Some Fractional Integral Inequalities for Stochastic Processes whose First and Second Derivatives are Quasi-Convex
Abstract
This work contains some Riemman-Liouville fractional integral inequalities of Hermite-Hadamard and Ostrowski type involving first and second derivatives of stochastic processes that are quasi convex. Also, from the attained results are deduced similar inequalities corresponding to the integral of Riemman.Visitas al artículo
Downloads
References
bibitem{Alo2009} M. Alomari and M. Darus. textit{On The Hadamard's Inequality for
Log-Convex Functions on the Coordinates}, J. Ineq. Appl., 2009 (2009), Art.
ID 283147, 13 pages
bibitem{Alo2010} M. Alomari, M. Darus , S.S. Dragomir and P. Cerone.
textit{Otrowski type inequalities for functions whose derivatives
are s-convex in the second sense}, Appl.Math. Lett., 23 (2010),
-1076
bibitem{Bai2009} A. Bain and D. Crisan. textit{Fundamentals of Stochastic Filtering.
Stochastic Modelling and Applied Probability}, 60., Springer, New York. 2009.
bibitem{Bar2012} A. Barani, S. Barani and S. S. Dragomir. textit{Refinements of
Hermite-Hadamard Inequalities for Functions When a Power of the Absolute
Value of the Second Derivative Is P-Convex}, J. Appl. Math. Vol. 2012,
Art. ID 615737, 10 pages.
bibitem{Dev2015} P. Devolder, J. Janssen and R. Manca. textit{Basic stochastic
processes}. Mathematics and Statistics Series. ISTE, London; John Wiley and
Sons, Inc. 2015.
bibitem{Fen1953} W. Fenchel. textit{Convex cones, sets, and functions}, Mimeo book, Princeton
Mathemathics Departament. Pricenton, New Jersey (1953).
bibitem{Fin1949} B. de Finetti. textit{Sulle stratificazioni convesse}, Ann. Mat. Pura Appl.,
(1949), 173-183
bibitem{GDD2015} M.E. Gordji, S.S.Dragomir, M.R. Delavar. textit{An inequality
related to $eta -$convex functions (II)}, Int. J. Nonlinear Anal. Appl. 6
(2015) No. 2, 27-33.
bibitem{Gom2018} J. Gomez and J. Hern'{a}ndez. textit{Hermite Hadamard type inequalities for Stochastic
Processes whose Second Derivatives are $left( m,h_{1},h_{2}right) -$Convex
using Riemann-Liouville Fractional Integral}, Revista Matua - Universidad del
Atl'{a}ntico, 5 (2018), no. 1, 13-28
bibitem{Gor1997} R. Gorenflo and F. Mainardi. textit{Fractional Calculus: Integral
and Differential Equations of Fractional Order}, Springer Verlag, Wien, (1997)
-276.
bibitem{Green1971} H. Greenberg and W. Pierskalla. textit{A Review of Quasi Convex Functions},
Reprinted from Operations Research, 19 (1971), no. 7, 1553-1569.
bibitem{Guerra2004} A. Guerraggio and E. Molho. Ttextit{he origins of quasi concavity: a development
between mathematics and economics}, Historia Matematica, 31 (2004), 62-75
bibitem{Kot2012} D. Kotrys. textit{Hermite-Hadamard inequality for convex
stochastic processes}, Aequationes Mathematicae, 83 (2012),143-151
bibitem{Kot2015} D. Kotrys and K. Nikodem. textit{Quasiconvex stochastic processes and a separation theorem},
Aequat. Math. 89 (2015), 41-48
bibitem{Liu2016} W. Liu, W. Wen and J. Park. textit{Hermite-Hadamard type
inequalities for MT-convex functions via classical integrals and fractional
integrals}, J. Nonlinear Sci. Appl., 9 (2016), 766-777
bibitem{Mik2010} T, Mikosch. textit{Elementary stochastic calculus with finance in
view}, Advanced Series on Statistical Science and Applied Probability, 6.
World Scientific Publishing Co., Inc.,2010.
bibitem{Mill1993} S. Miller and B. Ross. textit{An introduction to the Fractional
Calculus and Fractional Differential Equations}, John Wiley & Sons, USA,
(1993).
bibitem{Mitri1985} D.S. Mitrinovic and I.B. Lackovic. textit{Hermite and convexity},
Aequationes Math., 28 (1985), no. 1, 229-232.
bibitem{Nag1974} B. Nagy. textit{On a generalization of the Cauchy equation},
Aequationes Math., 11 (1974). 165--171.
bibitem{New1928} J. von Neumann. textit{Zur Theorie der Gesellschaftspiele}, Math. Ann.,
(1928), 295-320.
bibitem{Nick1980} K. Nikodem. textit{On convex stochastic processes}, Aequationes
Math., 20 (1980), no. 2-3, 184-197.
%bibitem{Ostrowski} Ostrowski A. textit{~AÂber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert}, Comment. Math. Helv., 10(1938), 226-227
bibitem{Pod1999} I. Podlubni. textit{Fractional Differential Equations}, Academic
Press, San Diego, (1999).
bibitem{Sar2013} M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak.
textit{Hermite-Hadamard's inequalities for fractional integrals and related
fractional inequalities}, Mathematical and Computer Modelling, 57 (2013),
-2407.
bibitem{Set2014} E. Set, M. Tomar and S. Maden. textit{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.
bibitem{Set2016} E. Set, A. Akdemir and N. Uygun. textit{On New Simpson Type
Inequalities for Generalized Quasi-Convex Mappings}, Xth International
Statistics Days Conference, 2016, Giresun, Turkey.
bibitem{Shak1985} M. Shaked and J. Shantikumar. textit{Stochastic Convexity and its
Applications}, Arizona Univ. Tuncson. 1985.
bibitem{Shy2013} J.J. Shynk. textit{Probability, Random Variables, and Random
Processes: Theory and Signal Processing Applications}, Wiley & Sons, 2013.
bibitem{Skr1992} A. Skowronski. textit{On some properties of $J-$convex stochastic
processes}, Aequationes Mathematicae, 44 (1992), 249-258.
bibitem{Skr1995} A. Skowronski. textit{On Wright-Convex Stochastic Processes}, Ann.
Math. Sil., 9 (1995), 29-32.
bibitem{Sobczyk} K. Sobczyk. textit{Stochastic Differential Equations with applications to Physics and Engineering}, Kluwer Academic Publishers, London, (1991)
bibitem{Varo2007} S. Varou{s}anec. textit{On h-convexity}, J. Math. Anal. Appl., 326
(2007), no. 1, 303-311.
bibitem{Vivas2017} M. Vivas-Cortez and J. Hern'{a}ndez. textit{On $ (m,h_{1},h_{2}) $-convex Stochastic Processes}, Appl. Math. Inf. Sci., 11 (2017), no. 3, 649-657
