Random variable inequalities involving (k, s)-integration

https://doi.org/10.26637/MJM0504/0005

Random variable inequalities involving

(k,s)-integration

M. Houas

* and Z. Dahmani

Abstract

In this paper, we use the (k, s) −Riemann-Liouville operator to establish new results on integral inequalities by using fractional moments of continuous random variables.

Keywords

(k, s) −Riemann-Liouville integral, random variable, (k, s) −fractional expectation, (k, s) −fractional variance, (k, s) −fractional moment.

AMS Subject Classification

26D15, 26A33, 60E15.

1_{Laboratory FIMA, UDBKM, University of Khemis Miliana, Algeria.} 2_{Laboratory LPAM, UMAB, University of Mostaganem, Algeria.}

*Corresponding author: ı zzdahmani@yahoo.fr; ı2_{houasmed@yahoo.fr}

Article History: Received 6 June 2017; Accepted 7 August 2017 2017 MJM.c

Contents

1 Introduction. . . 641

2 Preliminaries. . . 641

3 Inequalities for (k, s) − operator. . . 643

References. . . 645

1. Introduction

The study of integral inequalities is an important research subject in mathematical analysis. These inequalities have many applications in differential equations, probability theory and statistical problems, but one of the most useful appli-cations is to establish uniqueness of solutions in fractional boundary value problems. For detailed applications on the subject, one may refer to [9–12], and the references cited therein. Moreover, the integral inequalities involving frac-tional integration are also of great importance. For some earlier work on the topic, we refer to [2–5,7,8,14,16,17]. In [9], P. Kumar presented new results involving higher moments for continuous random variables. Also the author established some estimations for the central moments. Other results based on Gruss inequality and some applications of the truncated exponential distribution have been also discussed by the au-thor. In [10], Ostrowski type integral inequalities involving moments of a continuous random variable defined on a finite interval, is established. In [3], the author established several inequalities for the fractional dispersion and the fractional

variance functions of continuous random variables. Recently, A. Akkurt et al. [1] proposed new generalizations of the re-sults in [3]. Very recently, Z. Dahmani [4,6] presented new fractional integral results for the fractional moments of contin-uous random variables by correcting some results in [3]. In a very recent work, M. Tomar et al. [17] proposed new integral inequalities for the (k, s)−fractional expectation and variance functions of a continuous random variable.

Motivated by the results presented in [3,4,6,14,17], in this paper, we present some random variable integral inequalities for the (k, s)−fractional operator.

2. Preliminaries

We recall the notations and definitions of the (k, s) −fractional integration theory [13,14,17].

Definition 2.1. The Riemann–Liouville fractional integral of order α ≥ 0, for a continuous function f on [a, b] is defined by Jα a[ f (t)] =Γ(α )1 Rt a(t − τ) α −1 f(τ) dτ; α ≥ 0, a < t ≤ b , (2.1) where Γ (α) =R∞ 0 e −u_uα −1_du.

Definition 2.2. The k−Riemann–Liouville fractional integral of order α > 0, for a continuous function f on [a, b] is defined

by kJaα[ f (t)] =kΓk1(α) Rt a(t − τ) α k−1_{f(τ) dτ; α > 0, a < t ≤ b ,} (2.2) where Γk(α) = R∞ 0 e −uk k_uα −1_du.

Definition 2.3. The (k, s) −Riemann-Liouville fractional in-tegral of order α > 0, for a continuous function f on [a, b] is defined as k sJaα[ f (t)] = (s+1)1− αk kΓk(α) Rt a ts+1− τs+1
α_k−1 τsf(τ) dτ; α > 0, a < t ≤ b , (2.3) where k_{> 0, s ∈ R\ {−1} .}

Theorem 2.4. Let f be continuous on [a, b] , k > 0, and s ∈ R\ {−1} . Then, k sJaα h k sJ β a[ f (t)] i = k_sJaα +β[ f (t)] = ksJ β a _k sJaα[ f (t)] , k > 0, s ∈ R\{−1} , (2.4) for all α > 0, β > 0, a < t ≤ b.

Theorem 2.5. Let α > 0, β > 0, k > 0 and s ∈ R\ {−1} . Then, we have k sJaα  ts+1− as+1
β_k−1  = Γk(β ) (s+1)αkΓk(α+β ) ts+1− as+1
β +α_k −1 , (2.5) Remark 2.6. (i) : Taking s = 0, k > 0 in (5) , we obtain

kJaα  (t − a)βk−1  = Γk(β ) Γk(α+β )(t − a) β +α k −1_{, α, β > 0, a > 0, k > 0 .} (2.6) (ii) : The formula (5) , for s = 0 and k = 1 becomes

Jα a h (t − a)β −1i= Γ(β ) Γ(α +β )(t − a) β +α −1 . (2.7)

Corollary 2.7. Let k > 0 and s ∈ R\ {−1} . Then, for any α > 0, we have k sJaα[1] = 1 (s+1)αkΓk(α+k) ts+1− as+1
α_k−2 . (2.8)

Remark 2.8. (i) : For s = 0, k > 0 in (8) , we get kJaα[1] =Γk(α+k)1 (t − a)

α

k−2 _{. (2.9)}

(ii) : For s = 0, k = 1 in (8) , we have Jα

a [1] =Γ(α +1)1 (t − a)

β −2 _{. (2.10)}

For more details on(k, s) −fractional integral, we refer the reader to [14,17].

We recall also the following definitions [3,4,17] Definition 2.9. The (k, s) −fractional expectation function of order α > 0, for a random variable X with a positive p.d. f .

f defined on[a, b] is defined as

k sEX,α(t) := (s + 1)1−αk kΓk(α) t Z a ts+1− τs+1
α_k−1 τs+1f(τ) dτ, (2.11) α > 0, k > 0, s ∈ R\ {−1} and a < t ≤ b.

Definition 2.10. The (k, s) −fractional expectation function of order α > 0 for the random variable X − E (X ) with a positive probability density function f defined on [a, b] is defined as k sEX−E(X),α(t) := (s + 1)1−αk kΓk(α) t Z a ts+1− τs+1
αk−1 τs(τ − E (X )) f (τ) dτ, (2.12) α > 0, k > 0, s ∈ R\ {−1} and a < t ≤ b.

Definition 2.11. The (k, s) −fractional variance function of order α > 0 for a random variable X having a positive p.d. f .

f on[a, b] is defined as k sσX,α2 (t) := (s + 1)1−αk kΓk(α) t Z a ts+1− τs+1
α_k−1 τs(τ − E (X ))2f(τ) dτ, (2.13) α > 0, k > 0, s ∈ R\ {−1} and a < t ≤ b.

We introduce also the following definition.

Definition 2.12. The (k, s) −fractional moment function of orders r> 0, α > 0 for a continuous random variable X having a p.d. f . f defined on [a, b] is defined as

k sMr,α(t) := (s + 1)1−αk kΓ_k(α) t Z a ts+1− τs+1
α_k−1 τs+rf(τ) dτ, (2.14) α > 0, k > 0, s ∈ R\ {−1} and a < t ≤ b.

Remark 2.13. If we take s = 0, α = k = 1 in Definition 12, we obtain the classical moment of order r> 0 given by Mr:= b

R

a

τrf(τ) dτ.

We define the quantities that will be used later:

H(τ, ρ) := (g (τ) − g (ρ)) (h (τ) − h (ρ)) , τ, ρ ∈ (a,t) , a < t ≤ b, (2.15)

and ϕ_k,sα (t, τ) :=(s + 1) 1−α k Γk(α) ts+1− τs+1
αk−1 τsp(τ) , k > 0, s ∈ R/ {−1} τ ∈ (a,t) , (2.16)

where p : [a, b] → R+is a continuous function.

3. Inequalities for (k, s) − operator

We prove:

Theorem 3.1. Let X be a continuous random variable having a p_{.d. f . f : [a, b] → R}+.

(i) : If f ∈ L∞[a, b], then for all a < t ≤ b and α > 0, k > 0, s ∈ R\ {−1} , k sJaα[ f (t)] ksEXr−1_{(X −E(X )),α}(t) − k_sE_X−E(X),α(t) k_sMr−1,α(t) ≤ k f k2_∞hk sJaα[1]ksJaα[tr] − ksJaα[t)]ksJaαtr−1 i , (3.1)

(ii) : for all a < t ≤ b, the inequality k sJaα[ f (t)] ksEXr−1(X −E(X )),α(t) − ksEX−E(X),α(t) ksMr−1,α(t) ≤ 1 2(t − a) t r−1_{− a}r−1
_k sJaα[ f (t)] 2 , (3.2)

is also valid for α > 0, k > 0, s ∈ R/ {−1} . Proof. Using (2.15) and (2.16), we can write

t Z a ϕ_k,sα (t, τ) H (τ, ρ) dτ = t Z a ϕ_k,sα (t, τ) (g (τ) − g (ρ)) (h (τ) − h (ρ)) dτ. (3.3) Then Z t a Z t a ϕ_k,sα (t, τ) ϕ_k,sα (t, ρ) H (τ, ρ) dτdρ (3.4) = Z t a Z t a ϕ_k,sα (t, τ) ϕ_k,sα (t, τ) (g (τ) − g (ρ)) (h (τ) − h (ρ)) dτdρ. Hence (s + 1)2(1−αk) k2_Γ2 k(α) t Z a t Z a ts+1− τs+1
α_k−1 ts+1− ρs+1
α_k−1 τsρs (3.5) ×p (τ) p (ρ) (g (τ) − g (ρ)) (h (τ) − h (ρ)) dτdρ = 2k_sJα a[p (t)] ksJ α a[pgh (t)] − 2ksJ α a[pg (t)] ksJ α a [ph (t)] . In (3.5), we choose p(t) = f (t), g(t) = t − E(X ) and h(t) = tr−1,t ∈ (a, b), we obtain (s + 1)2(1−αk) k2_Γ2 k(α) t Z a t Z a ts+1− τs+1
αk−1 _ts+1_{− ρ}s+1
αk−1 τsρs (3.6) ×(τ − ρ) τr−1− ρr−1
f (τ) f (ρ)dτdρ = 2k_sJα a[ f (t)] ksJaαtr−1(t − E(X )) f (t) − 2ksJaα[(t − E(X )) f (t)]ksJaαtr−1f(t)  = 2k_sJα a[ f (t)] ksEXr−1(X −E(X )),α(t) − 2  k sEX−E(X),α(t)  k sMr−1,α(t) .

Using the fact f ∈ L∞([a, b]), we can write (s + 1)2(1−αk) k2_Γ2 k(α) t Z a t Z a ts+1− τs+1
α_k−1 ts+1− ρs+1
α_k−1 τsρs (3.7) ×(τ − ρ) τr−1− ρr−1
f (τ) f (ρ)dτdρ ≤ k f k2_∞(s + 1) 2(α k−1) k2_Γ2_(α) t Z a t Z a ts+1− τs+1
αk−1 _ts+1_{− ρ}s+1
αk−1 τsρs ×(τ − ρ) τr−1− ρr−1
dτdρ = k f k2_∞h2k_sJα a [1]ksJaα[tr] − 2skJaα[t] ksJaαtr−1 i . By (3.6) and (3.7), we have k sJaα[ f (t)]ksEXr−1_{(X −E(X )),α}(t) −  k sEX−E(X),α(t)  k sMr−1,α(t) ≤ k f k2_∞hk sJaα[1]ksJaα[tr] − ksJaα[t]ksJaαtr−1 i . (3.8) Since sup τ ,ρ ∈[a,t] |τ − ρ| τr−1− ρr−1 = (t − a) tr−1− ar−1
, then we observe that

(s + 1)2(1−αk) k2_Γ2_(α) t Z a t Z a ts+1− τs+1
αk−1 _ts+1_{− ρ}s+1
αk−1 τsρs ×(τ − ρ) τr−1− ρr−1
f (τ) f (ρ)dτdρ (3.9) ≤ sup τ ,ρ ∈[a,t] |τ − ρ| τr−1− ρr−1  (s + 1) 2(α k−1) k2_Γ2_(α) t Z a t Z a ts+1− τs+1
α_k−1 × ts+1− ρs+1
α_k−1 τsρsf(τ) f (ρ)dτdρ = (t − a) tr−1− ar−1
(k sJaα[ f (t)])2. Thanks to (3.6) and (3.9), we obtain

k sJaα[ f (t)]ksEXr−1(X −E(X )),α(t) −  k sEX−E(X),α(t)  k sMr−1,α(t) ≤ (t − a) tr−1− ar−1
(k sJaα[ f (t)])2. (3.10)

We prove also the following result:

Theorem 3.2. Let X be a continuous random variable having a p_{.d. f . f : [a, b] → R}+. Then we have:

(i∗_{) For any k > 0, s ∈ R/ {−1} and α > 0, β > 0,} k sJaα[ f (t)] ksEXr−1_{(X −E(X )),β}(t) +k_sJ_aβ[ f (t)] k_sE_Xr−1_{(X −E(X )),α}(t) −k sEX,α(t)ksMr−1,β(t) − skEX,β(t) ksMr−1,α(t) (3.11) ≤ k f k2_∞hk sJaα[1]ksJaβ[tr] +ksJaβ[1]ksJaα[tr] −k_sJα a [t]ksJaβtr−1 −ksJaβ[t] skJ α atr−1 i , a < t ≤ b, where f∈ L∞[a, b] .

(ii∗) The inequality k sJaα[ f (t)]ksEXr−1_{(X −E(X )),β}(t) + k_sJ_aβ[ f (t)]k_sE_Xr−1_{(X −E(X )),α}(t) −k sEX,α(t) ksMr−1,β(t) − skEX,β(t) ksMr−1,α(t) (3.12) ≤ (t − a) tr−1− ar−1
k sJaα[ f (t)] ksJaβ[ f (t)] , a < t ≤ b is also valid for any k> 0, s ∈ R/ {−1} and α > 0, β > 0. Proof. Multiplying both sides of (3.4) by ϕ_k,sβ (t, ρ) , where

ϕ_k,sβ (t, ρ) :=(s + 1) 1−α k Γk(α) ts+1− ρs+1
α_k−1 τsp(τ) , ρ ∈ (a,t) , a < t ≤ b, (3.13) we can obtain Z t a Z t a ϕ_k,sα (t, τ) ϕ_k,sβ (t, ρ) H (τ, ρ) dτdρ (3.14) = Z t a Z t a ϕ_k,sα (t, τ) ϕ_k,sβ (t, ρ) (g (τ) − g (ρ)) (h (τ) − h (ρ)) dτdρ. Hence, (s + 1)2  1−α +β_k  k2_Γ k(α) Γk(β ) t Z a t Z a ts+1− τs+1
α_k−1 ts+1− ρs+1
β_k−1 (3.15) ×τsρsp(τ) p (ρ) (g (τ) − g (ρ)) (h (τ) − h (ρ)) dτdρ = k_sJα a [p (t)]ksJaβ[pgh (t)] +ksJaβ[p (t)] ksJaα[pgh (t)] −k sJaα[ph (t)] ksJaβ[pg (t)] − ksJaβ[ph (t)] ksJaα[pg (t)] . In (3.15), we take p(t) = f (t), g(t) = t − E(X ), h(t) = tr−1. So, we get (s + 1)2  1−α +β k  k2_Γ k(α) Γk(β ) t Z a t Z a ts+1− τs+1
αk−1 _ts+1_{− ρ}s+1
βk−1 τsρs ×(τ − ρ) τr−1− ρr−1
) f (τ) f (ρ) dτdρ (3.16) = k_sJα a [ f (t)]ksJaβtr−1(t − E(X )) f (t) + ksJaβ[ f (t)] ksJaαtr−1(t − E(X )) f (t)  −k_sJα a[(t − E(X )) f (t)] ksJaβtr−1f(t) − ksJaβ[(t − E(X )) f (t)]ksJaαtr−1f(t)  = k_sJα a [ f (t)]ksEXr−1_{(X −E(X )),β}(t) + k_sJ_aβ[ f (t)]k_sE_Xr−1_{(X −E(X )),α}(t) −k sEX,α(t) ksMr−1,β(t) − ksEX,β(t) ksMr−1,α(t). We have also (s + 1)2  1−α +β k  k2_Γ k(α) Γk(β ) t Z a t Z a ts+1− τs+1
αk−1 _ts+1_{− ρ}s+1
βk−1 τsρs ×(τ − ρ) τr−1− ρr−1
f (τ) f (ρ)dτdρ (3.17) ≤ k f k2_∞(s + 1) 21−α +β k  k2_Γ k(α) Γk(β ) t Z a t Z a ts+1− τs+1
αk−1 _ts+1_{− ρ}s+1
βk−1 τsρs ×(τ − ρ) τr−1− ρr−1
dτdρ = k f k2_∞hk sJ α a[1]ksJ β a[tr] + ksJ β a[1]ksJ α a [tr] − k sJaα[t]ksJaβtr−1 −ksJaβ[t] ksJaαtr−1 i . By (3.16) and (3.17), we obtain (3.11). To prove (3.12), we remark that

(s + 1)2  1−α +β k  k2_Γ k(α) Γk(β ) t Z a t Z a ts+1− τs+1
α_k−1 ts+1− ρs+1
β_k−1 τsρs ×(τ − ρ) τr−1− ρr−1
f (τ) f (ρ)dτdρ (3.18) ≤ sup τ ,ρ ∈[a,t] |(τ − ρ| τr−1− ρr−1 (s + 1)2  1−α +β_k  k2_Γ k(α) Γk(β ) t Z a t Z a ts+1− τs+1
α_k−1 ts+1− ρs+1
β_k−1 ×τsρsf(τ) f (ρ)dτdρ = (t − a) tr−1− ar−1
_k sJaα[ f (t)] ksJaβ[ f (t)] . Therefore, by (3.16) and (3.18), we get (3.12).

Remark 3.3. If we take α = β in Theorem 15, we obtain Theorem 14.

We give also the following (k, s) −fractional integral in-equality:

Theorem 3.4. Let X be a continuous random variable hav-ing a p.d. f . f : [a, b] → R+_{. Assume that there exist} con-stants ϕ, φ such that ϕ ≤ f (t) ≤ φ . Then, for all k > 0, s ∈ R/ {−1} , α > 0, we have k sJaα[ f (t)] ksM2r,α(t)− ksMr,α2 (t) ≤ 1 4(b r_{− a}r₎2k sJaα[ f (t)] 2 , a < t ≤ b. (3.19)

Proof. Using Theorem 2.4 of [17], we can write    k sJaα[p(t)] ksJaα[phg(t)] − ksJaα[ph(t)] ksJaα[pg(t)]   ≤ 1 4  k sJaα[p(t)] 2 (φ − ϕ)2. (3.20)

In (3.20), we replace h by g, we will have    k sJaα[p (t)]ksJaα pg2(t) − (ksJaα[pg (t)])2   ≤ 1 4  k sJaα[p(t)] 2 (φ − ϕ)2. (3.21) Taking p(t) = f (t), g(t) = tr, a < t ≤ b in (3.21), we obtain    k sJaα[ f (t)]ksJaαt2rf(t) − (ksJaα[trf(t)])2   ≤ 1 4  k sJaα[ f (t)] 2 (φ − ϕ)2. (3.22)

In (3.22), we take φ = brand ϕ = ar, then we have 0 ≤k_sJα a[ f (t)]ksJaαt2rf(t) −(ksJaα[trf(t)])2≤ 1 4(b r_{− a}r₎2k sJaα[ f (t)] 2 . (3.23)

This implies that k sJaα[ f (t)]ksM2r,α(t)−ksMr,α2 (t) ≤ 1 4(b r_{− a}r₎2k sJaα[ f (t)] 2 . (3.24)

Finally, we present the following result:

Theorem 3.5. Let X be a continuous random variable hav-ing a p_{.d. f . f : [a, b] → R}+. Assume that there exist con-stants ϕ, φ such that ϕ ≤ f (t) ≤ φ . Then, for all k > 0, s ∈ R/ {−1} , α > 0, β > 0, we have k sJaα[ f (t)] ksM2r,β(t) + skJaβ[ f (t)]ksM2r,α(t) (3.25) +2arbr k_sJα a[ f (t)] ksJaβ[ f (t)] ≤ (ar+ br)k_sJα a[ f (t)] ksMr,β(t) + (ar+ br)ksJaβ[ f (t)] ksMr,α(t), a < t ≤ b. Proof. We take p(t) = f (t), g(t) = tr, a < t ≤ b and by

The-orem 2.5 of [17], we can write h k sJaα[ f (t)] ksJaβt2rf(t) +ksJaβ[ f (t)] ksJaαt2rf(t)  (3.26) −2k sJaα[trf(t)]ksJaβ[trf(t)] i2 ≤ hφk_sJα a[ f (t)] − ksJaα[trf(t)]   k sJaβ[trf(t)] − ϕksJaβ[ f (t)]  +k_sJα a [trf(t)] − ϕksJaα[ f (t)]   φk_sJβ a[ f (t)] − ksJaβ[trf(t)] i2 . Combining (3.21) and (3.26) and taking into account the fact

that the left-hand side of (3.21) is positive, we can write k sJaα[ f (t)] ksJaβt2rf(t) +ksJaβ[ f (t)] skJaαt2rf(t)  −2s kJ α a [trf(t))]ksJ β a[trf(t)] (3.27) ≤ φ k_sJ_aα[ f (t)] − k_sJ_aα[trf(t)]   k sJaβ[trf(t)] − ϕksJaβ[ f (t)]  +k_sJα a[trf(t)] − ϕksJaα[ f (t)]   φk_sJ_aβ[ f (t)] −k_sJ_aβ[trf(t)]  . Therefore k sJaα[ f (t)] ksM2r,β(t) + ksJaβ[ f (t)] ksM2r,α(t) − 2ksMr,β(t)ksMr,α(t) ≤ φ k_sJα a[ f (t)] − ksMr,α(t)   k sMr,β(t) − ϕksJaβ[ f (t)]  (3.28) +k_sMr,α(t) − ϕksJaα[ f (t)]   φ k_sJβ a[ f (t)] −ksMr,β(t)  . This implies that

k

sJaα[ f (t)] ksM2r,β(t) + ksJaβ[ f (t)] ksM2r,α(t) + 2ϕφskJaα[ f (t)]ksJaβ[ f (t)] ≤ (φ + ϕ)ksJaα[ f (t)] ksMr,β(t) + ksJaβ[ f (t)] ksMr,α(t)



. (3.29)

In (3.29), we take φ = br, ϕ = ar_{, then we have} k sJaα[ f (t)]ksM2r,β(t) + ksJaβ[ f (t)] skM2r,α(t) + 2arbr skJaα[ f (t)] ksJaβ[ f (t)] ≤ (ar+ br)k_sJα a[ f (t)] ksMr,β(t) + ksJaβ[ f (t)] ksMr,α(t)  . (3.30)

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ISSN(P):2319 − 3786

Malaya Journal of Matematik

ISSN(O):2321 − 5666