SOME RESULTS ASSOCIATED WITH FRACTIONAL
CALCULUS OPERATORS INVOLVING APPELL
HYPERGEOMETRIC FUNCTION
Fractional Calculus Operators
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
R. K. RAINA
10/11 Ganpati Vihar, Opposite Sector 5
Udaipur 313002, Rajasthan, India
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EMail: rkraina_7@hotmail.com
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Received:
14 August, 2008
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Accepted:
11 January, 2009
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Communicated by:
H.M. Srivastava
2000 AMS Sub. Class.:
26A33, 30C45.
Key words:
Analytic functions, Hardy space, Fractional derivatives and fractional integrals,
Appell hypergeometric function, Inclusion relation.
Abstract:
A class of fractional derivative operators (with the Appell hypergeometric function in the kernel) is used here to define a new subclass of analytic functions and
a coefficient bound inequality is established for this class of functions. Also, an
inclusion theorem for a class of fractional integral operators involving the Hardy
space of analytic functions is proved. The concluding remarks briefly mentions
the relevances of the main results and possibilities of further work by using these
new classes of fractional calculus operators.
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Contents
1
Introduction, Definitions and Preliminaries
3
2
A Set of Coefficient Bounds
7
3
Inclusion Relations
9
Fractional Calculus Operators
4
Concluding Remarks
14
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
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1.
Introduction, Definitions and Preliminaries
Let A(n) denote the class of functions f (z) normalized by
f (z) = z +
(1.1)
∞
X
ak z k
(n ∈ N),
k=n+1
Fractional Calculus Operators
which are analytic in the open unit disk
R. K. Raina
U = {z : z ∈ C
and |z| < 1}.
vol. 10, iss. 1, art. 14, 2009
(α,α0 ,β,β 0 ,γ)
We denote by ∆n
(σ) the subclass of functions in A(n) which also satisfy
the inequality:
n
o
0
(α,α0 ,β,β 0 ,γ)
(1.2)
Re χ1 (α, α0 , β, β 0 , γ)z α+α +γ−1 D0,z
f (z) > σ
(z ∈ U),
(α,α0 ,β,β 0 ,γ)
D0,z
where
is the generalized fractional derivative operator (defined below),
and (for convenience)
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0
0
(1.3) χm (α, α , β, β , γ)
Γ(1 + m + β 0 )Γ(1+m−α−α0 −γ)Γ(1+m−α0 −β −γ)
=
Γ(1+m)Γ(1+m−α0 + β 0 )Γ(1 + m − α − α0 − β − γ)
(m ∈ N),
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provided that
(1.4)
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0 ≤ σ < 1; 0 ≤ γ < 1;
γ< min (−α − α0 , −α0 − β, −α − α0 − β) + m + 1;
β 0 > max(0, α0 ) − m − 1.
Following [8], a function f (z) is said to be in the class Vn (θk ) if f (z) ∈ A(n)
satisfies the condition that
arg(ak ) = θk
(k ≥ n + 1; n ∈ N)
and if there exists a real number ρ such that
(1.5)
θk + (k − 1)ρ ≡ π(mod2π)
(k ≥ n + 1; n ∈ N),
Fractional Calculus Operators
then we say that f (z) is in the class Vn (θk ; ρ). Suppose Vn = ∪Vn (θk ; ρ) over all pos(α,α0 ,β,β 0 ,γ)
sible sequences θk with ρ satisfying (1.5), then we denote by ∇n
(σ) the sub(α,α0 ,β,β 0 ,γ)
class of Vn which consists of functions f (z) belonging to the class ∆n
(σ).
We present here the following family of fractional integral (and derivative) operators which involve the familiar Appell hypergeometric function F3 (see also Kiryakova
[4] and Saigo and Maeda [9]).
Definition 1.1. Let γ > 0 and α, α0 , β, β 0 ∈ R. Then the fractional integral operator
(α,α0 ,β,β 0 ,γ)
I0,z
of a function f (z) is defined by
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(α,α0 ,β,β 0 ,γ)
(1.6) I0,z
f (z)
−α Z z
z
ζ
z
γ−1 −α0
0
0
=
(z − ζ) ζ F3 α, α , β, β ; γ; 1 − , 1 −
f (ζ)dζ
Γ(γ) 0
z
ζ
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
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(γ> 0),
where the function f (z) is analytic in a simply-connected region of the complex zplane containing the origin, and it is understood that (z−ζ)γ−1 denotes the principal
value for 0 5 arg(z − t) < 2π. The function F3 occurring in the kernel of (1.6) is
the familiar Appell hypergeometric function of third type (also known as Horn’s F3
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- function; see, for example, [10]) defined by
(1.7) F3 (α, α0 , β, β 0 ; γ; z, ξ)
∞ X
∞
X
(α)m (α0 )n (β)m (β 0 )n z m ξ n
=
(γ)m+n
m! n!
m=0 n=0
(|z| < 1, |ξ| < 1) ,
which is related to the Gaussian hypergeometric function 2 F1 (α, β; γ; z) by the following relationship:
2 F1 (α, β; γ; z)
= F3 (α, α0 , β, β 0 ; γ; z, 0)
= F3 (α, 0, β, β 0 ; γ; z, ξ) = F3 (α, α0 , β, 0; γ; z, ξ) .
Fractional Calculus Operators
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
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Definition 1.2. The fractional derivative operator
defined by
(1.8)
(α,α0 ,β,β 0 ,γ)
D0,z
f (z)
(α,α0 ,β,β 0 ,γ)
D0,z
dn (α,α0 ,β−n,β 0 ,n−γ)
= n I0,z
f (z)
dz
of a function f (z) is
(n − 1 ≤ γ < n; n ∈ N).
It may be observed that for
(1.9)
α = λ + µ,
α0 = β 0 = 0,
β = −η,
γ = λ,
(λ+µ,0,−η,0,λ)
I0,z
α = µ − λ,
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λ,µ,η
= I0,z
λ,µ,η
in terms of the Saigo type fractional integral operator I0,z
([12]). On the other
hand, if
(1.11)
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we obtain the relationship
(1.10)
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α0 = β 0 = 0,
β = −η,
γ = λ,
then we get
(µ−λ,0,−η,0,λ)
D0,z
(1.12)
(λ,µ,η
where J0,z
ther, when
(1.13)
λ,µ,η
= J0,z
,
is the Saigo type fractional derivative operator ([6]; see also [7]). Furα = β 0 = 0,
α0 = 1 − µ,
γ = λ (or − λ),
Fractional Calculus Operators
R. K. Raina
(0,1−µ,0,0,λ)
(0,1−µ,0,0,−λ)
then the operators I0,z
and D0,z
correspond to the differentialλ
integral operators Qµ due to Dziok [2].
Let Hp (0 ≤ p < ∞) be the class of analytic functions in U such that
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kf kp = lim {Mp (r, f )} < ∞,
(1.14)
Contents
r→1−
where
(1.15)
vol. 10, iss. 1, art. 14, 2009
kf kp =
 R 1
2π 1

iθ p p
 2π
f
(re
)
(0 < p < ∞),
0

 sup |f (z)| .
|z|≤r
In this paper we first define a new function class in terms of the fractional derivative operators (with the Appell hypergeometric function in the kernel) and then establish a coefficient bound inequality for this function class. Also, we prove an inclusion theorem for a class of fractional integral operators involving the Hardy space
of analytic functions. The relevance of the main results and possibilities of further
work by using the new classes of fractional calculus operators are briefly pointed out
in the concluding section of this paper.
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2.
A Set of Coefficient Bounds
We begin by proving the following coefficient bounds inequality for a function f (z)
(α,α0 ,β,β 0 ,γ)
to be in the class ∆n
(σ).
(α,α0 ,β,β 0 ,γ)
Theorem 2.1. Let f (z) defined by (1.1) be in the class ∆n
(σ), then
∞
X
|ak |
1−σ
≤
,
0
0
χk (α, α , β, β , γ)
χ1 (α, α0 , β, β 0 , γ)
k=n+1
(2.1)
Fractional Calculus Operators
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
where χm (α, α0 , β, β 0 , γ) is defined by (1.3). The result is sharp.
Proof. Assume that
n
o
0
(α,α0 ,β,β 0 ,γ)
Re χ1 (α, α0 , β, β 0 , γ)z α+α −γ−1 D0,z
f (z) > σ
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(z ∈ U).
Using (1.1) and the formula (see, e.g. [9, p. 394]):
(2.2)
(α,α0 ,β,β 0 ,γ)
D0,z
zq
Γ(1 + q)Γ(1 + q − α0 + β 0 )Γ(1 + q − α − β − γ)
0
=
z q−α−α −γ ,
0
0
0
Γ(1 + q + β )Γ(1 + q − α − β − γ)Γ(1 + q − α − α − γ)
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(0 ≤ γ < 1; α, α0 , β, β 0 ∈ R; q > max (0, α0 − β 0 , α + β + γ) − 1)
we obtain
(
(2.3)
∞
X
χ1 (α, α0 , β, β 0 , γ)
Re 1 +
a z k−1
0 , β, β 0 , γ) k
χ
(α,
α
k
k=n+1
)
>σ
(z ∈ U),
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and for f (z) ∈ Vn (θk ; ρ) (z = reiθ ), the inequality thus obtainable from (2.3) on
letting r → 1− therein, readily yields
(
)
∞
X
χ1 (α, α0 , β, β 0 , γ)
(2.4)
Re 1 +
|ak | exp (i (θk + (k − 1) ρ)) > σ.
χk (α, α0 , β, β 0 , γ)
k=n+1
If we apply (1.5), then (2.4) gives
(2.5)
1−
Fractional Calculus Operators
∞
X
χ1 (α, α0 , β, β 0 , γ)
|ak | > σ,
χk (α, α0 , β, β 0 , γ)
k=n+1
which leads to the desired inequality (2.1). We also observe that the equality sign in
(2.1) is attained for the function f (z) defined by
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
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(2.6)
(1 − σ)χk (α, α0 , β, β 0 , γ) k
f (z) = z +
z exp (iθk )
χ1 (α, α0 , β, β 0 , γ)
and this completes the proof of Theorem 2.1.
(k ≥ n + 1; n ∈ N),
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3.
Inclusion Relations
Under the hypotheses of Definition 1.1, let
(3.1)
γ > 0; min (γ − α − α0 , γ − α0 − β, β 0 , γ − α − α0 − β, β 0 − α0 ) > −2;
α, α0 , β, β 0 ∈ R,
then the fractional integral operator
0 ,β,β 0 ,γ)
Ω(α,α
z
Fractional Calculus Operators
:A→A
(A(1) = A)
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
is defined by
(3.2)
0
0
0
(α,α0 ,β,β 0 ,γ)
,β,β ,γ)
Ω(α,α
f (z) = χ1 (α, α0 , β, β 0 , −γ)z α+α +γ I(0,z)
z
f (z).
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where χ1 (α, α0 , β, β 0 , −γ) is given by (1.3).
By using the formula ([9, p. 394]; see also [4, p. 170, Lemma 9])
(α,α0 ,β,β 0 ,γ) q
(3.3) I0,z
z
Γ(1 + q)Γ(1 + q − α0 + β 0 )Γ(1 + q − α − α0 − β + γ) q−α−α0 +γ
=
z
,
Γ(1 + q + β 0 )Γ(1 + q − α0 − β + γ)Γ(1 + q − α − α0 + γ)
(γ > 0; α, α0 , β, β 0 ∈ R; q > max (0, α0 − β 0 , α + β − γ) − 1)
it follows from (1.1), (3.2) and (3.3) that
(3.4)
(α,α0 ,β,β 0 ,γ)
Ωz
f (z) = z + χ1 (α, α0 , β, β 0 , −γ)
∞
X
k=2
0
0
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ak
zk ,
0
χk (α, α , β, β 0 , −γ)
where (as before) χk (α, α , β, β , −γ) is given by (1.3).
Before stating and proving our main inclusion theorem, we recall here the following known results concerning the class R(ρ) in A which satisfies the inequality
that <{f 0 (z)} > ρ(0 ≤ ρ < 1), where R(1) is denoted by R.
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Lemma 3.1 ([3, p. 141]). Let f (z) ∈ R, then
f (z) ∈ Hp
(3.5)
: (0 < p < ∞).
Lemma 3.2 ([5, p. 533]). Let f (z) defined by (1.1) be in the class R(ρ) (0 ≤ ρ < 1),
then
|ak | ≤
(3.6)
2
k
(k = 2, 3, 4, ...).
R. K. Raina
Theorem 3.3. Let f (z) ∈ R, then (under the constraints stated in (3.1))
0
0
,β,β ,γ)
Ω(α,α
f (z) ∈ Hp
z
(3.7)
Fractional Calculus Operators
vol. 10, iss. 1, art. 14, 2009
(0 < p < ∞)
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and
0
0
Ωz(α,α ,β,β ,γ) f (z) ∈ H∞
(3.8)
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(γ > 1).
Proof. In view of (1.6) and (3.2), we obtain
0
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0
,β,β ,γ)
(3.9) Ω(α,α
f (z) = χ1 (α, α0 , β, β 0 , −γ)
z
Z 1
1
γ−1 −α0
0
0
×
(1 − t) t F3 α, α , β, β ; γ; 1 − t, 1 −
f (zt)dt.
t
0
This implies that
d (α,α0 ,β,β 0 ,γ)
(3.10) Re
Ω
f (z) = χ1 (α, α0 , β, β 0 , −γ)
dz z
Z 1
1
γ−1 1−α0
0
0
×
(1 − t) t
F3 α, α , β, β ; γ; 1 − t, 1 −
< {f 0 (zt)} dt.
t
0
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Since f (z) ∈ R, therefore, we infer from (3.10) that
(3.11)
0
0
,β,β ,γ)
Ω(α,α
f (z) ∈ R,
z
and applying Lemma 3.1, (3.11) gives the inclusion relation (3.7) under the conditions stated in (3.1).
To prove the result (3.8), we observe the following three-term recurrence relation:
Fractional Calculus Operators
d (α,α0 ,β,β 0 ,γ)
(3.12)
Ω
f (z)
dz z
n
0 ,β,β 0 ,γ−1)
−1
=z
(γ − α0 − β + 1) Ω(α,α
f (z)
z
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
o
0 ,β,β 0 ,γ)
− (γ − α0 − β) Ω(α,α
f
(z)
,
z
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which yields the inequality
p
d (α,α0 ,β,β 0 ,γ)
(3.13) Ωz
f (z)
dz
p
n
p (α,α0 ,β,β 0 ,γ−1)
−p
0
≤r
(γ − α − β + 1) Ωz
f (z)
p o
0 ,β,β 0 ,γ)
p
− (γ − α0 − β) Ω(α,α
f
(z)
z
and 0 < p < ∞.
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(|z| = r) ,
provided that
(3.14) γ > 1;
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min 1 + γ − α − α0 , 1 + γ − α0 − β, 1 + β 0 ,
1 + γ − α − α0 − β, 1 + β 0 − α0 > −1;
α, α0 , β, β 0 ∈ R
Making use of (1.14) and (1.15), the above inequality (3.13) (with p = 1) yields
d (α,α0 ,β,β 0 ,γ)
(3.15) M1 r, Ωz
f (z)
dz
n
0 ,β,β 0 ,γ−1)
f
(z)
≤ r−1 (γ − α0 − β + 1) M1 r, Ω(α,α
z
o
0
(α,α0 ,β,β 0 ,γ)
− (γ − α − β) M1 r, Ωz
f (z)
Fractional Calculus Operators
R. K. Raina
and
(3.16)
vol. 10, iss. 1, art. 14, 2009
d (α,α0 ,β,β 0 ,γ)
(α,α0 ,β,β 0 ,γ−1)
Ωz
≤ (γ − α0 − β + 1) f
(z)
Ω
f
(z)
z
dz
1
1
(α,α0 ,β,β 0 ,γ)
0
− (γ − α − β) Ωz
f (z) .
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1
Applying (3.7), we infer (under the constraints stated in (3.14)) that
(3.17)
0
0
Ωz(α,α ,β,β ,γ−1) f (z) ∈ H1
and
0
0
,β,β ,γ)
Ω(α,α
f (z) ∈ H1
z
(γ > 1),
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and consequently (3.16) implies that
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d (α,α0 ,β,β 0 ,γ)
Ω
f (z) ∈ H1 ,
dz z
provided that the conditions stated in (3.14) are satisfied. By appealing to a known
(α,α0 ,β,β 0 ,γ)
result [1, p. 42, Theorem 3.11], we infer from (3.17) that Ωz
f (z) is continuous in U∗ = {z : z ∈ C and |z| ≥ 1} . But U∗ being compact, we finally conclude
(α,α0 ,β,β 0 ,γ)
that Ωz
f (z) is a bounded analytic function in U, and the proof of the second
assertion (3.8) of Theorem 3.3 is complete.
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The assertion (3.8) of Theorem 3.3 can also be proved by applying Lemma 3.2
(see also [3, p. 145]). Indeed, it follows from (3.4) and (3.6) that
∞
X
(α,α0 ,β,β 0 ,γ)
0
0
Ω
f
(z)
≤
|z|
+
χ
(α,
α
,
β,
β
,
−γ)
z
1
k
|ak |
z χk (α, α0 , β, β 0 , −γ)
k=2
∞
X
≤ 1 + 2 : χ1 (α, α0 , β, β 0 , −γ)
k=2
Γ(k)
Γ(k + 1)χk (α, α0 , β, β 0 , −γ)
2(2 − α0 + β 0 )(2 − α − α0 − β + γ)
=1+
(2 + β 0 )(2 − α − α0 + γ)(2 − α0 − β + γ)
1, 2, 3 − α0 + β 0 , 3 − α − α0 − β + γ;
× 4 F3
3 + β 0 , 3 − α − α0 + γ, 3 − α0 − β + γ;
Fractional Calculus Operators
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vol. 10, iss. 1, art. 14, 2009
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in terms of the generalized hypergeometric function.
Now, for fixed values of the parameters α, α0 , β, β 0 , γ satisfying the conditions
stated in (3.1), we observe that by using the asymptotic formula [10, p. 109],
Γ(k)
= o k −γ−1
0
0
Γ(k + 1)χk (α, α , β, β , −γ)
and since γ > 1, this proves our assertion (3.8).
(k → ∞),
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4.
Concluding Remarks
In view of the relationships (1.10) and (1.12), the main results (Theorems 2.1 and
3.3) of this paper would correspond to the results due to Raina and Srivastava [8,
p. 75, Theorem 1; p. 79, Theorem 7]. Furthermore, in view of the relationship
(1.13), we can easily apply Theorems 2.1 and 3.3 to obtain the corresponding results
associated with Dziok’s differential-integral operators [2]. The family of fractional
calculus operators (fractional integrals and fractional derivatives) defined by (1.6)
and (1.8) can fruitfully be used in Geometric Function Theory. Several new analytic,
multivalent (or meromorphic) function classes can be defined and the various properties of coefficient estimates, distortion bounds, radii of starlikeness, convexity and
close to convexity for such contemplated classes investigated.
Fractional Calculus Operators
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vol. 10, iss. 1, art. 14, 2009
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References
[1] P.L. DUREN, Theory of H p Spaces, Vol. 38, A series of monographs and textbooks in pure and applied mathematics, Academic Press, New York, 1970.
[2] J. DZIOK, Applications of the Jack lemma, Acta Math. Hungar., 105 (2004),
93–102.
[3] I. B. JUNG, Y. C. KIM AND H. M. SRIVASTAVA, The Hardy space of analytic
functions associated with certain one-parameter families of integral operators,
J. Math. Anal. Appl., 176 (1993), 138–147.
[4] V. KIRYAKOVA, On two Saigo’s fractional integal operators in the class of
univalent functions, Fracl. Cal. Appld. Math., 9 (2006), 159–176.
[5] T.H. MACGREGOR, Functions whose derivative has a positive real part, Trans.
Amer. Math. Soc., 104 (1962), 532–537.
[6] R.K. RAINA AND T.S. NAHAR, Characterization properties for starlikeness
and convexity of some subclasses of analytic functions involving a class of
fractional derivative operators, Acta Math. Univ. Comenianae, 69 (2000), 1–8.
[7] R.K. RAINA AND H.M. SRIVASTAVA, A certain subclass of analytic functions associated with operators of fractional calculus, Comput. Math. Appl., 32
(1996), 13–19.
[8] R.K. RAINA AND H.M. SRIVASTAVA, Some subclasses of analytic functions
associated with fractional calculus operators, Comput. Math. Appl., 37 (1999),
73–84.
[9] M. SAIGO AND N. MAEDA, More generalization of fractional calculus, in:
Transform Methods and Special Functions, Varna’96 (Proc. Second Internat.
Fractional Calculus Operators
R. K. Raina
vol. 10, iss. 1, art. 14, 2009
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Workshop), Science Culture Technology Publishing, Singapore (1998), 386400.
[10] H.M. SRIVASTAVA AND P.W. KARLSSON, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and
Sons, New York, 1985.
[11] H.M. SRIVASTAVA AND S. OWA, Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and
Hongkong, 1992.
[12] H.M. SRIVASTAVA, M. SAIGO AND S. OWA, A class of distortion theorems
involving certain operators of fractional calculus, J. Math. Anal. Appl., 131
(1988), 412–420.
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vol. 10, iss. 1, art. 14, 2009
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