Artículo Original
Desigualdades de tipo Hermite-Hadamard, procesos estocásticos convexos y la integral fraccionaria de Katugampola
Publicado 2018-12-12
Palabras clave
- Desigualdad de Hermite-Hadamard,
- Procesos Estocásticos,
- Integral fraccionaria de Katugampola
Cómo citar
Hernández H., J. E., & Gómez, J. F. (2018). Desigualdades de tipo Hermite-Hadamard, procesos estocásticos convexos y la integral fraccionaria de Katugampola. Revista Integración, Temas De matemáticas, 36(2), 133–149. https://doi.org/10.18273/revint.v36n2-2018005
Resumen
En este trabajo se presentan algunas desigualdades de tipo Hermite-Hadamard para procesos estocásticos convexos usando la integral fraccional de Katugampola, y de estos resultados se deducen casos específicos para la integral fraccionaria de Riemann-Liouville y la integral de Riemann.
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Referencias
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[2] Alomari M. and Darus M., “On The Hadamard’s Inequality for Log-Convex Functions on the Coordinates”, J. Inequal. Appl. (2009), Article ID 283147, 13 pp.
[3] Alomari M., Darus M., Dragomir S.S. and Cerone P., “Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense”, Appl. Math. Lett. 23 (2010), No. 9, 1071–1076.
[4] Bain A. and Crisan D., Fundamentals of Stochastic Filtering, Springer-Verlag, New York, 2009.
[5] Chen H. and Katugampola U.N., “Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals”, J. Math. Anal. Appl. 446 (2017), No. 2, 1274–1291.
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[7] Dahmani Z., “New inequalities in fractional integrals”, Int. J. Nonlinear Sci. 9 (2010), No. 4, 493–497.
[8] Dahmani Z., Tabharit L. and Taf S., “New generalisations of Gruss inequality using Riemann-Liouville fractional integrals”, Bull. Math. Anal. Appl. 2 (2010), No. 3, 93–99.
[9] Devolder P., Janssen J. and Manca R., Basic stochastic processes. Mathematics and Statistics Series, ISTE, London, John Wiley and Sons, Inc., 2015.
[10] Gorenflo R. and Mainardi F., “Fractional Calculus: Integral and Differential Equations of Fractional Order”, in CISM Courses and Lect. 378, Springer, Vienna (1997), 223–276.
[11] Hadamard J., “Essai sur l’etude des fonctions donnees par leur developpment de Taylor”, J. Math. Pures Appl. 8 (1892), 101–186.
[12] Katugampola U.N., “New approach to a generalized fractional integral”, Appl. Math. Comput. 218 (2011), No. 3, 860–865.
[13] Katugampola U.N., “A new approach to generalized fractional derivatives”, Bull. Math. Anal. Appl. 6 (2014), No. 4, 1–15.
[14] Katugampola U.N., “Mellin transforms of generalized fractional integrals and derivatives”, Appl. Math. Comput. 257 (2015), 566–580.
[15] Kilbas A.A., Srivastava H.M. and Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier, Amsterdam (2006).
[16] Kotrys D., “Hermite-Hadamard inequality for convex stochastic processes”, Aequationes Math. 83 (2012), No. 1-2, 143–151.
[17] Kotrys D., “Remarks on strongly convex stochastic processes”, Aequationes Math. 86 (2013), No. 1-2, 91–98.
[18] Kumar P., “Hermite-Hadamard inequalities and their applications in estimating moments”, in Inequality Theory and Applications, Volume 2, (Edited by Y.J. Cho, J.K. Kim and S.S. Dragomir), Nova Science, (2003).
[19] Liu W., Wen W. and J. Park, “Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals”, J. Nonlinear Sci. Appl. 9 (2016), No. 3, 766–777.
[20] Mikosch T., Elementary stochastic calculus with finance in view, Advanced Series on Statistical Science and Applied Probability, World Scientific Publishing Co., Inc., 2010.
[21] Miller S. and Ross B., An introduction to the Fractional Calculus and Fractional Diferential Equations, John Wiley & Sons, USA, (1993).
[22] Mitrinovic D.S. and Lackovic I.B., “Hermite and convexity”, Aequationes Math., 28 (1985), No. 1, 229–232.
[23] Nagy B., “On a generalization of the Cauchy equation”, Aequationes Math. 10 (1974), No. 2-3, 165–171.
[24] Nikodem K., “On convex stochastic processes”, Aequationes Math. 20 (1980), No. 1, 184–197.
[25] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, (1999).
[26] Sarikaya M.Z., Set E., Yaldiz H. and Basak N., “Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities”, Math. Comput. Modelling 57 (2013), No. 9-10, 2403–2407.
[27] Set E., Tomar M. and Maden S., “Hermite Hadamard Type Inequalities for s−Convex Stochastic Processes in the Second Sense”, Turkish Journal of Analysis and Number Theory 2 (2014), No. 6, 202–207.
[28] Set E., Akdemir A. and Uygun N., “On New Simpson Type Inequalities for Generalized Quasi-Convex Mappings”, Xth International Statistics Days Conference, Giresun, Turkey, 571–581, 2016.
[29] Shaked M. and Shantikumar J. Stochastic Convexity and its Applications, Arizona Univ., Tucson, 1985.
[30] Shynk J.J., Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications, John Wiley and Sons, Inc., 2013.
[31] Skowronski A., “On some properties of J−convex stochastic processes”, Aequationes Math. 44 (1992), No. 2-3, 249–258.
[32] Skowronski A., “On Wright-Convex Stochastic Processes”, Ann. Math. Sil. 9 (1995), 29–32.
[33] Thaiprayoon Ch., Ntouyas S., Tariboon J., “On the nonlocal Katugampola fractional integral conditions for fractional Langevin equation”, Adv. Difference Equ. (2015) 2015:374, 16 pp.
[34] Tunç T., Budak H., Usta F. and Sarikaya M.Z, “On new generalized fractional integral operators and related inequalities”, Submitted article, ResearchGate. https://www.researchgate.net/publication/313650587 [12 November 2018].
[35] Youness E.A., “E-convex sets, E-convex functions, and Econvex programming”, J. Optim. Theory Appl. 102 (1999), No. 2, 439–450.
[2] Alomari M. and Darus M., “On The Hadamard’s Inequality for Log-Convex Functions on the Coordinates”, J. Inequal. Appl. (2009), Article ID 283147, 13 pp.
[3] Alomari M., Darus M., Dragomir S.S. and Cerone P., “Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense”, Appl. Math. Lett. 23 (2010), No. 9, 1071–1076.
[4] Bain A. and Crisan D., Fundamentals of Stochastic Filtering, Springer-Verlag, New York, 2009.
[5] Chen H. and Katugampola U.N., “Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals”, J. Math. Anal. Appl. 446 (2017), No. 2, 1274–1291.
[6] Cortés J.C., Jódar L. and Villafuerte L., “Numerical solution of random differential equations: A mean square approach”, Math. Comput. Modelling 45 (2007), No. 7-8, 757–765.
[7] Dahmani Z., “New inequalities in fractional integrals”, Int. J. Nonlinear Sci. 9 (2010), No. 4, 493–497.
[8] Dahmani Z., Tabharit L. and Taf S., “New generalisations of Gruss inequality using Riemann-Liouville fractional integrals”, Bull. Math. Anal. Appl. 2 (2010), No. 3, 93–99.
[9] Devolder P., Janssen J. and Manca R., Basic stochastic processes. Mathematics and Statistics Series, ISTE, London, John Wiley and Sons, Inc., 2015.
[10] Gorenflo R. and Mainardi F., “Fractional Calculus: Integral and Differential Equations of Fractional Order”, in CISM Courses and Lect. 378, Springer, Vienna (1997), 223–276.
[11] Hadamard J., “Essai sur l’etude des fonctions donnees par leur developpment de Taylor”, J. Math. Pures Appl. 8 (1892), 101–186.
[12] Katugampola U.N., “New approach to a generalized fractional integral”, Appl. Math. Comput. 218 (2011), No. 3, 860–865.
[13] Katugampola U.N., “A new approach to generalized fractional derivatives”, Bull. Math. Anal. Appl. 6 (2014), No. 4, 1–15.
[14] Katugampola U.N., “Mellin transforms of generalized fractional integrals and derivatives”, Appl. Math. Comput. 257 (2015), 566–580.
[15] Kilbas A.A., Srivastava H.M. and Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier, Amsterdam (2006).
[16] Kotrys D., “Hermite-Hadamard inequality for convex stochastic processes”, Aequationes Math. 83 (2012), No. 1-2, 143–151.
[17] Kotrys D., “Remarks on strongly convex stochastic processes”, Aequationes Math. 86 (2013), No. 1-2, 91–98.
[18] Kumar P., “Hermite-Hadamard inequalities and their applications in estimating moments”, in Inequality Theory and Applications, Volume 2, (Edited by Y.J. Cho, J.K. Kim and S.S. Dragomir), Nova Science, (2003).
[19] Liu W., Wen W. and J. Park, “Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals”, J. Nonlinear Sci. Appl. 9 (2016), No. 3, 766–777.
[20] Mikosch T., Elementary stochastic calculus with finance in view, Advanced Series on Statistical Science and Applied Probability, World Scientific Publishing Co., Inc., 2010.
[21] Miller S. and Ross B., An introduction to the Fractional Calculus and Fractional Diferential Equations, John Wiley & Sons, USA, (1993).
[22] Mitrinovic D.S. and Lackovic I.B., “Hermite and convexity”, Aequationes Math., 28 (1985), No. 1, 229–232.
[23] Nagy B., “On a generalization of the Cauchy equation”, Aequationes Math. 10 (1974), No. 2-3, 165–171.
[24] Nikodem K., “On convex stochastic processes”, Aequationes Math. 20 (1980), No. 1, 184–197.
[25] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, (1999).
[26] Sarikaya M.Z., Set E., Yaldiz H. and Basak N., “Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities”, Math. Comput. Modelling 57 (2013), No. 9-10, 2403–2407.
[27] Set E., Tomar M. and Maden S., “Hermite Hadamard Type Inequalities for s−Convex Stochastic Processes in the Second Sense”, Turkish Journal of Analysis and Number Theory 2 (2014), No. 6, 202–207.
[28] Set E., Akdemir A. and Uygun N., “On New Simpson Type Inequalities for Generalized Quasi-Convex Mappings”, Xth International Statistics Days Conference, Giresun, Turkey, 571–581, 2016.
[29] Shaked M. and Shantikumar J. Stochastic Convexity and its Applications, Arizona Univ., Tucson, 1985.
[30] Shynk J.J., Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications, John Wiley and Sons, Inc., 2013.
[31] Skowronski A., “On some properties of J−convex stochastic processes”, Aequationes Math. 44 (1992), No. 2-3, 249–258.
[32] Skowronski A., “On Wright-Convex Stochastic Processes”, Ann. Math. Sil. 9 (1995), 29–32.
[33] Thaiprayoon Ch., Ntouyas S., Tariboon J., “On the nonlocal Katugampola fractional integral conditions for fractional Langevin equation”, Adv. Difference Equ. (2015) 2015:374, 16 pp.
[34] Tunç T., Budak H., Usta F. and Sarikaya M.Z, “On new generalized fractional integral operators and related inequalities”, Submitted article, ResearchGate. https://www.researchgate.net/publication/313650587 [12 November 2018].
[35] Youness E.A., “E-convex sets, E-convex functions, and Econvex programming”, J. Optim. Theory Appl. 102 (1999), No. 2, 439–450.