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http://dx.doi.org/10.5831/HMJ.2021.43.4.587

SOME INTEGRAL INEQUALITIES IN THE FRAMEWORK OF GENERALIZED K-PROPORTIONAL FRACTIONAL INTEGRAL OPERATORS WITH GENERAL KERNEL  

Valdes, Juan E. Napoles (UNNE, FaCENA and UTN-FRRE)
Publication Information
Honam Mathematical Journal / v.43, no.4, 2021 , pp. 587-596 More about this Journal
Abstract
In this article, using the concept proposed reciently by the author, of a Generalized k-Proportional Fractional Integral Operators with General Kernel, new integral inequalities are obtained for convex functions. It is shown that several known results are particular cases of the proposed inequalities and in the end new directions of work are provided.
Keywords
Generalized k-Proportional Fractional Integral Operators with General Kernel; Fractional derivatives and integrals; Integral inequalities;
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1 R. Gorenflo, F. Mainardi, Fractals and fractional calculus in continuum mechanics, 223-276, Springer, Vienna, 1997.
2 D. Baleanu, COMMENTS ON: Ortigueira M., Martynyuk V., Fedula M., Machado J.A.T., The failure of certain fractional calculus operators in two physical models, in Fract. Calc. Appl. Anal. 22(2)(2019), Fract. Calc. Appl. Anal., Volume 23: Issue 1, DOI: https://doi.org/10.1515/fca-2020-0012.   DOI
3 D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics, 7(830)(2019).
4 S. Bermudo, P. Korus, Juan E. Napoles, On q-Hermite-Hadamard inequalities for general convex functions, Acta Math. Hungar. 162, 364-374 (2020).   DOI
5 V. L. Chinchane, D. B. Pachpatte, On new fractional integral inequalities involving convex functions using Hadamard fractional integral, Bull. Allahabad Math. Soc., 2016, 31(2), 183-192.
6 O. M. Duarte, Fractional Calculus for Scientists and Engineers, Dordrecht Heidelberg London New York: Springer, 2011.
7 J. D. Galeano, J. E. Napoles, E. Perez, A note on some integral inequalities in a generalized framework, Int. J. Appl. Math. Stat.; Vol. 60; Issue No. 1; Year 2021, 45-52.
8 J. D. Galeano, J. E. Napoles, E. Perez, Concerning to the generalized Hermite-Hadamard integral inequality, submitted.
9 Z. Dahmani, A note on some new fractional results involving convex functions, Acta Math. Univ. Comenianae, 81(2)(2012), 241-246.
10 J. D. Galeano, J. E. Napoles, E. Perez, On a general formulation of the fractional operator Riemann-Liouville and related inequalities, submitted.
11 S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, Y. M. Chu, Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function, Mathematics 2020, 7, 1225; doi:10.3390/math7121225   DOI
12 J. Hadamard, Essai sur l'etude des fonctions donnees par leur developpement de Taylor, J. Math. Pures Appl. (4) 8, 101-186 (1892)
13 I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in science and engineering; v. 198, San Diego: Academic Press, 1999.
14 M. A. Ali, J. E. Napoles, A. Kashuri, Z. Zhang, Fractional non conformable Hermite-Hadamard inequalities for generalized ϕ-convex functions, Fasciculi Mathematici, Nr 64 2020, 5-16 DOI: 10.21008/j.0044-4413.2020.0007   DOI
15 G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Diff. Eqs. 2019, 2019, Article ID 454, 10 pages.
16 S. Rashid, Z. Hammouch, F. Jarad, Y. M. Chu, New Estimates of Integral Inequalities via Generalized Proportional Fractional Integral Operator with Respect to Another Function, Fractals, doi: 10.1142/S0218348X20400277   DOI
17 B. Ahmad, A. Alsaedi, M. Kirane, B. T. Toberek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals, ArXiv: 1701.00092   DOI
18 A. Atangana, Derivative with a New Parameter Theory, Methods and Applications, Academic Press, 2016.
19 J. E. Napoles Valdes, J. M. Rodriguez, J. M. Sigarreta, New Hermite-Hadamard Type Inequalities Involving Non-Conformable Integral Operators, Symmetry 2019, 11, 1108; doi:10.3390/sym11091108   DOI
20 K. Oldham, J. Spanier, Applications of Differentiation and Integration to Arbitrary Order, Volume 111, Elsevier Science, 1974.
21 V. E. Tarasov, Fractional Dynamics; Applications of the Fractional Calculus to Dynamics of Particles, fields and Media, Dordrecht Heidelberg London New York: Springer, 2010.
22 W. J. Liu, Q. A. Ngo, V. N. Huy, Several interesting integral inequalities, Journal of Math. Inequal., 3(2) (2009), 201-212.
23 J. E. Napoles Valdes, F. Rabossi, A. D. Samaniego, Convex functions: Ariadne's thread or Charlotte's spiderweb?, Advanced Mathematical Models & Applications Vol.5, No.2, 2020, 176-191   DOI
24 H. U. Rehman, M. Darus, J. Salah, A Note on Caputo's Derivative Operator Interpretation in Economy, Journal of Applied Mathematics, 2018, Article ID 1260240, 7 pages https://doi.org/10.1155/2018/1260240   DOI
25 E. Kacar, Z. Kacar, H. Yildirim, Integral inequalities for Riemann-Liouville fractional integrals of a function with respect to another function, Iran. J. Math. Sci. Inform. 2018, 13, 1-13.   DOI
26 U. N. Katugampola, New Approach Generalized Fractional Integral, Applied Math and Comp. 218(2011),860-865.   DOI
27 A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integrals and Derivatives, Theory and Applications, Gordon & Breach, Switzerland (1993).
28 A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Netherlands, Elsevier, Febrary 2006.
29 P. Korus, L. M. Lugo, J. E. Napoles Valdes, Integral inequalities in a generalized context, Studia Scientiarum Mathematicarum Hungarica 57 (3), 312-320 (2020)   DOI
30 F. Mainardi, Fractional Calculus and Waves in Linear Viscoelsticity, Ed. Imperial College Press, 2010.
31 S. Mubeen, G. M. Habibullah, k-Fractional Integrals and Application, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 2, 89 - 94
32 J. E. Napoles V., A Generalized k-Proportional Fractional Integral Operators with General Kernel, submited
33 J. E. Napoles V., Hermite-Hadamard inequality in generalized context, VI COLLOQUIUM ON APPLIED MATHEMATICS and II INTERNATIONAL MEETING OF APPLIED MATHEMATICS, UNIMILITAR, BOGOTA, COLOMBIA, NOVEMBER 11-13, 2020
34 S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon & Breach Science, Yverdon, 1993.
35 H. Yildirim, Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat. 2014, 2, 322-329.
36 J. E. Napoles V., New generalized fractional integral inequalities of Hermite-Hadamard type for harmonically convex functions, XVI International Meeting of Mathematics, Barranquilla, Colombia NOVEMBER 17-20, 2020
37 J. Hadamard, Etude sur les proprietes des fonctions enti'eres et en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl. 58, 171-215 (1893)
38 C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, V. Feliu-Batle, Fractional Order Systems and Controls, Fundamentals and Applications, Londres: Springer-Verlag London Limited, 2010.
39 T. F. Nonnenmacher, R. Metzler, Applications of Fractional Calculus Ideas to Biology, World Scientific, 1998.
40 S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order-function and diffusion with changing modes, Z. Anal. Anwend. 28 (4) (2009) 431-450.