Generalized Fractional Integral Inequalities for Continuous Random Variables

Abstract

Some generalized integral inequalities are established for the fractional expectation and the fractional variance for continuous random variables. Special cases of integral inequalities in this paper are studied by Barnett et al. and Dahmani.

1. 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 [1–8] 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 and the variance of a random variable having a probability density function (p.d.f.) . 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 , for the quantity , , 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 [20–22].

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.) by using these concepts. Our results are extension of [12, 14, 17].

2. Preliminaries

Definition 1 (see [23]). Let . The Riemann-Liouville fractional integrals and of order are defined by respectively, where is Gamma function and .

We give the following properties for the :

Definition 2 (see [24]). Consider the space () of those real-valued Lebesgue measurable functions on for which

Definition 3 (see [24]). Consider the space of those real-valued Lebesgue measurable functions on for which and for the case In particular, when () the space coincides with the -space and also if we take () the space coincides with the classical -space.

Definition 4 (see [24]). Let . The generalized Riemann-Liouville fractional integrals and of orders and are defined by Here is Gamma function and . Integral formulas (7) and (8) are called right generalized Riemann-Liouville integral and left generalized Riemann-Liouville fractional integral, respectively.

Definition 5. The fractional expectation function of orders and , for a random variable with a positive p.d.f. defined on , is defined as In the same way, we define the fractional expectation function of by what follows.

Definition 6. The fractional expectation function of orders , , and , for a random variable , is defined as where is the p.d.f. of .

For , we introduce the following concept.

Definition 7. The fractional expectation of orders , , and , 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 , , and , for a random variable having a p.d.f. , is defined as where is the classical expectation of .

Definition 9. The fractional variance of order , for a random variable with a p.d.f. , is defined as We give the following important properties.(1)If we take and in Definition 5, we obtain the classical expectation .(2)If we take and in Definition 7, we obtain the classical variance .(3)If we take in Definitions 5–9, we obtain Definitions  2.2–2.6 in [17].(4)For , the p.d.f. satisfies .(5)For , we have the well known property .

3. Main Results

Theorem 10. Let be a continuous random variable having a p.d.f. . Then(a) for all , , and ,  provided that ;(b) the inequality  is also valid for all , , and .

Proof. Let us define the quantity for p.d.f. and : Taking a function , multiplying (16) by , and then integrating the resulting identity with respect to from to , we have Similarly, multiplying (17) by , , and integrating the resulting identity with respect to over , we can write If, in (18), we take and , , then we have On the other hand, we have Thanks to (19) and (20), we obtain part (a) of Theorem 10.
For part (b), we have Then, by (19) and (21), we get the desired inequality (14).

We give also the following corollary.

Corollary 11. Let be a continuous random variable with a p.d.f. defined on . Then(i)if , then for any and , one has (ii)the inequality is also valid for any and .

Remark 12. (r1) Taking and in (i) of Corollary 11, we obtain the first part of Theorem 1 in [14].
(r2) Taking and in (ii) of Corollary 11, we obtain the last part of Theorem 1 in [14].

We will further generalize Theorem 10 by considering two fractional positive parameters.

Theorem 13. Let X be a continuous random variable having a p.d.f. . Then one has the following.(a) For all , , , and ,  where .(b) The inequality  is also valid for any , , , and .

Proof. Using (15), we can write Taking and , , in the above identity, yields We have also Thanks to (27) and (28), we obtain (a).
To prove (b), we use the fact that . We obtain And, by (27) and (29), we get (25).

Remark 14. (r1) Applying Theorem 13 for , we obtain Theorem 10.

We give also the following fractional integral result.

Theorem 15. Let be the p.d.f. of on . Then for all , , and , one has

Proof. Using Theorem 1 of [25], we can write Taking and , , then and . Hence, (30) allows us to obtain This implies that Theorem 15 is thus proved.

For , we propose the following interesting inequality.

Corollary 16. Let be the p.d.f. of on . Then for any and , one has

Remark 17. Taking in Corollary 16, we obtain Theorem 2 of [14].

We also present the following result for the fractional variance function with two parameters.

Theorem 18. Let be the p.d.f. of the random variable on . Then for all , , , and , one has

Proof. Thanks to Theorem 4 of [25], we can state that In (35), we take and , . We obtain Combining (27) and (37) and taking into account the fact that the left-hand side of (27) is positive, we get Therefore, 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.

Corollary 19. Let be the p.d.f. of on . Then for all , , and , the inequality is valid.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.