Minkowski and Hölder type inequalities for a new generalized fractional integral
Keywords:
Fractional integral inequalities generalized fractional integral operator Desigualdades fraccionales integrales Operador integral fraccionario generalizado
Abstract
The present study is concerning about some inequalities of Minkoveski and Hölder type using a new generalized fractional integral operator of Raina's type. Using the Raina generalized function model, $ \mathcal{F}_{\rho,\lambda}^{\sigma} $, which involve certain parameters and a bounded sequence of positive real numbers, a new definition of a generalized fractional integral is given and some others classical fractional integral operators are deduced from this. Also the validity of the main results in the setting of Riemman--Liouville, Hadamard, Katugampola, Prabhakar and Salim fractional integrals is proved.
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References
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Annals of Functional Analysis 1(1) (2010), 51–58.
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Inequalities Involving Two Functions. Journal of Inequalities and Applications 2010:148102 (2010),
1–9.
[14] R. S. Dubey and P. Goswami, Some fractional integral inequalities for the Katugampola integral operator,
AIMS Mathematics 4(2) (2019), 193–198.
[15] A. Fernandez, D. Baleanu and H. M. Srivastava, Series representations for models of fractional calculus
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[16] R. Garra, R. Gorenflo, F. Polito and Z. Tomovski, Hilfer–Prabhakar derivative and some applications,
Appl. Math.Comput. 242 (2014), 576-?589.
[17] R. Garra, R. Garrappa, The Prabhakar or three parameter Mittag-Le er function: Theory and application,
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Fractional Integral Operator using convex functions, Rev. Mat. Teor. Apl. 26(1) (2019), 1–19.
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[23] U. N. Katugampola, A New Approach to Generalized Fractional Derivatives, Bull. Math. Anal. Appl.,
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Equations, North-Holland Mathematics Studies Vol 204., Elsevier, Amsterdam, The Netherlands,
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Wiley, New York, 1993.
[26] P. O. Mohammed and T. Abdeljawad, Integral inequalities for a fractional operator of a function with
respect to another function with nonsingular kernel, Advances in Di erence Equations 2020 (363)
(2020), 1–19
[27] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, San Diego, 1974.
[28] T. R. Prabhakar, A Singular Integral Equation With a Generalized Mittag-Le er function in the kernel,
Yokohama Math. J. 19 (1971), 7–15
[29] F. Polito and ˘ Z. Tomovski, Some properties of Prabhakar-type fractional calculus operators, Fractional
Di erential Calculus 6(1) (2016), 73–94.
[30] X. Qiang, G. Farid and J. Pe˘cari c, Generalized fractional integral inequalities for exponentially
(s;m) convex functions , Journal of inequalities and applications 2020:70 (2020), 1–18.
[31] R. K. Raina, On Generalized Wright’s Hypergeometric functions and Fractional Calculus Operator,
East Asian Mathematical Journal 21(2) (2005), 191–203.
[32] T.O. Salim and A. W. Faraj, A generalization of Mittag-Le er function and integral operator associated
with fractional calculus, J. Fract. Calc. Appl. 3(2012), 1–13.
[33] T. Sandev, Generalized Langevin Equation and the Prabhakar Derivative, Mathematics 5 (2017), 66-76.
[34] S. G.Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications,
Taylor & Francis, London, 2002.
[35] J. V.Sousa and E. C. Oliveira, The Minkowski?s inequality by means of a generalized fractional integral,
AIMS Mathematics 3(1) (2018), 131?-147
[36] T. Tunu uc , H. Budak, F. Usta and M.Z. Sarikaya, On new generalized fractional integral operators
and related inequalities, Konuralp Journal of Mathematics 8(2) (2020), 268–278.
[37] M. Vivas-Cortez, A. Kashuri and J. E. Hern andez Hern andez, Trapezium-Type Inequalities for Raina’s
Fractional Integrals Operator Using Generalized Convex Functions, Symmetry 12 (2020), 1–17.
91
frame of di erent type kernels, Chaos Solitons Fractals 130:109438 (2020), 1–19
[2] R. P. Agarwal, M-J. Luo and R. K. Raina, On Ostrowski type inequalities, Fasciculli Mathematica 56
(2016), 5–27.
[3] A. Alaria, A. M. Khan, D. L. Suthar and D. Kumar, Application of fractional operators in modelling
for charge carrier transport in amorphous semiconductor with multiple trapping, International Journal
of Applied and Computational Mathematics 50(6) (2019), 1–10
[4] A. Atangana, Derivative with a new parameter. Theory,Methods and Applications, Academic Press
Elsevier, London, 2016.
[5] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel:
theory and applications to heat transfer model, Therm. Sci. 20 (2016), 763-?769.
[6] V. Berdnikov, and V. Lokhin, Synthesis of guaranteed stability regions of a nonstationary nonlinear
system with a fuzzy controller, Civil Engineering Journal 50(1) (2019), 107–116
[7] L. Bougo a, On Minkowski and Hardy Integral Inequalities. Jounal of Inequalities in Pure and Applied
Mathematics 7(2) (2006), 1–15.
[8] Z. Chen, G. Farid, A. U. Rehman and N. Latif, Estimations of fractional integral operators for convex
functions and related results, Advances in Di erence equations 2020:163 (2020, 1–18.
[9] H. Chen and U.N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fej er type Inequalities
for Generalized Fractional Integrals, J. Math. Anal. Appl. 446 (2017), 1274–1291
[10] S-B. Chen, S. Rashid, Z. Hammouch, M. A. Noor, R. Ashraf and Y-M. Chu, Integral inequalities via
Raina?s fractional integrals operator with respect to a monotone function, Adv. Di er. Equ. 2020:647
(2020), 1–20
[11] J. Choi and P. Agarwal, Certain Fractional Integral Inequalities Involving Hypergeometric Operators,
East Asian Math. J. 30(3) (2014), 283–291.
[12] Z. Dahmani, On Minkowski and Hermite-Hadamard Integral Inequalities Via Fractional Integration.
Annals of Functional Analysis 1(1) (2010), 51–58.
[13] S.S. Dragomir, E. Set and M.E. O zdemir, On the Hermite-Hadamard Inequality and Other Integral
Inequalities Involving Two Functions. Journal of Inequalities and Applications 2010:148102 (2010),
1–9.
[14] R. S. Dubey and P. Goswami, Some fractional integral inequalities for the Katugampola integral operator,
AIMS Mathematics 4(2) (2019), 193–198.
[15] A. Fernandez, D. Baleanu and H. M. Srivastava, Series representations for models of fractional calculus
involving generalised Mittag–Le er functions, Commun. Nonlinear Sci. Numer. Simul. 67 (2019),
517-?527.
[16] R. Garra, R. Gorenflo, F. Polito and Z. Tomovski, Hilfer–Prabhakar derivative and some applications,
Appl. Math.Comput. 242 (2014), 576-?589.
[17] R. Garra, R. Garrappa, The Prabhakar or three parameter Mittag-Le er function: Theory and application,
Commun. Nonlinear Sci. Numer. Simul. 56 (2018), 314-?329.
[18] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin,Mittag-Le er functions, related topics and
applications, Springer, Berlin, 2014.
[19] J. Hadamard, Essai sur l etude des fonctions donnees par leur developpement de Taylor, Journal Math.
Pures et Appl. Ser 4 8 (1892), 101–186.
[20] J. E. Hern andez Hern andez and M. Vivas-Cortez, Hermite–Hadamard Inequalities type for Raina’s
Fractional Integral Operator using convex functions, Rev. Mat. Teor. Apl. 26(1) (2019), 1–19.
[21] T. U. Khan, M.A. Khan, Generalized conformable fractional integral operators, J. Comput. Appl.
Math. 346 (2019), 378?389.
[22] M. A. Khan and T. U. Khan, Generalized Conformable Fractional Operators and Their Applications,In
: Fractional Order Analysis: Theory, Methods and Applications (Chapter 2) , JohnWiley & Sons, Ltd.,
43–89, 2020.
[23] U. N. Katugampola, A New Approach to Generalized Fractional Derivatives, Bull. Math. Anal. Appl.,
6 (2014), 1–15.
[24] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Di erential
Equations, North-Holland Mathematics Studies Vol 204., Elsevier, Amsterdam, The Netherlands,
2006.
[25] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Di erential Equations,
Wiley, New York, 1993.
[26] P. O. Mohammed and T. Abdeljawad, Integral inequalities for a fractional operator of a function with
respect to another function with nonsingular kernel, Advances in Di erence Equations 2020 (363)
(2020), 1–19
[27] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, San Diego, 1974.
[28] T. R. Prabhakar, A Singular Integral Equation With a Generalized Mittag-Le er function in the kernel,
Yokohama Math. J. 19 (1971), 7–15
[29] F. Polito and ˘ Z. Tomovski, Some properties of Prabhakar-type fractional calculus operators, Fractional
Di erential Calculus 6(1) (2016), 73–94.
[30] X. Qiang, G. Farid and J. Pe˘cari c, Generalized fractional integral inequalities for exponentially
(s;m) convex functions , Journal of inequalities and applications 2020:70 (2020), 1–18.
[31] R. K. Raina, On Generalized Wright’s Hypergeometric functions and Fractional Calculus Operator,
East Asian Mathematical Journal 21(2) (2005), 191–203.
[32] T.O. Salim and A. W. Faraj, A generalization of Mittag-Le er function and integral operator associated
with fractional calculus, J. Fract. Calc. Appl. 3(2012), 1–13.
[33] T. Sandev, Generalized Langevin Equation and the Prabhakar Derivative, Mathematics 5 (2017), 66-76.
[34] S. G.Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications,
Taylor & Francis, London, 2002.
[35] J. V.Sousa and E. C. Oliveira, The Minkowski?s inequality by means of a generalized fractional integral,
AIMS Mathematics 3(1) (2018), 131?-147
[36] T. Tunu uc , H. Budak, F. Usta and M.Z. Sarikaya, On new generalized fractional integral operators
and related inequalities, Konuralp Journal of Mathematics 8(2) (2020), 268–278.
[37] M. Vivas-Cortez, A. Kashuri and J. E. Hern andez Hern andez, Trapezium-Type Inequalities for Raina’s
Fractional Integrals Operator Using Generalized Convex Functions, Symmetry 12 (2020), 1–17.
91
Published
2021-08-17
How to Cite
Vivas-Cortez, M., & Hernández Hernández, J. E. (2021). Minkowski and Hölder type inequalities for a new generalized fractional integral. Revista MATUA ISSN: 2389-7422, 8(1), 74-91. Retrieved from https://investigaciones.uniatlantico.edu.co/revistas/index.php/MATUA/article/view/3008
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