In this present study, firstly, some necessary definitions and some results related to Riemann-Liouville fractional and new fractional conformable integral operators defined by Jarad et al. ^{[13]} are given. As a second, a new identity has been proved. By using this identity, new Ostrowski type inequalities has obtained involving fractional conformable integral operators. Also, some new inequalities has established for AG-convex functions via fractional conformable integrals in this study. Relevant connections of the results presented here with those earlier ones are also pointed out.
Citation: Erhan Set, Ahmet Ocak Akdemir, Abdurrahman Gözpınar, Fahd Jarad. Ostrowski type inequalities via new fractional conformable integrals[J]. AIMS Mathematics, 2019, 4(6): 1684-1697. doi: 10.3934/math.2019.6.1684
In this present study, firstly, some necessary definitions and some results related to Riemann-Liouville fractional and new fractional conformable integral operators defined by Jarad et al. ^{[13]} are given. As a second, a new identity has been proved. By using this identity, new Ostrowski type inequalities has obtained involving fractional conformable integral operators. Also, some new inequalities has established for AG-convex functions via fractional conformable integrals in this study. Relevant connections of the results presented here with those earlier ones are also pointed out.
[1] | M. Alomari, M. Darus, S. S. Dragomir, et al. Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23 (2010), 1071-1076. doi: 10.1016/j.aml.2010.04.038 |
[2] | G. A. Anastassiou, Ostrowski type inequalities, Proc. Amer. Math. Soc., 123 (1995), 3775-3781. doi: 10.1090/S0002-9939-1995-1283537-3 |
[3] | G. Anastassiou, M. R. Hooshmandasl, A. Ghasemi, et al. Montogomery identities for fractional integrals and related fractional inequalities, Journal of Inequalities in Pure and Applied Mathematics, 10 (2009), 1-6. |
[4] | G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., 335 (2007), 1294-1308. doi: 10.1016/j.jmaa.2007.02.016 |
[5] | S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, Journal of Inequalities in Pure and Applied Mathematics, 10 (2009), 86. |
[6] | Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlin. Sci. Num., 9 (2010), 493-497. |
[7] | Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1 (2010), 51-58. doi: 10.15352/afa/1399900993 |
[8] | Z. Dahmani, L. Tabharit, S. Taf, Some fractional integral inequalities, Nonl. Sci. Lett. A, 1 (2010), 155-160. |
[9] | Z. Dahmani, L. Tabharit, S. Taf, New generalizations of Gruss inequality usin Riemann-Liouville fractional integrals, Bulletin of Mathematical Analysis and Applications, 2 (2010), 93-99. |
[10] | S. S. Dragomir, Ostrowski type inequalities for lebesque integral: A survey of recent results, Australian Journal of Mathematical Analysis and Applications, 14 (2017), 1-287. |
[11] | S. S. Dragomir, General Lebesgue integral inequalities of Jensen and Ostrowski type for differentiable functions whose derivatives in absolute value are h-convex and applications, Annales Universitatis Mariae Curie-Sklodowska, Sectio A-Mathematica, 69 (2015), 17-45. doi: 10.17951/a.2015.69.2.17-45 |
[12] | A. Gözpınar, Some Hermite-Hadamard Type Inequalities For Convex Functions Via New Fractional Conformable Integrals And Related Inequalities, AIP Conference Proceedings, 1991 (2018), 20006. |
[13] | F. Jarad, E. Uğurlu, T. Abdeljawad, et al. On a new class of fractional operators, Adv. Differ. Equ-NY, 2017 (2017), 247. |
[14] | A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. |
[15] | D. S. Mitrinoviç, J. E. Peèariæ, A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dortrecht, 1991. |
[16] | C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl., 3 (2000), 155-167. |
[17] | M. A. Noor, K. A. Noor, M. A. Awan, Fractional Ostrowski inequalities for (s,m)-Godunova-Levin functions, Facta Universitatis, Series: Mathematics and Informatics, 30 (2015), 489-499. |
[18] | A. M. Ostrowski, Über die absolutabweichung einer differentierbaren Funktion von ihren Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227. |
[19] | M. E. Özdemir, H. Kavurmaci, E. Set, Ostrowski's type inequalities for (α, m)-convex functions, Kyungpook Mathematical Journal, 50 (2010), 371-378. doi: 10.5666/KMJ.2010.50.3.371 |
[20] | M. E. Özdemir, A. O. Akdemir, E. Set, A new Ostrowski type inequality for double integrals, J. Inequal. Spec. Funct., 2 (2011), 27-34. |
[21] | M. E. Özdemir, A. O. Akdemir, E. Set, On the Ostrowski-Grüss type inequality for twice differentiable functions, Hacet. J. Math. Stat., 41 (2012), 651-655. |
[22] | M. Z. Sarıkaya, E. Set, On new Ostrowski type integral inequalities, Thai J. Math., 12 (2014), 145-154. |
[23] | M. Z. Sarıkaya, E. Set, M. E. Özdemir, Some Ostrowski's type inequalities for functions whose second derivatives are s-convex in the second sense, Demonstratio Mathematica, 47 (2014), 37-47. |
[24] | E. Set, J. Choi, A. Gözpınar, Hermite-Hadamard type inequalities for new fractional conformable integral operators, 2018. Available from: https://www.researchgate.net/publication/322936389. |
[25] | E. Set, A. Gözpınar, F. Demirci, Hermite-Hadamard type inequalities for quasi-convex functions via new fractional conformable integrals, AIP Conference Proceedings, 1991 (2018), 20002. |
[26] | E. Set, A. Karaoğlan, A. Gözpınar, Some inequalities related to different convex functions via new fractional conformable integrals, 2018. |
[27] | E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), 1147-1154. doi: 10.1016/j.camwa.2011.12.023 |
[28] | H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. |
[29] | A. Atangana and Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors?, Eur. Phys. J. Plus, 134 (2019), 429. |
[30] | F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20. doi: 10.1016/j.chaos.2018.10.006 |
[31] | M. A. Imran, M. Aleem, M. B. Riaz, et al. A comprehensive report on convective flow of fractional (ABC) and (CF) MHD viscous fluid subject to generalized boundary conditions, Chaos, Solitons and Fractals, 118 (2019), 274-289. doi: 10.1016/j.chaos.2018.12.001 |
[32] | N. A. Asif, Z. Hammouch, M. B. Riaz, et al. Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, Eur. Phys. J. Plus, 133 (2018), 272. |
[33] | M. B. Riaz and A. Zafar, Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo-Fabrizio derivatives, Math. Model. Nat. Pheno., 13 (2018), 8. |
[34] | M. B. Riaz, N. A. Asif, A. Atangana, et al. Couette flows of a viscous fluid with slip effect and non-integer order derivative without singular kernel, Discrete and Continuous Dynamical Systems Series-S, 12 (2019), 645-664. doi: 10.3934/dcdss.2019041 |
[35] | H. Yepez-Martinez and J. F. Gomez-Aguilar, Optical solitons solution of resonance nonlinear Schrodinger type equation with Atangana's-conformable derivative using sub-equation method, Waves in Random and Complex Media, (2019), 1-24. |
[36] | F. Gomez and B. Ghanbari, Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity, Rev. Mex. Fis., 65 (2018), 73-81. |
[37] | V. F. Morales-Delgado, J. F. Gomez-Aguilar, R. F. Escobar-Jimenez, Fractional conformable attractors with low fractality, Math. Method. Appl. Sci., 41 (2018), 6378-6400. doi: 10.1002/mma.5146 |
[38] | V. F. Morales-Delgado, J. F. Gomez-Aguilar, R. F. Escobar-Jimenez, et al. Fractional conformable derivatives of Liouville-Caputo type with low-fractionality, Physica A: Statistical Mechanics and its Applications, 503 (2018), 424-438. doi: 10.1016/j.physa.2018.03.018 |
[39] | J. E. S. Perez, J. F. Gomez-Aguilar, D. Baleanu, et al. Chaotic Attractors with Fractional Conformable Derivatives in the Liouville-Caputo Sense and Its Dynamical Behaviors, Entropy, 20 (2018), 384. |
[40] | H. Yepez-Martinez and J. F. Gomez-Aguilar, Fractional sub-equation method for Hirota-Satsumacoupled KdV equation and coupled mKdV equation using the Atangana's conformable derivative, Waves in Random and Complex Media, 29 (2019), 678-693. doi: 10.1080/17455030.2018.1464233 |
[41] | H. Yepez-Martinez, J. F. Gomez-Aguilar and A. Atangana, First integral method for non-linear differential equations with conformable derivative, Math. Model. Nat. Pheno., 13 (2018), 14. |
[42] | V. F. Morales-Delgado, J. F. Gomez-Aguilar and M. A. Taneco-Hernandez, Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense, AEU-Int. J. Electron. C., 85 (2018), 108-117. doi: 10.1016/j.aeue.2017.12.031 |