Abstract
In this paper, we introduce some new concepts to the field of probability theory: \(\left( k,s\right) \)-Riemann–Liouville fractional expectation and variance functions. Some generalized integral inequalities are established for \(\left( k,s\right) \)-Riemann–Liouville expectation and variance functions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Agarwal, R.; Barnett, N.; Cerone, P.; Dragomir, S.: A survey on some inequalities for expectation and variance. Comput. Math. Appl. 49, 429–480 (2005)
Anastassiou, G.A.; Hooshmandasl, M.R.; Ghasemi, A.; Moftakharzadeh, F.: Montogomery identities for fractional inregrals and related fractional inequalities. J. Inequal. Pure Appl. Math. 10(4), 1–6 (2009)
Anastassiou, G.A.: Fractional Differentiation Inequalities. Springer, New York (2009)
Barnett, N.S.; Cerone, P.; Dragomir, S.S.; Roumeliotis, J.: Some inequalities for the dispersion of a random variable whose p.d.f. is defined on a finite interval. J. Inequal Pure Appl. Math. 2(1), 1–29 (2001)
Barnett, N.; Dragomir, S.; Agarwal, R.: Some inequalities for probability, expectation and variance of random variables defined over a finite interval. Comput. Math. Appl. 43, 1319–1357 (2002)
Barnett, N.; Dragomir, S.: Some further inequalities for univariate moments and some new ones for the covariance. Comput. Math. Appl. 47, 23–36 (2004)
Dahmani, Z.: Fractional integral inequalities for continuous random variables. Malaya J. Mat. 2(2), 172–179 (2014)
Diaz, R.; Pariguan, E.: On hypergeometric functions and Pochhammer k-symbol. Divulg. Mat. 15, 179–192 (2007)
Gavrea, B.: A Hermite–Hadamard type inequality with applications to the estimation of moments of continuous random variables. Appl. Math. Comput. 254, 92–98 (2015)
Hadamard, J.: Essai sur l’etude des fonctions donnees par leur developpement de Taylor. J. Pure Appl. Math. 4(8), 101–186 (1892)
Hwang, D.Y.: Essai sur l’etude des fonctions donnees par leur developpement de Taylor. Appl. Math. Comput. 217, 9598–9605 (2011)
Hwang, D.Y.: Some inequalities for differentiable convex mapping with application to weighted midpoint formula and higher moments of random variables. Appl. Math. Comput. 232, 68–75 (2014)
Katugompola, U.N.: New approach generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)
Kumar, P.: Inequalities involving moments of a continuous random variable defined over a finite interval. Comput. Math. Appl. 48, 257–273 (2004)
Kumar, P.: The Ostrowski type moment integral inequalities and moment-bounds for continuous random variables. Comput. Math. Appl. 49, 1929–1940 (2005)
Miller, S.; Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, USA (1993)
Mubeen, S.; Habibullah, G.M.: \(k\)-Fractional integrals and application. Int. J. Contemp. Math. Sci. 7, 89–94 (2012)
Niezgoda, M.: New bounds for moments of continuous random variables. Comput. Math. Appl. 60, 3130–3138 (2010)
Romero, L.G.; Luque, L.L.; Dorrego, G.A.; Cerutti, R.A.: On the k-Riemann–Liouville fractional derivative. Int. J. Contemp. Math. Sci. 8(1), 41–51 (2013)
Sarıkaya, M.Z.; Dahmani, Z.; Kiriş, M.E.; Ahmad, F.: \((k, s)\)-Riemann–Liouville fractional integral and applications. Hacet. J. Math. Stat. 45(1), 77–89 (2016)
Set, E.; Sarikaya, M.Z.; Tomar, M.: On weighted Grüss type inequalities via \((k,s)\)-fractional integrals (submitted)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Tomar, M., Maden, S. & Set, E. \({\varvec{(k,s)}}\)-Riemann–Liouville fractional integral inequalities for continuous random variables. Arab. J. Math. 6, 55–63 (2017). https://doi.org/10.1007/s40065-016-0158-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-016-0158-9