JPS Journal of Probability and Statistics 1687-9538 1687-952X Hindawi Publishing Corporation 10.1155/2015/958980 958980 Research Article Generalized Fractional Integral Inequalities for Continuous Random Variables Akkurt Abdullah Kaçar Zeynep Yildirim Hüseyin Bai Z. D. Department of Mathematics, Faculty of Science and Arts University of Kahramanmaraş Sütçü İmam 46000 Kahramanmaraş Turkey ksu.edu.tr 2015 112015 2015 05 10 2014 09 12 2014 0 0 0 2015 Copyright © 2015 Abdullah Akkurt et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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  and the references therein. Moreover, the study of fractional type inequalities is also of vital importance. Also see  for further information and applications. The first one is given in ; 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  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.  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 .

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

Let fL1[a,b]. The Riemann-Liouville fractional integrals Ja+αfx and Jb-αfx of order α0 are defined by (1)Ja+αfx=1Γαaxx-tα-1ftdtx>a,(2)Jb-αfx=1Γαxbt-xα-1ftdtx<b, respectively, where Γα=0e-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,α0,β0,Ja+αJa+βft=Ja+βJa+αft,α0,β0.

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

Consider the space Lp,k(a,b) (k0,1p<) of those real-valued Lebesgue measurable functions f on [a,b] for which (4)fLp,ka,b=abfxpxkdx1/p<,1p<,k0.

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

Consider the space Xcp(a,b)(cR,1p<) of those real-valued Lebesgue measurable functions f on [a,b] for which (5)fXcp=abxcfxpdxx1/p<,1p<,cR and for the case p=(6)fXc=esssupaxb[xcf(x)],cR. In particular, when c=(k+1)/p (1p<,k0) the space Xcp(a,b) coincides with the Lp,k(a,b)-space and also if we take c=(1/p) (1p<) 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 fL1,k[a,b]. The generalized Riemann-Liouville fractional integrals Ja+α,kfx and Jb-α,kfx of orders α0 and k0 are defined by (7)Ja+α,kfx=k+11-αΓαaxxk+1-tk+1α-1tkftdtx>a,(8)Jb-α,kfx=k+11-αΓαxbtk+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 α0 and k0, for a random variable X with a positive p.d.f. f defined on [a,b], is defined as (9)EX,αtJa+α,ktft=k+11-αΓαattk+1-τk+1α-1τk+1fτdτ,a<tb. 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 α0, k0, and a<tb, for a random variable X-E(X), is defined as (10)EX-EX,αtk+11-αΓαattk+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 α0, a<tb, and k0, for a random variable X with a positive p.d.f. f defined on [a,b], is defined as (11)EX,α=k+11-αΓαabbk+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 α0, a<tb, and k0, for a random variable X having a p.d.f. f:[a,b]R+, is defined as (12)σX,α2Ja+α,kt-E(X)2f(t)=k+11-αΓαattk+1-τk+1α-1τ-E(X)2τkfτdτ, where E(X)abτfτdτ is the classical expectation of X.

Definition 9.

The fractional variance of order α0, for a random variable X with a p.d.f. f:[a,b]R+, is defined as (13)σX,α2=k+11-αΓαabbk+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 59, we obtain Definitions  2.2–2.6 in .

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 Results Theorem 10.

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

for all a<tb, α0, and k0, (14)Ja+α,kftσX,α2t-EX-E(X),α(t)2f2k+11-αtk+1-ak+1αΓα+1Ja+α,kt2k+2k+11-αtk+1-ak+1αΓα+1-Ja+α,kt2,

provided that fL[a,b];

the inequality (15)Ja+α,kftσX,α2t-EX-EX,αt212tk+1-ak+12Ja+α,kt2

is also valid for all a<tb, α0, and k0.

Proof.

Let us define the quantity for p.d.f. g and h: (16)Hτ,ρgτ-gρhτ-hρ;τ,ρa,t,a<tb,α0. Taking a function p:[a,b]R+, multiplying (16) by (tk+1-τk+1)α-1/Γαpττk,τa,t, and then integrating the resulting identity with respect to τ from a to t, we have (17)k+11-αΓαattk+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, ρa,t, and integrating the resulting identity with respect to ρ over (a,t), we can write (18)k+12-2αΓ2αatattk+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×atattk+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×atattk+1-τk+1α-1tk+1-ρk+1α-1×fτfρτk+1-ρk+12τkρkfτdτdρf22k+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×atattk+1-τk+1α-1tk+1-ρk+1α-1×fτfρτk+1-ρk+12τkρkfτdτdρsupτ,ρa,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 fL[a,b], then for any α0 and k0, one has (22)bk+1-ak+1α-1ΓασX,α2-EX,α2f2bk+1-ak+12α+2Γα+1Γα+3-bk+1-ak+1α+1Γα+12;

the inequality (23)bk+1-ak+1α-1ΓασX,α2-EX,α212bk+1-ak+12αΓ2α is also valid for any α0 and k0.

Remark 12.

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

(r2) Taking α=1 and k=0 in (ii) of Corollary 11, we obtain the last part of Theorem 1 in .

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<tb, α0, β0, and k0, (24)Ja+α,kftσX,β2t+Ja+β,kftσX,α2t-2EX-EX,αtEX-EX,βtf2k+11-αtk+1-ak+1αΓα+1Ja+β,kt2k+2+f2k+11-βtk+1-ak+1βΓβ+1Ja+α,kt2k+2k+11-βtk+1-ak+1βΓβ+1-2Ja+α,ktJa+β,kt,

where fL[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<tb, α0, β0, and k0.

Proof.

Using (15), we can write (26)k+12-α-βΓαΓβatattk+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×atattk+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×atattk+1-τk+1α-1tk+1-ρk+1α-1×pτpρτk+1-ρk+12τkρkfτdτdρf2k+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,ρa,t(τk+1-ρk+1)2=tk+1-ak+12. We obtain (29)k+12-α-βΓαΓβ×atattk+1-τk+1α-1tk+1-ρk+1α-1×fτfρτk+1-ρk+12τkρkfτdτdρtk+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<tb, α0, and k0, one has (30)Ja+α,kftσX,α2t-EX-EX,αt214bk+1-ak+12Ja+α,kt2.

Proof.

Using Theorem 1 of , we can write (31)Ja+α,kptJa+α,kpg2t-Ja+α,kpgt214Ja+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)0Ja+α,kftJa+α,kfttk+1-EX2-Ja+α,kf(t)tk+1-E(X)2214Ja+α,kf(t)2bk+1-ak+12. This implies that (33)Ja+α,kftσX,α2t-EX-EX,αt214bk+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 α0 and k0, one has (34)bk+1-ak+1α-1ΓασX,α2-EX-EX,αt214Γ2αbk+1-ak+12α.

Remark 17.

Taking α=1 in Corollary 16, we obtain Theorem 2 of .

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<tb, α0, β0, and k0, one has (35)Ja+α,kftσX,β2t+Ja+β,kftσX,α2t+2ak+1-EXbk+1-EX×Ja+α,kftJa+β,kftak+1+bk+1-2E(X)×Ja+α,kftEX-EX,βt+Ja+β,kftEX-E(X),α(t).

Proof.

Thanks to Theorem 4 of , 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)2MJa+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)2MJa+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)2MJa+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<tb, α0, and k0, the inequality (40)σX,α2t+ak+1-E(X)bk+1-E(X)Ja+α,kftak+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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