Aug 9, 2017 - Fractional Integrals and Related Inequalities .... inequality for Riemann-Liouville fractional integral for stochastic processes and the same.
Description of the article: On Generalized Stochastic Fractional Integrals and Related Inequalities Jorge E. Hernández and Miguel Vivas August 9, 2017
1 Introduction This is a description and review of the article whose autors are HÜSEYIN BUDAK and MEHMET ZEKI SARIKAYA and entitled: ON GENERALIZED STOCHASTIC FRACTIONAL INTEGRALS AND RELATED INEQUALITIES.
2 Structure of the article 2.1
Abstract
The authors wrote: "In this paper, we rst introduce the generalized mean-square fractional integrals J ; ;a+;w and J ; ;b ;w of the stochastic process X. Then, we establish generalized fractional Hermite-Hadamard inequality for Jensen-convex and strongly convex stochastic proceses via generalized stochastic fractional integrals". 2.2
Introduction
First, the authors gave a theorical framework starting with the historical origin of convex stochastic processes, making mention to Nickodem [5] and Skrowronski [6]. Below are well-known de nitions and properties of stochastic processes: the de nition of stochastic processes, the continuity in probability and mean square continuity. Also gave the de nition of mean square integral of a stochastic processes and wrote: "For the existence of the mean-square integral it is enough to assume the mean-square continuity of the stochastic process X", and the monotonicity property of the same. Subsequently, gave the de nition of Convex Stochastic Processes referring to the articles of Nickodem [5] and Sobczyk [7]. They show some fundamentals found by Kotrys in [2], about Hermite Hadamard Inequality. Lemma 1 If X : I
! R is a Stochastic processes of the form X (t; ) = A ( ) t + B ( ) 1
where A; B : then
! R are random variables with E A2 < 1 , E B 2 < 1 and [a; b] Z
b
X (t; ) dt = A ( )
b2
a2 2
a
+ B ( ) (b
I,
a):
Proposition 2 Let X : I ! R be a convex stochastic processes and t0 2 int(I): Then there exist a random variable A : ! R such that X is supported in t0 by the stochastic processes A ( ) (t t0 ) + X (t0 ; ) : That is X (t; )
t0 ) + X (t0 ; ) a.e.
A ( ) (t
for all t 2 I: The following is the Hermite-Hadamard inequality for Jensen convex stochastic processes. Theorem 3 Let X : I ! R be Jensen-convex, mean-square continuous in the interval I , stochastic process. Then for any u; v 2 I , with u < v , we have Z v u+v 1 X (u; :) + X (v; :) X ;: ; a.e. X (t; ) dt 2 v u u 2 Also, referring to the article by Kotrys [3] gave a de nition of strongly convex stochastic processes, and referring to Ha z [4] gave the de nition of stochastic mean-square fractional integrals, also gave the Hermite-Hadamard version for this last kind of fractional integral in [1] and Ha z[4]. De nition 1 For the stochastic proces X : I ! R, the concept of stochastic meansquare fractional integrals Ia+ and Ib of X of order > 0 is de ned by Z t 1 1 Ia+ [X] (t) = (t s) X (s; ) ds ( ) a and Z b 1 1 (t s) X (s; ) ds: Ib [X] (t) = ( ) t Theorem 4 Let X : I ! R be a Jensen-convex stochastic process that is mean-square continuous in the interval I.Then for any u; v 2 I; u < v; the following Hermite-Hadamard inequality ( + 1) X (u; :) + X (v; :) u+v X ;: Iu+ [X] (v) + Iv [X] (u) 2 2 2 (v u) holds, where > 0:
2
2.3
Main Results.
In this section the authors started referring to the article of R.K. Raina to take the de nition of the function F ; , as a generalized special function. With this, gave a de nition of the generalized mean-square fractional integrals (de nition 5) De nition 2 Let X : I ! R be a stochastic process. The generalized mean-square fractional integrals operator J ; ;a+;w and J ; ;b ;w for stochastic processes are de ned by Z t 1 J ; ;a+;w [X] (t) = (t s) F ; [w (t s) ] X (s; ) ds; a.e. t > a a
and
J
; ;b+;w
[X] (t) =
Z
b
(s
t)
t
1
F
;
[w (s
t) ] X (s; ) ds; a.e. t < b:
As a rst result they proved the Hermite-Hadamard inequality version using generalized mean-square fractional integrals operator for convex stochastic processes. (Theorem 3) Theorem 5 Let X : I ! R be a Jensen-convex stochastic process that is mean-square continuous in the interval I: For every u; v 2 I; (u < v), we have the following Hermite– Hadamard inequality u+v 1 X J ; ;u+;w [X] (v) + J ; ;v ;w [X] (u) ;: 2 2 (v u) F ; +1 [w (v u) ] X (u; :) + X (v; :) 2
As a consequence of this result, in Remark 1, particularize given some convenientes values to the parameters ; and w; with which they obtain the Hermite-Hadamard inequality for Riemann-Liouville fractional integral for stochastic processes and the same inequality for ordinary integral. The next result is a theorem that establishes the inequality of Hermite Hadamard for strongly convex stochastic processes. Theorem 6 Let X : I ! R be a stochastic process, which is strongly Jensen-convex with modulus C( )and mean-square continuous in the interval I so that E[C 2 ] < 1. Then for any u; v 2 I; we have n u+v +2 X ;: C( ) 2 (v u) F ;2 [w (v u) ] 2 (v u) F ;1 [w (v u) ] 2 ) 2 u + v + u2 + v 2 (v u) F ; [w (v u) ] 2 1 J ; ;u+;w [X] (v) + J ; ;v ;w [X] (u) 2 (v u) F ; +1 [w (v u) ] 3
X (u; :) + X (v; :) 2 2 (v
u) F
1
;
C( )
[w (v
u2 + v 2 + 2 (v 2
u) ] + u2 + v 2 (v
u)
+2
u) F
;
F
2
;
[w (v
The authors do not specify the values of the terms of the sequences There are no conclusions or acknowledgments. 2.4 [1]
[2] [3] [4] [5] [6] [7]
[w (v
1
u) ] o u) ]
and
2:
Referencias H. Agahi and A. Babakhani, On fractional stochastic inequalities related to Hermite–Hadamard and Jensen types for convex stochastic processes, Aequat. Math. 90 (2016), 1035–1043. D. Kotrys, Hermite-Hadamard inequality for convex stochastic processes, Aequat. Math. 83 (2012), 143-151. D. Kotrys, Remarks on strongly convex stochastic processes. Aequat. Math. ,86, 91–98 (2013). F.M. Ha z, The fractional calculus for some stochastic processes. Stoch. Anal. Appl. 22,507–523 (2004). K. Nikodem, On convex stochastic processes, Aequat. Math. 20, 184-197 (1980). A. Skowronski, On some properties of J-convex stochastic processes, Aequat. Math. 44, 249-258 (1992). K. Sobczyk, Stochastic dix o erential equations with applications to physics and engineering, Kluwer, Dordrecht (1991).
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