MONTGOMERY IDENTITIES FOR FRACTIONAL INTEGRALS AND RELATED FRACTIONAL INEQUALITIES

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Montgomery Identities for Fractional Integrals G. Anastassiou, M. R. Hooshmandasl,

A. Ghasemi and F. Moftakharzade vol. 10, iss. 4, art. 97, 2009

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MONTGOMERY IDENTITIES FOR FRACTIONAL INTEGRALS AND RELATED FRACTIONAL

INEQUALITIES

G. ANASTASSIOU

Department of Mathematical Sciences The University of Memphis

Memphis,TN 38152, USA EMail:ganastss@memphis.edu

M. R. HOOSHMANDASL, A. GHASEMI AND F. MOFTAKHARZADE

Department of Mathematical Sciences Yazd University, Yazd, Iran

EMail:hooshmandasl@yazduni.ac.ir esfahan.ghasemi@yahoo.com f_moftakhar@yahoo.com

Received: 06 August, 2009

Accepted: 29 November, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26A33.

Key words: Montgomery identity; Fractional integral; Ostrowski inequality; Grüss inequality.

Abstract: In the present work we develop some integral identities and inequalities for the fractional integral. We have obtained Montgomery identities for fractional integrals and a generalization for double fractional integrals. We also produced Ostrowski and Grüss inequalities for fractional integrals.

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1. Introduction

Letf : [a, b] → R be differentiable on[a, b], and f⁰ : [a, b] → Rbe integrable on [a, b], then the following Montgomery identity holds [1]:

(1.1) f(x) = 1

b−a Z b

a

f(t)dt+ Z b

a

P₁(x, t)f⁰(t)dt,

whereP₁(x, t)is the Peano kernel

(1.2) P₁(x, t) =

( t−a

b−a, a≤t ≤x,

t−b

b−a, x < t≤b.

Suppose now thatw : [a, b] → [0,∞)is some probability density function, i.e. it is a positive integrable function satisfyingRb

a w(t)dt = 1, andW(t) =Rx

a w(x)dx fort ∈ [a, b],W(t) = 0 fort < aandW(t) = 1fort > b. The following identity (given by Peˇcari´c in [4]) is the weighted generalization of the Montgomery identity:

(1.3) f(x) =

Z b

a

w(t)f(t)dt+ Z b

a

P_w(x, t)f⁰(t)dt,

where the weighted Peano kernel is P_w(x, t) =

( W(t), a≤t≤x, W(t)−1, x < t≤b.

In [2, 3], the authors obtained two identities which generalized (1.1) for functions of two variables. In fact, for a functionf : [a, b]×[c, d] → R such that the partial

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derivatives ^∂f_∂s^(s,t), ^∂f_∂t^(s,t) and ^∂²_∂s∂t^f^(s,t) all exist and are continuous on[a, b]×[c, d], so for all(x, y)∈[a, b]×[c, d]we have:

(1.4) (d−c)(b−a)f(x, y) = Z d

c

Z b

a

f(s, t)ds dt

+ Z d

c

Z b

a

∂f(s, t)

∂s p(x, s)ds dt+ Z b

a

Z d

c

∂f(s, t)

∂t q(y, t)dt ds +

Z d

c

Z b

a

∂²f(s, t)

∂s∂t p(x, s)q(y, t)ds dt, where

(1.5) p(x, s) =

( s−a, a≤s≤x,

s−b, x < s≤b, and q(y, t) =

( t−c, c≤t≤y, t−d, y < t≤d.

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2. Fractional Calculus

We give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used further in this paper.

Definition 2.1. The Riemann-Liouville integral operator of orderα >0witha ≥0 is defined as

J_a^αf(x) = 1 Γ(α)

Z x

a

(x−t)^α−1f(t)dt, (2.1)

J_a⁰f(x) = f(x).

Properties of the operator can be found in [8]. In the case ofα = 1, the fractional integral reduces to the classical integral.

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3. Montgomery Identities for Fractional Integrals

Montgomery identities can be generalized in fractional integral forms, the main re- sults of which are given in the following lemmas.

Lemma 3.1. Let f : [a, b] → Rbe differentiable on [a, b], and f⁰ : [a, b] → R be integrable on[a, b], then the following Montgomery identity for fractional integrals holds:

(3.1) f(x) = Γ(α)

b−a(b−x)^1−αJ_a^αf(b)

−J_a^α−1(P₂(x, b)f(b)) +J_a^α(P₂(x, b)f⁰(b)), α≥1, whereP2(x, t)is the fractional Peano kernel defined by:

(3.2) P₂(x, t) =

( t−a

b−a(b−x)^1−αΓ(α), a≤t≤x,

t−b

b−a(b−x)^1−αΓ(α), x < t≤b.

Proof. In order to prove the Montgomery identity for fractional integrals in relation (3.1), by using the properties of fractional integrals and relation (3.2), we have

Γ(α)J_a^α(P₁(x, b)f⁰(b)) (3.3)

= Z b

a

(b−t)^α−1P₁(x, t)f⁰(t)dt

= Z x

a

t−a

b−a(b−t)^α−1f⁰(t)dt+ Z b

x

t−b

b−a(b−t)^α−1f⁰(t)dt

= Z x

a

(b−t)^α−1f⁰(t)dt− 1 b−a

Z b

a

(b−t)^αf⁰(t)dt.

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Next, integrating by parts and using (3.3), we have Γ(α)J_a^α(P1(x, b)f⁰(b))

(3.4)

= (b−x)^α−1f(x)− α

b−aΓ(α)J_a^αf(b) + (α−1) Z x

a

(b−t)^α−2f(t)dt

= (b−x)^α−1f(x)− 1

b−aΓ(α)J_a^αf(b) + Γ(α)J_a^α−1(P₁(x, b)f(b)).

Finally, from (3.4) forα≥1, we obtain f(x) = Γ(α)

b−a(b−x)^1−αJ_a^αf(b)−J_a^α−1(P₂(x, b)f(b)) +J_a^α(P₂(x, b)f⁰(b)), and the proof is completed.

Remark 1. Lettingα= 1, formula (3.1) reduces to the classic Montgomery identity (1.1).

Lemma 3.2. Letw: [a, b]→[0,∞)be a probability density function, i.e.Rb

a w(t)dt = 1, and setW(t) = Rt

aw(x)dxfora ≤ t ≤ b, W(t) = 0 fort < aandW(t) = 1 fort > b, α ≥ 1. Then the generalization of the weighted Montgomery identity for fractional integrals is in the following form:

(3.5) f(x) = (b−x)^1−αΓ(α)J_a^α(w(b)f(b))

−J_a^α−1(Q_w(x, b)f(b)) +J_a^α(Q_w(x, b)f⁰(b)), where the weighted fractional Peano kernel is

(3.6) Q_w(x, t) =

( (b−x)^1−αΓ(α)W(t), a≤t≤x, (b−x)^1−αΓ(α)(W(t)−1), x < t≤b.

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Proof. From fractional calculus and relation (3.6), we have J_a^α(Q_w(x, b)f⁰(b))

(3.7)

= 1

Γ(α) Z b

a

(b−t)^α−1Q_w(x, t)f⁰(t)dt

= (b−x)^1−α Z b

a

(b−t)^α−1W(t)f⁰(t)dt− Z b

x

(b−t)^α−1f⁰(t)dt

.

Using integration by parts in (3.7) andW(a) = 0, W(b) = 1, we have (3.8)

Z b

a

(b−t)^α−1W(t)f⁰(t)dt

=−Γ(α)J_a^α(w(b)f(b)) + (α−1) Z b

a

(b−t)^α−2W(t)f(t)dt, and

(3.9) Z b

x

(b−t)^α−1f⁰(t)dt=−(b−x)^α−1f(x) + (α−1) Z b

x

(b−t)^α−2f(t)dt.

We apply (3.8) and (3.9) to (3.7), to get J_a^α(Q_w(x, b)f⁰(b))

(3.10)

= (b−x)^1−α

−Γ(α)J_a^α(w(b)f(b))−(α−1) Z b

x

(b−t)^α−2f(t)dt

+(b−x)^α−1f(x) + (α−1) Z b

a

(b−t)^α−2W(t)f(t)dt

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=f(x)−Γ(α)(b−x)^1−αJ_a^α(w(b)f(b)) + (b−x)^1−α(α−1)

× Z x

a

(b−t)^α−2W(t)f(t)dt+ Z b

x

(b−t)^α−2(W(t)−1)f(t)dt

=f(x)−Γ(α)(b−x)^1−αJ_a^α(w(b)f(b)) +J_a^α−1(Q_w(x, b)f(b)).

Finally, we have obtained that

(3.11) f(x) = (b−x)^1−αΓ(α)J_a^α(w(b)f(b))

−J_a^α−1(Q_w(x, b)f(b)) +J_a^α(Q_w(x, b)f⁰(b)), proving the claim.

Remark 2. Lettingα = 1, the weighted generalization of the Montgomery identity for fractional integrals in (3.5) reduces to the weighted generalization of the Mont- gomery identity for integrals in (1.3).

Lemma 3.3. Let a functionf : [a, b]×[c, d]→Rhave continuous partial derivatives

∂f(s,t)

∂s , ^∂f(s,t)_∂t and ^∂²_∂s∂t^f(s,t) on[a, b]×[c, d],for all(x, y)∈[a, b]×[c, d]andα, β ≥2.

Then the following two variables Montgomery identity for fractional integrals holds:

(d−c) (b−a)f(x, y)

= (b−x)^1−α(d−y)^1−βΓ(α)Γ(β)

J_a,c^α,β

q(y, d) ∂

∂tf(b, d)

+J_c,a^β,α

f(b, d) +p(x, b)∂f(b, d)

∂s +p(x, b)q(y, d)∂²f(b, d)

∂s ∂t

−J_c,a^β,α−1

p(x, b)f(b, d) +p(x, b)q(y, d)∂f(b, d)

∂t

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−J_c,a^β−1,α

q(y, d)f(b, d) +p(x, b)q(y, d)∂f(b, d)

∂s

+J_c,a^{β−1,α−1}

p(x, b)q(y, d)f(b, d)i ,

where

J_c,a^β,αf(x, y) = 1 Γ(α)Γ(β)

Z y

c

Z x

a

(x−s)^α−1(y−t)^β−1f(s, t)ds dt.

Also,p(x, s)andq(y, t)are defined by (1.5).

Proof. Put into (1.4), instead of f, the function g(x, y) = f(x, y)(b− x)^α−1(d − y)^β−1.

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4. An Ostrowski Type Fractional Inequality

In 1938, Ostrowski proved the following interesting integral inequality [5]:

(4.1)

f(x)− 1 b−a

Z b

a

f(t)dt

≤

"

1

4 + 1

(b−a)²

x− a+b 2

2#

(b−a)M,

wheref : [a, b] → Ris a differentiable function such that|f⁰(x)| ≤ M, for every x∈[a, b]. Now we extend it to fractional integrals.

Theorem 4.1. Let f : [a, b] → R be differentiable on[a, b]and |f⁰(x)| ≤ M, for every x ∈ [a, b] and α ≥ 1. Then the following Ostrowski fractional inequality holds:

(4.2)

f(x)− Γ(α)

b−a(b−x)^1−αJ_a^αf(b) +J_a^α−1P₂(x, b)f(b)

≤ M α(α+ 1)

(b−x)

2α

b−x b−a

−α−1

+ (b−a)^α(b−x)^1−α

.

Proof. From Lemma3.1we have (4.3)

f(x)− Γ(α)

b−a(b−x)^1−αJ_a^αf(b) +J_a^α−1(P₂(x, b)f(b))

=

J_a^α(P2(x, b)f⁰(b)) .

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Therefore, from (4.3) and (2.1) and|f⁰(x)| ≤M, we have 1

Γ(α)

Z b

a

(b−t)^α−1P₂(x, t)f⁰(t)dt (4.4)

≤ 1 Γ(α)

Z b

a

(b−t)^α−1

P₂(x, t)

f⁰(t) dt

≤ M Γ(α)

Z b

a

(b−t)^α−1

P₂(x, t) dt

≤M(b−x)^1−α b−a

Z x

a

(b−t)^α−1(t−a)dt+ Z b

x

(b−t)^αdt

= M

α(α+ 1)

(b−x)

2α

b−x b−a

−α−1

+ (b−a)^α(b−x)^1−α

.

This proves inequality (4.2).

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5. A Grüss Type Fractional Inequality

In 1935, Grüss proved one of the most celebrated integral inequalities [6], which can be stated as follows

(5.1)

1 b−a

Z b

a

f(x)g(x)dx− 1 (b−a)²

Z b

a

f(x)dx Z b

a

g(x)dx

≤ 1

4(M −m)(N −n), provided thatf andgare two integrable functions on[a, b]and satisfy the conditions

m ≤f(x)≤M, n ≤g(x)≤N, for allx∈[a, b], wherem, M, n, N are given real constants.

A great deal of attention has been given to the above inequality and many papers dealing with various generalizations, extensions, and variants have appeared in the literature [7].

Proposition 5.1. Given thatf(x)andg(x)are two integrable functions for allx ∈ [a, b], and satisfy the conditions

m ≤(b−x)^α−1f(x)≤M, n ≤(b−x)^α−1g(x)≤N,

whereα > 1/2, andm, M, n, N are real constants, the following Grüss fractional inequality holds:

(5.2)

Γ(2α−1)

(b−a)Γ²(α)J_a^2α−1(f g)(b)− 1

(b−a)²J_a^αf(b)J_a^αg(b)

≤ 1

4 Γ²(α)(M −m)(N −n).

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Proof. If substituteh(x) = (b−x)^α−1f(x)andk(x) = (b−x)^α−1g(x)in (5.1), we will obtain (5.2).

In [10] some related fractional inequalities are given.

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References

[1] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Inequalities for Func- tions and their Integrals and Derivatives, Kluwer Academic Publishers, Dor- drecht, 1994.

[2] N.S. BARNETT AND S.S. DRAGOMIR, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27(1) (2001), 1–10.

[3] S.S. DRAGOMIR, P. CERONE, N.S. BARNETTANDJ. ROUMELIOTIS, An inequlity of the Ostrowski type for double integrals and applications for cuba- ture formulae, Tamsui Oxf. J. Math. Sci., 16(1) (2000), 1–16.

[4] J.E. PE ˇCARI ´C, On the ˇCebyšev inequality, Bul. ¸Sti. Tehn. Inst. Politehn. "Tra- ian Vuia" Timi¸soara, 25(39)(1) (1980), 5–9.

[5] A. OSTROWSKI, Über die absolutabweichung einer differentiebaren funktion von ihren integralmittelwert, Comment. Math. Helv., 10 (1938), 226–227.

[6] G. GRÜSS, Über das maximum des absoluten betrages von [1/(b − a)]Rb

af(x)g(x)dx−[1/(b−a)²]Rb

a f(x)dxRb

a g(x)dx, Math. Z., 39 (1935), 215–226.

[7] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C ANDA.M. FINK, Classical and New In- equalities in Analysis, Kluwer Academic, Dordrecht, 1993.

[8] S. MILLER AND B. ROSS, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 199, p. 2.

[9] J. PE ˇCARI ´C AND A. VUKELI ´C, Montgomery identities for function of two variables, J. Math. Anal. Appl., 332 (2007), 617–630.

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[10] G. ANASTASSIOU, Fractional Differentiation Inequalities, Springer, N.Y., Berlin, 2009.