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 E(X) and the variance σ2(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 σ2(X)+(t-E(X))2, t∈[a,b], and for the Lp absolute deviation of a random variable X. 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 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].

2. PreliminariesDefinition 1 (see [<xref ref-type="bibr" rid="B26">23</xref>]).

Let f∈L1[a,b]. The Riemann-Liouville fractional integrals Ja+αfx and Jb-αfx of order α≥0 are defined by
(1)Ja+αfx=1Γα∫axx-tα-1ftdtx>a,(2)Jb-αfx=1Γα∫xbt-xα-1ftdtx<b,
respectively, where Γα=∫0∞e-uuα-1du is Gamma function and Ja+0f(x)=Jb-0f(x)=f(x).

We give the following properties for the Ja+α:
(3)Ja+αJa+βft=Ja+α+βft,α≥0,β≥0,Ja+αJa+βft=Ja+βJa+αft,α≥0,β≥0.

Definition 2 (see [<xref ref-type="bibr" rid="B25">24</xref>]).

Consider the space Lp,k(a,b) (k≥0,1≤p<∞) of those real-valued Lebesgue measurable functions f on [a,b] for which
(4)fLp,ka,b=∫abfxpxkdx1/p<∞,1≤p<∞,k≥0.

Definition 3 (see [<xref ref-type="bibr" rid="B25">24</xref>]).

Consider the space Xcp(a,b)(c∈R,1≤p<∞) of those real-valued Lebesgue measurable functions f on [a,b] for which
(5)fXcp=∫abxcfxpdxx1/p<∞,1≤p<∞,c∈R
and for the case p=∞(6)fXc∞=esssupa≤x≤b[xcf(x)],c∈R.
In particular, when c=(k+1)/p (1≤p<∞,k≥0) the space Xcp(a,b) coincides with the Lp,k(a,b)-space and also if we take c=(1/p) (1≤p<∞) the space Xcp(a,b) coincides with the classical Lp(a,b)-space.

Definition 4 (see [<xref ref-type="bibr" rid="B25">24</xref>]).

Let f∈L1,k[a,b]. The generalized Riemann-Liouville fractional integrals Ja+α,kfx and Jb-α,kfx of orders α≥0 and k≥0 are defined by
(7)Ja+α,kfx=k+11-αΓα∫axxk+1-tk+1α-1tkftdtx>a,(8)Jb-α,kfx=k+11-αΓα∫xbtk+1-xk+1α-1tkftdtb>x.
Here Γα is Gamma function and Ja+0,kfx=Jb-0,kfx=f(x). 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 α≥0 and k≥0, for a random variable X with a positive p.d.f. f defined on [a,b], is defined as
(9)EX,αt≔Ja+α,ktft=k+11-αΓα∫attk+1-τk+1α-1τk+1fτdτ,a<t≤b.
In the same way, we define the fractional expectation function of X-E(X) by what follows.

Definition 6.

The fractional expectation function of orders α≥0, k≥0, and a<t≤b, for a random variable X-E(X), is defined as
(10)EX-EX,αt≔k+11-αΓα∫attk+1-τk+1α-1×τ-E(X)τkfτdτ,
where f:[a,b]→R+ is the p.d.f. of X.

For t=b, we introduce the following concept.

Definition 7.

The fractional expectation of orders α≥0, a<t≤b, and k≥0, for a random variable X with a positive p.d.f. f defined on [a,b], is defined as
(11)EX,α=k+11-αΓα∫abbk+1-τk+1α-1τk+1fτdτ.

For the fractional variance of X, we introduce the following two definitions.

Definition 8.

The fractional variance function of orders α≥0, a<t≤b, and k≥0, for a random variable X having a p.d.f. f:[a,b]→R+, is defined as
(12)σX,α2≔Ja+α,kt-E(X)2f(t)=k+11-αΓα∫attk+1-τk+1α-1τ-E(X)2τkfτdτ,
where E(X)≔∫abτfτdτ is the classical expectation of X.

Definition 9.

The fractional variance of order α≥0, for a random variable X with a p.d.f. f:[a,b]→R+, is defined as
(13)σX,α2=k+11-αΓα∫abbk+1-τk+1α-1×τ-E(X)2τkfτdτ.
We give the following important properties.

If we take α=1 and k=0 in Definition 5, we obtain the classical expectation EX,1=E(X).

If we take α=1 and k=0 in Definition 7, we obtain the classical variance σX,12=σ2(X)=∫ab(τ-E(X))2f(τ)dτ.

If we take k=0 in Definitions 5–9, we obtain Definitions 2.2–2.6 in [17].

For α>0, the p.d.f. f satisfies Jα[f(b)]=(b-a)α-1/Γ(α).

For α=1, we have the well known property Jα[f(b)]=1.

3. Main ResultsTheorem 10.

Let X be a continuous random variable having a p.d.f. f:[a,b]→R+. Then

for all a<t≤b, α≥0, and k≥0,
(14)Ja+α,kftσX,α2t-EX-E(X),α(t)2≤f∞2k+11-αtk+1-ak+1αΓα+1Ja+α,kt2k+2k+11-αtk+1-ak+1αΓα+1-Ja+α,kt2,

provided that f∈L∞[a,b];

the inequality
(15)Ja+α,kftσX,α2t-EX-EX,αt2≤12tk+1-ak+12Ja+α,kt2

is also valid for all a<t≤b, α≥0, and k≥0.

Proof.

Let us define the quantity for p.d.f. g and h:
(16)Hτ,ρ≔gτ-gρhτ-hρ;τ,ρ∈a,t,a<t≤b,α≥0.
Taking a function p:[a,b]→R+, multiplying (16) by (tk+1-τk+1)α-1/Γαpττk,τ∈a,t, and then integrating the resulting identity with respect to τ from a to t, we have
(17)k+11-αΓα∫attk+1-τk+1α-1pτHτ,ρτkfτdτ=Ja+α,kpght-hρJa+α,kpgt-gρJa+α,kph(t)+gρhρJa+α,kp(t).
Similarly, multiplying (17) by ((tk+1-ρk+1)α-1/Γα)pρρk, ρ∈a,t, and integrating the resulting identity with respect to ρ over (a,t), we can write
(18)k+12-2αΓ2α∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×pτpρHτ,ρτkρkfτdτdρ=2Ja+α,kptJa+α,kpght-2Ja+α,kpg(t)Ja+α,kph(t).
If, in (18), we take p(t)=f(t) and g(t)=h(t)=tk+1-E(X), t∈(a,b), then we have
(19)k+12-2αΓ2α×∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×fτfρτk+1-ρk+12τkρkfτdτdρ=2Ja+α,kftJa+α,kfttk+1-EX2-2Ja+α,kf(t)tk+1-E(X)2.
On the other hand, we have
(20)k+12-2αΓ2α×∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×fτfρτk+1-ρk+12τkρkfτdτdρ≤f∞22k+11-αtk+1-ak+1αΓα+1Ja+α,kt2k+2k+11-αtk+1-ak+1αΓα+1-2Ja+α,kt2.
Thanks to (19) and (20), we obtain part (a) of Theorem 10.

For part (b), we have
(21)k+12-2αΓ2α×∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×fτfρτk+1-ρk+12τkρkfτdτdρ≤supτ,ρ∈a,tτk+1-ρk+12Ja+α,kft2=tk+1-ak+12Ja+α,kft2.
Then, by (19) and (21), we get the desired inequality (14).

We give also the following corollary.

Corollary 11.

Let X be a continuous random variable with a p.d.f. f defined on [a,b]. Then

if f∈L∞[a,b], then for any α≥0 and k≥0, one has
(22)bk+1-ak+1α-1ΓασX,α2-EX,α2≤f∞2bk+1-ak+12α+2Γα+1Γα+3-bk+1-ak+1α+1Γα+12;

the inequality
(23)bk+1-ak+1α-1ΓασX,α2-EX,α2≤12bk+1-ak+12αΓ2α
is also valid for any α≥0 and k≥0.

Remark 12.

(r1) Taking α=1 and k=0 in (i) of Corollary 11, we obtain the first part of Theorem 1 in [14].

(r2) Taking α=1 and k=0 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. f:[a,b]→R+. Then one has the following.

For all a<t≤b, α≥0, β≥0, and k≥0,
(24)Ja+α,kftσX,β2t+Ja+β,kftσX,α2t-2EX-EX,αtEX-EX,βt≤f∞2k+11-αtk+1-ak+1αΓα+1Ja+β,kt2k+2+f∞2k+11-βtk+1-ak+1βΓβ+1Ja+α,kt2k+2k+11-βtk+1-ak+1βΓβ+1-2Ja+α,ktJa+β,kt,

where f∈L∞[a,b].

The inequality
(25)Ja+α,kftσX,β2t+Ja+β,kftσX,α2t-2EX-E(X),α(t)EX-E(X),β(t)≤tk+1-ak+12Ja+α,ktJa+β,kt

is also valid for any a<t≤b, α≥0, β≥0, and k≥0.

Proof.

Using (15), we can write
(26)k+12-α-βΓαΓβ∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×pτpρHτ,ρτkρkfτdτdρ=Ja+α,kptJa+β,kpgh(t)+Ja+β,kptJa+α,kpgh(t)-Ja+α,kphtJa+β,kpgt-Ja+β,kph(t)Ja+α,kpg(t).
Taking p(t)=f(t) and g(t)=h(t)=tk+1-E(X), t∈(a,b), in the above identity, yields
(27)k+12-α-βΓαΓβ×∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×pτpρτk+1-ρk+12τkρkfτdτdρ=Ja+α,kftJa+β,kfttk+1-EX2+Ja+β,kftJa+α,kf(t)tk+1-E(X)2-2Ja+α,kfttk+1-EXJa+β,k×f(t)tk+1-E(X).
We have also
(28)k+12-α-βΓαΓβ×∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×pτpρτk+1-ρk+12τkρkfτdτdρ≤f∞2k+11-αtk+1-ak+1αΓα+1Ja+β,kt2k+2+k+11-βtk+1-ak+1βΓβ+1Ja+α,kt2k+2k+11-αtk+1-ak+1αΓα+1-2Ja+α,ktJa+β,kt.
Thanks to (27) and (28), we obtain (a).

To prove (b), we use the fact that supτ,ρ∈a,t(τk+1-ρk+1)2=tk+1-ak+12. We obtain
(29)k+12-α-βΓαΓβ×∫at∫attk+1-τk+1α-1tk+1-ρk+1α-1×fτfρτk+1-ρk+12τkρkfτdτdρ≤tk+1-ak+12Ja+α,ktJa+β,kt.
And, by (27) and (29), we get (25).

Remark 14.

(r1) Applying Theorem 13 for α=β, we obtain Theorem 10.

We give also the following fractional integral result.

Theorem 15.

Let f be the p.d.f. of X on [a,b]. Then for all a<t≤b, α≥0, and k≥0, one has
(30)Ja+α,kftσX,α2t-EX-EX,αt2≤14bk+1-ak+12Ja+α,kt2.

Proof.

Using Theorem 1 of [25], we can write
(31)Ja+α,kptJa+α,kpg2t-Ja+α,kpgt2≤14Ja+α,kp(t)2M-m2.
Taking p(t)=f(t) and g(t)=tk+1-E(X), t∈(a,b), then M=bk+1-E(X) and m=ak+1-E(X). Hence, (30) allows us to obtain
(32)0≤Ja+α,kftJa+α,kfttk+1-EX2-Ja+α,kf(t)tk+1-E(X)22≤14Ja+α,kf(t)2bk+1-ak+12.
This implies that
(33)Ja+α,kftσX,α2t-EX-EX,αt2≤14bk+1-ak+12Ja+α,kt2.
Theorem 15 is thus proved.

For t=b, we propose the following interesting inequality.

Corollary 16.

Let f be the p.d.f. of X on [a,b]. Then for any α≥0 and k≥0, one has
(34)bk+1-ak+1α-1ΓασX,α2-EX-EX,αt2≤14Γ2αbk+1-ak+12α.

Remark 17.

Taking α=1 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 f be the p.d.f. of the random variable X on [a,b]. Then for all a<t≤b, α≥0, β≥0, and k≥0, one has
(35)Ja+α,kftσX,β2t+Ja+β,kftσX,α2t+2ak+1-EXbk+1-EX×Ja+α,kftJa+β,kft≤ak+1+bk+1-2E(X)×Ja+α,kftEX-EX,βt+Ja+β,kftEX-E(X),α(t).

Proof.

Thanks to Theorem 4 of [25], we can state that
(36)Ja+α,kptJa+β,kpg2(t)+Ja+β,kptJa+α,kpg2(t)-2Ja+α,kpg(t)Ja+β,kpg(t)2≤MJa+α,kpt-Ja+α,kpgt×Ja+β,kpgt-mJa+β,kpt+Ja+β,kpg(t)-mJa+β,kp(t)×MJa+β,kp(t)-Ja+β,kpg(t)2.
In (35), we take p(t)=f(t) and g(t)=tk+1-E(X), t∈(a,b). We obtain
(37)Ja+α,kftJa+β,kfttk+1-EX2+Ja+β,kftJa+α,kf(t)tk+1-E(X)2fttk+1-EX2-2Ja+α,kf(t)tk+1-E(X)Ja+β,kf(t)tk+1-E(X)2≤MJa+α,kft-Ja+α,kfttk+1-EX×Ja+β,kf(t)tk+1-E(X)-mJa+β,kf(t)+Ja+α,kfttk+1-EX-mJa+α,kftMJa+α,kft-Ja+α,kfttk+1-EX×MJa+β,kf(t)-Ja+β,kf(t)tk+1-E(X)2.
Combining (27) and (37) and taking into account the fact that the left-hand side of (27) is positive, we get
(38)Ja+α,kftJa+β,kfttk+1-EX2+Ja+β,kftJa+α,kf(t)tk+1-E(X)2-2Ja+α,kf(t)tk+1-E(X)Ja+β,kf(t)tk+1-E(X)≤MJa+α,kft-Ja+α,kfttk+1-EX×Ja+β,kfttk+1-EX-mJa+β,kft+Ja+α,kfttk+1-EX-mJa+α,kft×MJa+β,kf(t)-Ja+β,kf(t)tk+1-E(X).
Therefore,
(39)Ja+α,kftJa+β,kf(t)tk+1-E(X)2+Ja+β,kftJa+α,kf(t)tk+1-E(X)2≤MJa+α,kftEX-EX,βt+Ja+β,kftEX-EX,αt+mJa+α,kf(t)EX-E(X),β(t)Ja+β,kf(t)EX-E(X),α(t)+Ja+β,kf(t)EX-E(X),α(t).
Substituting the values of m and M 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 f be the p.d.f. of X on [a,b]. Then for all a<t≤b, α≥0, and k≥0, the inequality
(40)σX,α2t+ak+1-E(X)bk+1-E(X)Ja+α,kft≤ak+1+bk+1-2E(X)EX-E(X),α(t)
is valid.

Conflict of Interests

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

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