Generalized Fractional Integral Inequalities for Continuous Random Variables

Integral inequalities play a fundamental role in the theory of differential equations, functional analysis, and applied sciences. Important development in this theory has been achieved in the last two decades. For these, see [1–8] and the references therein. Moreover, the study of fractional type inequalities is also of vital importance. Also see [9–13] for further information and applications. The first one is given in [14]; in their paper, using Korkine identity and Holder inequality for double integrals, Barnett et al. established several integral inequalities for the expectation E(X) and the variance σ(X) of a random variable X having a probability density function (p.d.f.) f : [a, b] → R. In [15–17] the authors presented new inequalities for the moments and for the higher order central moments of a continuous random variable. In [17, 18] Dahmani and Miao and Yang gave new upper bounds for the standard deviation σ(X), for the quantity σ(X)+(t−E(X))2, t ∈ [a, b], and for the L absolute deviation of a random variableX. Recently, Anastassiou et al. [9] proposed a generalization of the weighted Montgomery identity for fractional integrals with weighted fractional Peano kernel. More recently, Dahmani and Niezgoda [17, 19] gave inequalities involvingmoments of a continuous random variable defined over a finite interval. Other papers dealing with these probability inequalities can be found in [20–22]. In this paper, we introduce new concepts on “generalized fractional random variables.” We obtain new generalized integral inequalities for the generalized fractional dispersion and the generalized fractional variance functions of a continuous random variable X having the probability density function (p.d.f.) f : [a, b] → R by using these concepts. Our results are extension of [12, 14, 17].


Introduction
Integral inequalities play a fundamental role in the theory of differential equations, functional analysis, and applied sciences.Important development in this theory has been achieved in the last two decades.For these, see [1][2][3][4][5][6][7][8] and the references therein.Moreover, the study of fractional type inequalities is also of vital importance.Also see [9][10][11][12][13] for further information and applications.The first one is given in [14]; in their paper, using Korkine identity and Holder inequality for double integrals, Barnett et al. established several integral inequalities for the expectation () and the variance  2 () of a random variable  having a probability density function (p.d.f.)  : [, ] → R + .In [15][16][17] the authors presented new inequalities for the moments and for the higher order central moments of a continuous random variable.In [17,18] Dahmani and Miao and Yang gave new upper bounds for the standard deviation (), for the quantity  2 ()+(−()) 2 ,  ∈ [, ], and for the   absolute deviation of a random variable .Recently, Anastassiou et al. [9] proposed a generalization of the weighted Montgomery identity for fractional integrals with weighted fractional Peano kernel.More recently, Dahmani and Niezgoda [17,19] gave inequalities involving moments of a continuous random variable defined over a finite interval.Other papers dealing with these probability inequalities can be found in [20][21][22].
In this paper, we introduce new concepts on "generalized fractional random variables." We obtain new generalized integral inequalities for the generalized fractional dispersion and the generalized fractional variance functions of a continuous random variable  having the probability density function (p.d.f.)  : [, ] → R + by using these concepts.Our results are extension of [12,14,17].
For  = , we introduce the following concept.
Definition 7. The fractional expectation of orders  ≥ 0,  <  ≤ , and  ≥ 0, for a random variable  with a positive p.d.f. defined on [, ], is defined as For the fractional variance of , we introduce the following two definitions.Definition 8.The fractional variance function of orders  ≥ 0,  <  ≤ , and  ≥ 0, for a random variable  having a p.d.f. : [, ] → R + , is defined as where () := ∫   () is the classical expectation of .
Definition 9.The fractional variance of order  ≥ 0, for a random variable  with a p.d.f. : [, ] → R + , is defined as We give the following important properties.
We give also the following corollary.

Corollary 11.
Let  be a continuous random variable with a p.d.f. defined on [, ].
then for any  ≥ 0 and  ≥ 0, one has (ii) the inequality is also valid for any  ≥ 0 and  ≥ 0.
We will further generalize Theorem 10 by considering two fractional positive parameters.
We give also the following fractional integral result.
Theorem 15.Let  be the p.d.f. of  on [, ].Then for all  <  ≤ ,  ≥ 0, and  ≥ 0, one has Proof.Using Theorem 1 of [25], we can write Substituting the values of  and  in (33), then a simple calculation allows us to obtain (35).Theorem 18 is thus proved.
To finish, we present to the reader the following corollary.