MONTGOMERY IDENTITIES FOR FRACTIONAL
INTEGRALS AND RELATED FRACTIONAL
INEQUALITIES
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
G. ANASTASSIOU
Department of Mathematical Sciences
The University of Memphis
Memphis,TN 38152, USA
vol. 10, iss. 4, art. 97, 2009
EMail: ganastss@memphis.edu
M. R. HOOSHMANDASL, A. GHASEMI AND F. MOFTAKHARZADE
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Department of Mathematical Sciences
Yazd University, Yazd, Iran
EMail: hooshmandasl@yazduni.ac.ir
esfahan.ghasemi@yahoo.com
Contents
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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Contents
1
Introduction
3
2
Fractional Calculus
5
3
Montgomery Identities for Fractional Integrals
6
4
An Ostrowski Type Fractional Inequality
11
5
A Grüss Type Fractional Inequality
13
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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1.
Introduction
Let f : [a, b] → R be differentiable on [a, b], and f 0 : [a, b] → R be integrable on
[a, b], then the following Montgomery identity holds [1]:
Z b
Z b
1
(1.1)
f (x) =
f (t) dt +
P1 (x, t)f 0 (t) dt,
b−a a
a
where P1 (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.
vol. 10, iss. 4, art. 97, 2009
Title Page
Suppose now that w : [a, b] → [0, ∞) is some probability density function, i.e. it
Rb
Rx
is a positive integrable function satisfying a w(t) dt = 1, and W (t) = a w(x) dx
for t ∈ [a, b], W (t) = 0 for t < a and W (t) = 1 for t > b. The following identity
(given by Pečarić in [4]) is the weighted generalization of the Montgomery identity:
Z
(1.3)
f (x) =
b
Z
w(t)f (t) dt +
a
where the weighted Peano kernel is
(
Pw (x, t) =
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
Contents
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b
0
Pw (x, t)f (t) dt,
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a
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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 function f : [a, b] × [c, d] → R such that the partial
Close
2
f (s,t)
(s,t) ∂f (s,t)
derivatives ∂f∂s
, ∂t and ∂ ∂s∂t
all exist and are continuous on [a, b] × [c, d], so
for all (x, y) ∈ [a, b] × [c, d] we have:
Z
d
Z
b
(1.4) (d − c)(b − a)f (x, y) =
f (s, t) ds dt
c
a
Z dZ b
Z bZ d
∂f (s, t)
∂f (s, t)
+
p(x, s) ds dt +
q(y, t) dt ds
∂s
∂t
c
a
a
c
Z dZ b 2
∂ f (s, t)
+
p(x, s)q(y, t) ds dt,
∂s∂t
c
a
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
vol. 10, iss. 4, art. 97, 2009
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 order α > 0 with a ≥ 0
is defined as
Z x
1
α
(2.1)
Ja f (x) =
(x − t)α−1 f (t) dt,
Γ(α) a
Ja0 f (x) = f (x).
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
Properties of the operator can be found in [8]. In the case of α = 1, the fractional
integral reduces to the classical integral.
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vol. 10, iss. 4, art. 97, 2009
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3.
Montgomery Identities for Fractional Integrals
Montgomery identities can be generalized in fractional integral forms, the main results of which are given in the following lemmas.
Lemma 3.1. Let f : [a, b] → R be differentiable on [a, b], and f 0 : [a, b] → R be
integrable on [a, b], then the following Montgomery identity for fractional integrals
holds:
(3.1) f (x) =
Γ(α)
(b − x)1−α Jaα f (b)
b−a
− Jaα−1 (P2 (x, b)f (b)) + Jaα (P2 (x, b)f 0 (b)),
vol. 10, iss. 4, art. 97, 2009
α ≥ 1,
where P2 (x, t) is the fractional Peano kernel defined by:
( t−a
(b − x)1−α Γ(α), a ≤ t ≤ x,
b−a
(3.2)
P2 (x, t) =
t−b
(b − x)1−α Γ(α), x < t ≤ b.
b−a
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
(3.3)
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
Γ(α)Jaα (P1 (x, b)f 0 (b))
Z b
=
(b − t)α−1 P1 (x, t)f 0 (t) dt
a
Z b
Z x
t−b
t−a
α−1 0
(b − t) f (t) dt +
(b − t)α−1 f 0 (t) dt
=
b
−
a
b
−
a
x
a
Z x
Z b
1
=
(b − t)α−1 f 0 (t) dt −
(b − t)α f 0 (t) dt.
b−a a
a
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Next, integrating by parts and using (3.3), we have
(3.4)
Γ(α)Jaα (P1 (x, b)f 0 (b))
Z x
α
α
= (b − x) f (x) −
Γ(α)Ja f (b) + (α − 1)
(b − t)α−2 f (t)dt
b−a
a
1
= (b − x)α−1 f (x) −
Γ(α)Jaα f (b) + Γ(α)Jaα−1 (P1 (x, b)f (b)).
b−a
α−1
Finally, from (3.4) for α ≥ 1, we obtain
f (x) =
Γ(α)
(b − x)1−α Jaα f (b) − Jaα−1 (P2 (x, b)f (b)) + Jaα (P2 (x, b)f 0 (b)),
b−a
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
vol. 10, iss. 4, art. 97, 2009
and the proof is completed.
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Remark 1. Letting α = 1, formula (3.1) reduces to the classic Montgomery identity
(1.1).
Rb
Lemma 3.2. Let w : [a, b] → [0, ∞) be a probability density function, i.e. a w(t) dt =
Rt
1, and set W (t) = a w(x) dx for a ≤ t ≤ b, W (t) = 0 for t < a and W (t) = 1
for t > b, α ≥ 1. Then the generalization of the weighted Montgomery identity for
fractional integrals is in the following form:
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(3.5) f (x) = (b − x)1−α Γ(α)Jaα (w(b)f (b))
− Jaα−1 (Qw (x, b)f (b)) + Jaα (Qw (x, b)f 0 (b)),
where the weighted fractional Peano kernel is
(
(b − x)1−α Γ(α)W (t),
a ≤ t ≤ x,
(3.6)
Qw (x, t) =
(b − x)1−α Γ(α)(W (t) − 1), x < t ≤ b.
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Proof. From fractional calculus and relation (3.6), we have
(3.7)
Jaα (Qw (x, b)f 0 (b))
Z b
1
=
(b − t)α−1 Qw (x, t)f 0 (t)dt
Γ(α) a
Z b
Z b
1−α
α−1
0
α−1 0
= (b − x)
(b − t) W (t)f (t)dt −
(b − t) f (t)dt .
a
x
Using integration by parts in (3.7) and W (a) = 0, W (b) = 1, we have
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
vol. 10, iss. 4, art. 97, 2009
Z
b
(b − t)α−1 W (t)f 0 (t) dt
(3.8)
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a
= −Γ(α)Jaα (w(b)f (b)) + (α − 1)
Z
b
(b − t)α−2 W (t)f (t)dt,
a
and
Z
(3.9)
b
(b − t)α−1 f 0 (t)dt = −(b − x)α−1 f (x) + (α − 1)
x
Z
b
(b − t)α−2 f (t) dt.
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We apply (3.8) and (3.9) to (3.7), to get
(3.10)
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Jaα (Qw (x, b)f 0 (b))
Z b
1−α
α
= (b − x)
−Γ(α)Ja (w(b)f (b)) − (α − 1)
(b − t)α−2 f (t) dt
x
Z b
α−1
α−2
+(b − x) f (x) + (α − 1)
(b − t) W (t)f (t) dt
a
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= f (x) − Γ(α)(b − x)1−α Jaα (w(b)f (b)) + (b − x)1−α (α − 1)
Z x
Z b
α−2
α−2
(b − t) W (t)f (t) dt +
(b − t) (W (t) − 1)f (t) dt
×
a
1−α
= f (x) − Γ(α)(b − x)
x
α
Ja (w(b)f (b))
+ Jaα−1 (Qw (x, b)f (b)).
Finally, we have obtained that
(3.11) f (x) = (b − x)1−α Γ(α)Jaα (w(b)f (b))
− Jaα−1 (Qw (x, b)f (b)) + Jaα (Qw (x, b)f 0 (b)),
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
vol. 10, iss. 4, art. 97, 2009
proving the claim.
Remark 2. Letting α = 1, the weighted generalization of the Montgomery identity
for fractional integrals in (3.5) reduces to the weighted generalization of the Montgomery identity for integrals in (1.3).
Lemma 3.3. Let a function f : [a, b]×[c, d] → R have continuous partial derivatives
2 f (s,t)
∂f (s,t) ∂f (s,t)
, ∂t and ∂ ∂s∂t
on [a, b] × [c, d], for all (x, y) ∈ [a, b] × [c, d] and α, β ≥ 2.
∂s
Then the following two variables Montgomery identity for fractional integrals holds:
(d − c) (b − a) f (x, y)
∂
α,β
= (b − x) (d − y) Γ(α)Γ(β) Ja,c q(y, d) f (b, d)
∂t
∂f (b, d)
∂ 2 f (b, d)
β,α
+ Jc,a f (b, d) + p(x, b)
+ p(x, b) q(y, d)
∂s
∂s ∂t
∂f (b, d)
β,α−1
− Jc,a
p(x, b) f (b, d) + p(x, b) q(y, d)
∂t
1−α
1−β
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∂f (b, d)
−
q(y, d) f (b, d) + p(x, b) q(y, d)
∂s
i
β−1,α−1
+ Jc,a
p(x, b) q(y, d) f (b, d) ,
β−1,α
Jc,a
where
β,α
Jc,a
f (x, y)
1
=
Γ(α)Γ(β)
Z
c
y
Z
x
(x − s)α−1 (y − t)β−1 f (s, t) ds dt.
a
Also, p(x, s) and q(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 .
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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4.
An Ostrowski Type Fractional Inequality
In 1938, Ostrowski proved the following interesting integral inequality [5]:
"
2 #
Z b
1
1
1
a
+
b
(4.1) f (x) −
x−
(b − a)M,
f (t) dt ≤
+
b−a a
4 (b − a)2
2
0
where f : [a, b] → R is a differentiable function such that |f (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 |f 0 (x)| ≤ M , for
every x ∈ [a, b] and α ≥ 1. Then the following Ostrowski fractional inequality
holds:
Γ(α)
1−α
α
α−1
(4.2) f (x) −
(b − x) Ja f (b) + Ja P2 (x, b)f (b)
b−a
M
b−x
α
1−α
≤
(b − x) 2α
− α − 1 + (b − a) (b − x)
.
α(α + 1)
b−a
Proof. From Lemma 3.1 we have
Γ(α)
1−α
α
α−1
f (x) −
(4.3)
(b
−
x)
J
f
(b)
+
J
(P
(x,
b)f
(b))
2
a
a
b−a
α
0
= Ja (P2 (x, b)f (b)).
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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Therefore, from (4.3) and (2.1) and |f 0 (x)| ≤ M , we have
Z
1 b
α−1
0
(4.4)
(b
−
t)
P
(x,
t)f
(t)
dt
2
Γ(α) a
Z b
1
≤
(b − t)α−1 P2 (x, t)f 0 (t) dt
Γ(α) a
Z b
M
≤
(b − t)α−1 P2 (x, t) dt
Γ(α) a
Z x
Z b
(b − x)1−α
α−1
α
≤M
(b − t) (t − a) dt +
(b − t) dt
b−a
a
x
b−x
M
α
1−α
=
(b − x) 2α
− α − 1 + (b − a) (b − x)
.
α(α + 1)
b−a
This proves inequality (4.2).
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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5.
A Grüss Type Fractional Inequality
In 1935, Grüss proved one of the most celebrated integral inequalities [6], which can
be stated as follows
Z b
Z b
Z b
1
1
(5.1) f (x)g(x) dx −
f
(x)
dx
g(x)
dx
b−a a
(b − a)2 a
a
1
≤ (M − m)(N − n),
4
provided that f and g are two integrable functions on [a, b] and satisfy the conditions
m ≤ f (x) ≤ M,
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
vol. 10, iss. 4, art. 97, 2009
n ≤ g(x) ≤ N,
Title Page
for all x ∈ [a, b], where m, 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 that f (x) and g(x) are two integrable functions for all x ∈
[a, b], and satisfy the conditions
m ≤ (b − x)α−1 f (x) ≤ M,
n ≤ (b − x)α−1 g(x) ≤ N,
where α > 1/2, and m, M, n, N are real constants, the following Grüss fractional
inequality holds:
Γ(2α − 1) 2α−1
1
α
α
(5.2) J
(f
g)(b)
−
J
f
(b)J
g(b)
a
(b − a)Γ2 (α) a
(b − a)2 a
1
≤
(M − m)(N − n).
2
4 Γ (α)
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Proof. If substitute h(x) = (b − x)α−1 f (x) and k(x) = (b − x)α−1 g(x) in (5.1), we
will obtain (5.2).
In [10] some related fractional inequalities are given.
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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References
[1] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities for Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 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. BARNETT AND J. ROUMELIOTIS, An
inequlity of the Ostrowski type for double integrals and applications for cubature formulae, Tamsui Oxf. J. Math. Sci., 16(1) (2000), 1–16.
Montgomery Identities for
Fractional Integrals
G. Anastassiou, M. R. Hooshmandasl,
A. Ghasemi and F. Moftakharzade
vol. 10, iss. 4, art. 97, 2009
Title Page
[4] J.E. PEČARIĆ, On the Čebyšev inequality, Bul. Şti. Tehn. Inst. Politehn. "Traian Vuia" Timişoara, 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 −
Rb
Rb
Rb
a)] a f (x)g(x) dx − [1/(b − a)2 ] a f (x) dx a g(x) dx, Math. Z., 39 (1935),
215–226.
[7] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New Inequalities 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. PEČARIĆ AND A. VUKELIĆ, 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.
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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