sevgi Kılıçer 0009-0009-4013-9115

Canmert Demır

Erken Görünüm Tarihi 29 Nisan 2024
Yayımlanma Tarihi 30 Nisan 2024
Gönderilme Tarihi 4 Aralık 2023
Kabul Tarihi 14 Aralık 2023
Yayımlandığı Sayı Yıl 2024 Cilt: 12 Sayı: 1

Kaynak Göster

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. Nisan 2024;12(1):62-73.
Chicago Erden, Samet, Burçin Gökkurt Özdemir, Sevgi Kılıçer, ve Canmert Demır. “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”. Konuralp Journal of Mathematics 12, sy. 1 (Nisan 2024): 62-73.
EndNote Erden S, Gökkurt Özdemir B, Kılıçer S, Demır C (01 Nisan 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, ve C. Demır, “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”, Konuralp J. Math., c. 12, sy. 1, ss. 62–73, 2024.
ISNAD Erden, Samet vd. “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”. Konuralp Journal of Mathematics 12/1 (Nisan 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 vd. “Ostrowski Type Inequalities Including Riemann-Liouville Fractional Integrals for Two Variable Functions”. Konuralp Journal of Mathematics, c. 12, sy. 1, 2024, ss. 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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