Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 264347, 23 pages
doi:10.1155/2010/264347
Research Article
OnanInequalityofH.G.Hardy
Sajid Iqbal,
1
Kristina Kruli´c,
2
and Josip P eˇcari´c
1, 2
1
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2
Faculty of Textile Technology , University of Zagreb, Prilaz baruna Filipovi
´
ca 28a, 10000 Zagreb, Croatia
Correspondence should be addressed to Sajid Iqbal, sajid
Received 18 June 2010; Accepted 16 October 2010
Academic Editor: Q. Lan
Copyright q 2010 Sajid Iqbal 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.
We state, prove, and discuss new general inequality for convex and increasing functions. As
a special case of that general result, we obtain new fractional inequalities involving fractional
integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H.
G. Har dy from 1918. We also obtain new results involving fractional derivatives of Canavati and
Caputo types as well as fractional integrals of a function with respect to another function. Finally,
we apply our main result to multidimensional settings to obtain new results involving mixed
. The Riemann-Liouville
fractional integrals I
α
a
f and I
α
b
−
f of order α>0aredefinedby
I
α
a
f
x
1
Γ
α
x
a
f
x
f
t
t − x
α−1
dt,
x<b
, 1.2
respectively. Here Γα is the Gamma function. These integrals are called the left-sided and
the right-sided fractional integrals. We denote some properties of the operators I
α
a
f and I
α
b
−
f
of order α>0, see also 4. The first result yields that the fractional integral operators I
α
a
f
and I
α
α
.
1.4
Inequality 1.3, that is the result involving the left-sided fractional integral, was proved by
H. G. Hardy in one of his first papers, see 5. He did not write down the constant, but the
calculation of the constant was hidden inside his proof.
Throughout this paper, all measures are assumed to be positive, all functions are
assumed to be positive and measurable, and expressions of the form 0 ·∞, ∞/∞,and0/0are
taken to be equal to zero. Moreover, by a weight u ux, we mean a nonnegative measurable
function on the actual interval or more general set.
The paper is organized in the following way. After this Introduction, in Section 2 we
state, prove, and discuss new general inequality for convex and increasing functions. As a
special case of that general result, we obtain new fractional inequalities involving fractional
integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality
of H. G. Hardy since 1918. We also obtain new results involving fractional derivatives
of Canavati and Caputo types as well as fractional integrals of a function with respect
to another function. We conclude this paper w ith new results involving mixed Riemann-
Liouville fractional integrals.
2. The Main Results
Let Ω
1
, Σ
1
,μ
1
and Ω
2
, Σ
Throughout this paper, we suppose that Kx > 0a.e.onΩ
1
, and by a weight function
shortly: a weight, we mean a nonnegative measurable function on the actual set. Let Uk
denote the class of functions g : Ω
1
→ with the representation
g
x
Ω
2
k
x, y
f
y
dμ
2
y
,
2.2
u
x
k
x, y
K
x
dμ
1
x
< ∞.
2.3
If φ : 0, ∞ →
is convex and increasing function, then the inequality
Ω
1
u
x
φ
φ
f
y
dμ
2
y
2.4
holds for all measurable functions f : Ω
2
→ and for all functions g ∈ Uk.
Proof. By using Jensen’s inequality and the Fubini theorem, since φ is increasing function, we
find that
Ω
1
u
x
φ
φ
1
K
x
Ω
2
k
x, y
f
y
dμ
2
y
φ
f
y
dμ
2
y
dμ
1
x
Ω
2
φ
2
y
Ω
2
v
y
φ
f
y
dμ
2
y
,
2.5
If φ : 0, ∞ →
is convex and increasing function, then the inequality
b
a
u
x
φ
Γ
α 1
x − a
α
I
α
a
f
x
1
xdx, dμ
2
ydy,
k
x, y
⎧
⎪
⎨
⎪
⎩
x − y
α−1
Γ
α
,a≤ y ≤ x,
0,x<y≤ b,
2.8
we get that Kxx − a
α
/Γα 1 and gxI
α
a
x
dx ≤
b
a
b − y
α
φ
f
y
dy.
2.9
Although 2.4 holds for all convex and increasing functions, some choices of φ are of
particular interest. Namely, we will consider power function. Let q>1andthefunction
φ :
b
a
b − y
α
fy
q
dy.
2.10
Since x ∈ a, b and α1 − q < 0, then we obtain that the left hand side of 2.10 is
b
a
x − a
α
Γα 1
x − a
α
|I
α
q
dx 2.11
and the right-hand side of 2.10 is
b
a
b − y
α
fy
q
dy ≤
b − a
α
b
a
fy
fy
q
dy.
2.13
Taking power 1/q on both sides, we obtain 1.3.
Corollary 2.4. Let u be a weight function on a, b and α>0. I
α
b
−
f denotes the Riemann-Liouville
fractional integral of f.Definev on a, b by
v
y
: α
y
a
u
x
y − x
α−1
α
b
−
f
x
dx ≤
b
a
v
y
φ
f
y
b
−
f
x
dx ≤
b
a
y − a
α
φ
f
y
dy.
q
dx ≤
b
a
y − a
α
fy
q
dy.
2.17
Since x ∈ a, b and α1 − q < 0, then we obtain that the left hand side of 2.17 is
b
a
b − x
α
Γα 1
b − x
I
α
b
−
fx
q
dx
2.18
and the right-hand side of 2.17 is
b
a
y − a
α
fy
q
dy ≤
b − a
b − a
α
Γα 1
q
b
a
fy
q
dy.
2.20
Taking power 1/q on both sides, we obtain 1.3.
Theorem 2.6. Let p, q > 1, 1/p 1/q 1, α>1/q, I
α
a
f and I
α
b
−
f denote the Riemann-Liouville
fractional integral of f, then the following inequalities
2.21
b
a
I
α
b
−
fx
q
dx ≤ C
b
a
fy
q
dy
2.22
hold, where C b − a
qα
f
t
x − t
α−1
dt.
2.23
Then by the H
¨
older inequality, the right-hand side of the above inequality is
≤
1
Γ
α
x
a
x − t
1/p
x
a
|ft|
q
dt
1/q
≤
1
Γ
α
x − a
α−11/p
p
α − 1
1
1/p
x − a
α−11/p
p
α − 1
1
1/p
b
a
|ft|
q
dt
1/q
, for every x ∈
a, b
.
2.25
Consequently, we find
b
a
ft
q
dt
, 2.26
and we obtain
b
a
I
α
a
fx
q
dx ≤
b − a
qα−1q/p1
2.27
Remark 2.7. For α ≥ 1, inequalities 2.21 and 2.22 are refinements of 1.3 since
qα
p
α − 1
1
q−1
≥ qα
q
>α
q
, so C<
b − a
α
αΓα
q
.
2.28
We proved that Theorem 2.6 is a refinement of 1.3, and Corollaries 2.2 and 2.4 are
generalizations of 1.3.
Next, we give results with respect to the generalized Riemann-Liouville fractional
derivative. Let us recall the definition, for details see 1, page 448.
y
dy,
2.29
where n α1,x∈ a, b.
For a, b ∈
, we say that f ∈ L
1
a, b has an L
∞
fractional derivative D
α
a
f α>0 in
a, b, if and only if
1 D
α−k
a
f ∈ Ca, b, k 1, ,nα1,
2 D
α−1
a
f ∈ ACa, b,
3 D
α
a
∈ L
∞
a, b.
Next, lemma is very useful in the upcoming corollary see 1, page 449 and 2.
1
Γ
β − α
x
a
x − y
β−α−1
D
β
a
f
y
dy,
2.31
for all a ≤ x ≤ b.
Corollary 2.9. Let u be a weight function on a, b, and let assumptions in Lemma 2.8 be satisfied.
Define v on a, b by
v
y
:
Γ
β − α 1
x − a
β−α
D
α
a
f
x
dx ≤
b
a
v
y
φ
⎧
⎪
⎨
⎪
⎩
x − y
β−α−1
Γ
β − α
,a≤ y ≤ x,
0,x<y≤ b,
2.34
8 Journal of Inequalities and Applications
we get that Kxx − a
β−α
/Γβ − α 1.Replacef by D
β
a
f. Then, by Lemma 2.8, gx
D
α
a
fx and we get 2.33.
Remark 2.10. In particular for the weight function uxx −a
β−α
, x ∈ a, b in Corollary 2.9,
dx ≤
b
a
b − y
β−α
φ
D
β
a
f
y
dy. 2.35
Let q>1andthefunctionφ :
→ be defined by φxx
q
, then after some calculation,
a
fy
q
dy.
2.36
Next, we define Canavati-type fractional derivative ν-fractional derivative of f, for details
see 1, page 446.Weconsider
C
ν
a, b
f ∈ C
n
a, b
: I
n−ν1
a
f
n
∈ C
1
D
γ
a
f
x
1
Γ
ν − γ
x
a
x − t
ν−γ−1
D
ν
a
f
t
2.40
Journal of Inequalities and Applications 9
If φ : 0, ∞ →
is convex and increasing function, then the inequality
b
a
u
x
φ
Γ
ν − γ 1
x − a
ν−γ
D
γ
a
f
x
Remark 2.13. In particular for the weight function uxx−a
ν−γ
, x ∈ a, b in Corollary 2.12,
we obtain the inequality
b
a
x − a
ν−γ
φ
Γ
ν − γ 1
x − a
ν−γ
D
γ
a
f
x
→ be defined by φxx
q
,then2.42 reduces to
Γ
ν − γ 1
q
b
a
x − a
ν−γ 1−q
D
γ
a
fx
q
dx ≤
b
q
dx ≤
b − a
ν−γ
Γν − γ 1
q
b
a
D
ν
a
fy
q
dy.
2.44
Taking power 1/q on both sides of 2.44,weobtain
D
γ
a
f
b
a
x − a
ν1−q
f
x
q
dx ≤
b
a
b − y
ν
D
ν
a
f
2.47
In the next corollary, we give results with respect to the Caputo fractional derivative.Let
us recall the definition, for details see 1, page 449.
10 Journal of Inequalities and Applications
Let α ≥ 0, n α, g ∈ AC
n
a, b. The Caputo fractional derivative is given by
D
α
∗a
g
t
1
Γ
n − α
x
a
g
n
y
x − y
x − a
n−α
dx < ∞.
2.49
If φ : 0, ∞ →
is convex and increasing function, then the inequality
b
a
u
x
φ
Γ
n − α 1
x − a
n−α
D
α
∗a
g
holds.
Proof. Applying Theorem 2.1 with Ω
1
Ω
2
a, b, dμ
1
xdx, dμ
2
ydy,
k
x, y
⎧
⎪
⎨
⎪
⎩
x − y
n−α−1
Γ
n − α
,a≤ y ≤ x,
0,x<y≤ b,
n−α
D
α
∗a
g
x
dx ≤
b
a
b − y
n−α
φ
g
n
y
n−α
Γn − α 1
q
b
a
g
n
y
q
dy.
2.53
Taking power 1/q on both sides, we obtain
D
α
∗a
g
x
q
≤
∗a
fx
q
dx ≤
b − a
qn−α
Γ
n − α
q
p
n − α − 1
1
q/p
q
n − α
b
γ
∗a
f
x
1
Γ
α − γ
x
a
x − y
α−γ−1
D
α
∗a
f
y
dy,
2.56
for all a ≤ x ≤ b.
Corollary 2.18. Let u be a weight function on a, b and α>0. D
is convex and increasing function, then the inequality
b
a
u
x
φ
Γ
α − γ 1
x − a
α−γ
D
γ
∗a
f
x
a, b, dμ
1
xdx, dμ
2
ydy,
k
x, y
⎧
⎪
⎨
⎪
⎩
x − y
α−γ−1
Γ
α − γ
,a≤ y ≤ x,
0,x<y≤ b,
2.59
we get that Kxx − a
α−γ
/Γα − γ 1.Replacef by D
α
∗a
D
γ
∗a
f
x
dx ≤
b
a
b − y
α−γ
φ
D
α
∗a
f
y
Γα − γ 1
q
b
a
D
α
∗a
fy
q
dy.
2.61
For γ 0, we obtain
b
a
fx
q
dx ≤
I
α
a;g
f
x
1
Γ
α
x
a
g
t
f
t
dt
g
t
f
t
dt
g
t
− g
x
1−α
,x<b,
2.64
respectively.
Corollary 2.20. Let u be a weight function on a, b,andletg be an increasing function on a, b,
such that g
is a continuous function on a, b and α>0. I
α
a
;g
x
− g
a
α
dx < ∞.
2.65
If φ : 0, ∞ →
is convex and increasing function, then the inequality
b
a
u
x
φ
Γ
α 1
g
x
φ
f
y
dy 2.66
holds.
Journal of Inequalities and Applications 13
Proof. Applying Theorem 2.1 with Ω
1
Ω
2
a, b, dμ
1
xdx, dμ
2
ydy,
k
x, y
⎧
Remark 2.21. In particular for the weight function uxg
xgx − ga
α
, x ∈ a, b in
Corollary 2.20, we obtain the inequality
b
a
g
x
g
x
− g
a
α
φ
Γ
α 1
b
a
g
y
g
b
− g
y
α
φ
f
y
dy.
2.68
I
α
a
;g
fx
q
dx
≤
b
a
g
y
g
b
− g
y
I
α
a
;g
fx
q
dx ≤
g
b
− g
a
α
Γα 1
q
b
a
g
−
;g
f denotes the right-sided fractional
integral of a function f with respect to another function g in a, b.Definev on a, b by
v
y
: αg
y
y
a
u
x
g
y
− g
x
g
b
− g
x
α
I
α
b
−
;g
f
x
dx ≤
b
x
I
α
b
−
;g
fx
q
dx ≤
g
b
− g
a
α
−
fx Riemann-Liouville fractional
integral and 2.73 becomes 2.20.
The refinements of 2.70 and 2.73 for α>1/q are given in the following theorem.
Theorem 2.26. Let p, q > 1, 1/p 1/q 1, α>1/q, I
α
a
;g
f and I
α
b
−
;g
f denote the left-sided and
right-sided fractional integral of a function f with respect to another function g in a, b, then the
following inequalities:
b
a
I
α
a
;g
f
p
α − 1
1
q/p
b
a
f
y
q
g
y
dy,
b
a
αq
αq
Γ
α
q
p
α − 1
1
q/p
b
a
f
y
q
g
log
x
y
α−1
f
y
dy
y
,x>a,
2.75
J
α
b−
f
x
1
Γ
α
q
dx
x
≤
log
b/a
α
Γα 1
q
b
a
f
y
q
dy
α
Γα 1
q
b
a
f
y
q
dy
y
.
2.78
Also, from Theorem 2.26 we obtain refinements of 2.77 and 2.78,forα>1/q,
b
a
J
α
a
b
a
f
y
q
dy
y
,
b
a
J
α
b−
fx
q
dx
x
≤
q
dy
y
.
2.79
Some results involving Hadamard type fractional integrals are given in 3, page 110.
Here, we mention the following result that can not be compared with our result.
Let α>0, 1 ≤ p ≤∞,and0≤ a<b≤∞, then the operators J
α
a
f and J
α
b−
f are bounded
in L
p
a, b as follows:
J
α
a
f
p
≤ K
1
f
p
, J
α
α
logb/a
0
t
α−1
e
−t/p
dt.
2.81
Now we present the definitions and some properties of the Erd
´
elyi-Kober type fractional
integrals. Some of these definitions and results were presented by Samko et al. in 4.
Let a, b, 0 ≤ a<b≤∞ be a finite or infinite interval of the half-axis
.Alsolet
α>0, σ>0, and η ∈
. We consider the left- and right-sided integrals of order α ∈ defined
by
I
α
a
;σ;η
f
α
b
−
;σ;η
f
x
σx
ση
Γ
α
b
x
t
σ1−η−α−1
f
t
dt
t
σ
− x
y
u
x
x
−ση
x
σ
− y
σ
α−1
x
σ
− a
σ
α
2
F
1
α, −η; α 1; 1 −
a/x
σ
α, −η; α 1; 1 −
a/x
σ
I
α
a
;σ;η
f
x
dx
≤
b
a
v
y
⎪
⎨
⎪
⎩
1
Γ
α
σx
−σαη
x
σ
− y
σ
1−α
y
σησ−1
,a≤ y ≤ x,
0,x<y≤ b,
2.86
we get that Kx1/Γα11 − a/x
σ
α
2
F
1
σ−1
x
σ
− a
σ
α
2
F
1
x
φ
Γ
α 1
1 −
a/x
σ
α
2
F
− y
σ
α
2
F
1
y
φ
f
y
dy,
2.87
where
2
F
1
y
2
F
x
σηα
y
σ
− x
σ
α−1
b
σ
− x
σ
α
2
F
1
α, α η; α 1; 1 −
b/x
σ
dx < ∞.
2.88
Journal of Inequalities and Applications 17
If φ : 0, ∞ →
is convex and increasing function, then the inequality
I
α
b
−
;σ;η
f
x
dx
≤
b
a
v
y
φ
Γ
α
σx
ση
y
σ
− x
σ
1−α
y
σ1−α−η−1
,x<y≤ b,
0,a≤ y ≤ x,
2.90
we get that Kx1/Γα1b/x
σ
− 1
α
2
F
1
α, αη; α1; 1−b/x
σ
,so2.89 follows.
Remark 2.30. In particular for the weight function uxx
σ−1
σ
α
2
F
1
x
φ
Γ
α 1
b/x
σ
− 1
α
2
F
1
x
F
1
y
φ
f
y
dy,
2.91
where
2
F
1
y
2
F
1
α, −α − η; α 1; 1 − b/y
σ
.
In the next corollary, we give some results related to the Caputo radial fractional
x
∂r
ν
:
1
Γ
n − ν
r
R
1
r − t
n−ν−1
∂
n
f
tω
∂r
n
dt,
2.92
where x ∈
A,thatis,x rω, r ∈ R
∂r
ν
∂
ν
f
x
∂r
ν
if ν ∈ , the usual radial derivative.
2.93
18 Journal of Inequalities and Applications
Corollary 2.31. Let u be a weight function on R
1
,R
2
,and∂
ν
∗R
1
fx/∂r
ν
denotes the Caputo radial
fractional derivative of f.Definev on R
1
,R
2
by
2
R
1
u
r
φ
Γ
n − ν 1
r − R
1
n−ν
∂
ν
∗R
1
f
x
∂r
n
dt 2.95
holds.
Proof. Apply Theorem 2.1 with Ω
1
Ω
2
R
1
,R
2
, dμ
1
xdr, dμ
2
ydt,and
k
r, t
⎧
⎪
2
,weobtain
the following inequality:
R
2
R
1
r − R
1
n−ν
φ
Γ
n − ν 1
r − R
1
n−ν
∂
∂
n
f
tω
∂r
n
dt. 2.97
Let q>1andφ :
→ be defined by φxx
q
,then2.97 becomes
Γn − ν 1
q
R
2
R
R
2
− t
n−ν
∂
n
ftω
∂r
n
q
dt.
2.98
Since r ∈ R
1
,R
2
and 1 − qn − ν ≤ 0, we obtain
R
2
q
R
2
R
1
∂
n
ftω
∂r
n
q
dt.
2.99
Taking power 1/q on both sides, we get
∂
ftω
∂r
n
q
.
2.100
Journal of Inequalities and Applications 19
If ν 0, then
f
x
q
≤
R
2
− R
1
n
Γ
n 1
q
≤
R
2
− R
1
n−ν
Γ
n − ν 1
∂
n
ftω
∂r
n
R
1
,R
2
× S
N−1
.
For a fixed ω ∈ S
N−1
,wedefine
g
ω
r
: f
rω
f
x
, 2.104
where
x ∈ A : B
0,R
2
x
∂r
β
D
β
R
1
f
rω
⎧
⎪
⎨
⎪
⎩
1
Γ
m − β
∂
∂r
m
r
,ω∈ S
N−1
,
K
f
:
ω ∈ S
N−1
: f
·ω
/
∈ L
1
R
1
,R
2
, B
R
1
,R
β
the Riemann-Liouville radial fractional derivative of f of order β.
The following result is given in 1, page 466.
Lemma 2.33. Let ν ≥ γ 1, γ ≥ 0, n : ν, f :
A → with f ∈ L
1
A. Assume that f·ω ∈
AC
n
R
1
,R
2
, for every ω ∈ S
N−1
,andthat∂
ν
R
1
f·ω/∂r
ν
is measurable on R
1
,R
2
for every
ω ∈ S
N−1
. Also assume that there exists ∂
ν
ω
∂r
ν
≤ M
1
, for e very
r, ω
∈
R
1
,R
2
× S
N−1
. 2.109
W e suppose that ∂
j
fR
1
r
R
1
r − t
ν−γ−1
D
ν
R
1
f
tω
dt
2.110
is valid for every x ∈
A, that is, true for every r ∈ R
1
,R
2
and for every ω ∈ S
N−1
, γ>0.
Corollary 2.34. Let u be a weight function on R
1
,R
ν−γ−1
r − R
1
ν−γ
dr < ∞.
2.111
If φ : 0, ∞ →
is convex and increasing, then the inequality
R
2
R
1
u
r
φ
Γ
ν − γ 1
r − R
1
D
ν
R
1
f
tω
dt 2.112
holds.
Proof. Applying Theorem 2.1 with Ω
1
Ω
2
R
1
,R
2
,
k
r, t
Lemma 2.33,wegetgrD
γ
R
1
frω. This will give us 2.112.
Remark 2.35. In particular for the weight function urr − R
1
ν−γ
, r ∈ R
1
,R
2
in above
Corollary 2.34 and for φxx
q
, q>1 we obtain, after some calculation, the following
inequality:
D
γ
R
1
frω
q
q
≤
R
2
− R
1
ν
Γ
ν 1
D
ν
R
1
ftω
q
.
2.115
In the previous corollaries, we derived only inequalities over some subsets of
···Γ
α
n
,
a, b
a
1
,b
1
×···×
a
n
,b
n
, 2.116
and by x > a,wemeanx
1
>a
1
, ,x
n
k
a
k
f
x
1
, ,x
k−1
,t
k
,x
k1
, ,x
n
x
k
− t
k
α
k
−1
dt
k
,
x
k
x
1
, ,x
k−1
,t
k
,x
k1
, ,x
n
t
k
− x
k
α
k
−1
dt
k
,
x
k
<b
k
, 2.118
x
1
a
1
···
x
n
a
n
f
t
x − t
α−1
dt,
x > a
,
I
α
b
−
f
.
2.119
22 Journal of Inequalities and Applications
Corollary 2.36. Let u be a weight function on a, b and α>0. I
α
a
f denotes the mixed partial
Riemann-Liouville fractional integral of f.Definev on a, b by
v
y
: α
b
1
y
1
···
b
n
y
n
u
x
α 1
x − a
α
I
α
a
f
x
dx ≤
b
1
a
1
···
b
n
⎧
⎪
⎨
⎪
⎩
x − y
α−1
Γ
α
, a ≤ y ≤ x,
0, x < y ≤ b,
2.122
we get that Kxx − a
α
/Γα 1 and gxI
α
a
fx,so2.121 follows.
Corollary 2.37. Let u be a weight function on a, b and α>0. I
α
b
−
f denotes the mixed partial
Riemann-Liouville fractional integral of f.Definev on a, b by
If φ : 0, ∞ →
is convex and increasing function, then the inequality
b
1
a
1
···
b
n
a
n
u
x
φ
Γ
α 1
b − x
α
f
y
dy
2.124
holds for all measurable functions f : a, b →
.
Remark 2.38. Analogous to Remarks 2.3 and 2.5, we obtain multidimensional version of
inequality 1.3 for q>1 as follows:
b
1
a
1
···
b
n
a
n
I
α
q
dy,
b
1
a
1
···
b
n
a
n
I
α
b
−
fx
g
dx ≤
b − a
References
1 G. A. Anastassiou, Fractional Differentiation Inequalities, Springer ScienceBusinness Media, LLC,
Dordrecht, the Netherlands, 2009.
2 G. D. Handley, J. J. Koliha, and J. Pe
ˇ
cari
´
c, “Hilbert-Pachpatte type integral inequalities for fractional
derivatives,” Fractional Calculus & Applied Analysis, vol. 4, no. 1, pp. 37–46, 2001.
3 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differ ential Equations,
vol. 204 of North-Holland Mathematics Studies, Elsevier, New York, NY, USA, 2006.
4 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivatives : Theory and Applications,
Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
5 H.G. Hardy, “Notes on some points in the integral calculus,” Messenger of Mathematics, vol. 47, no. 10,
pp. 145–150, 1918.
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