Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 4, Article 97, 2005
ON A CERTAIN CLASS OF p−VALENT FUNCTIONS WITH NEGATIVE
COEFFICIENTS
H.Ö. GÜNEY AND S. SÜMER EKER
U NIVERSITY OF D ICLE , FACULTY OF S CIENCE & A RT
D EPARTMENT OF M ATHEMATICS
21280-D IYARBAKıR - TURKEY
ozlemg@dicle.edu.tr
sevtaps@dicle.edu.tr
Received 09 March, 2005; accepted 04 October, 2005
Communicated by H.M. Srivastava
A BSTRACT. In this paper, we introduce the class A∗o (p, A, B, α ) of p-valent functions in the
unit disc U = { z : | z | < 1 }. We obtain coefficient estimate, distortion and closure theorems,
radii of close-to convexity, starlikeness and convexity of order δ ( 0 6 δ < 1 ) for this class.
We also obtain class preserving integral operators for this class. Furthermore, various distortion
inequalities for fractional calculus of functions in this class are also given.
Key words and phrases: p-valent, Coefficient, Distortion, Closure, Starlike, Convex, Fractional calculus, Integral operators.
2000 Mathematics Subject Classification. 30C45, 30C50.
1. I NTRODUCTION
Let A(n) be the class of functions f , analytic and p−valent in U = {z : |z| < 1} given by
(1.1)
f (z) = z p +
∞
X
ap+n z p+n ,
ap+n > 0.
n=1
A function f belonging to the class A(n) is said to be in the class A∗m (p, A, B, α) if and only if
(p)
zf (z)
(p − 1) + Re
> 0 for z ∈ U.
f (p−1) (z)
In the other words, f ∈ A∗m (p, A, B, α) if and only if it satisfies the condition
zf (p) (z)
(p
−
1)
+
−
p
f (p−1) (z)
h
i < 1
(p)
(A − B)(p − α) + pB − B (p − 1) + fzf(p−1)(z)
(z)
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
071-05
2
H.Ö. G ÜNEY AND S. S ÜMER E KER
where −1 ≤ B < A ≤ 1, −1 ≤ B < 0 and 0 ≤ α < p. Let Am denote the subclass of A(n)
consisting of functions analytic and p−valent which can be expressed in the form
f (z) = z p −
(1.2)
∞
X
ap+n z p+n ;
ap+n ≥ 0.
n=1
Let us define
A∗o (p, A, B, α) = A∗m (p, A, B, α)
\
Am .
In this paper, we obtain a coefficient estimate, distortion theorems, integral operators and radii
of close-to-convexity, starlikeness and convexity, closure properties and distortion inequalities
for fractional calculus. This paper is motivated by an earlier work of Nunokawa [1].
2. C OEFFICIENT E STIMATES
Theorem 2.1. If the function f is defined by (1.1), then f ∈ A∗o (p, A, B, α) if and only if
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(2.1)
n=1
(n + 1)!
ap+n ≤ (A − B)(p − α)p!.
The result is sharp.
Proof. Assume that the inequality (2.1) holds true and let |z| = 1. Then we obtain
(p)
zf (z) − f (p−1) (z) − (A − B)(p − α)f (p−1) − Bzf (p) + Bf (p−1) ∞
X n(p + n)!
= −
ap+n z n+1 − (A − B)(p − α)p!z
(n + 1)!
n=1
"
#
∞
∞
X
X
(p + n)!
n(p
+
n)!
− (A − B)(p − α)
ap+n z n+1 − B
ap+n z n+1 (n + 1)!
(n + 1)!
n=1
n=1
≤
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
(n + 1)!
ap+n − (A − B)(p − α)p! ≤ 0
by hypothesis. Hence, by the maximum modulus theorem, we have f ∈ A∗o (p, A, B, α). To
prove the converse, assume that
zf (p) (z)
(p − 1) + f (p−1) (z) − p
i h
(p)
(A − B)(p − α) + pB − B (p − 1) + fzf(p−1)(z)
(z)
∞
P
n(p+n)!
n+1
a
z
−
p+n
(n+1)!
n=1
< 1.
= ∞
∞
(A − B)(p − α) p!z − P (p+n)! a z n+1 + B P n(p+n)! a z n+1 p+n
p+n
(n+1)!
(n+1)!
n=1
n=1
Since Re(z) ≤ |z| for all z, we have

∞
P
n(p+n)!


−
a z n+1

(n+1)! p+n
n=1
(2.2) Re
∞
∞
P
P

(p+n)!

n+1
 (A − B)(p − α) p!z −
a
z
+
B
p+n
(n+1)!
n=1
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
n=1




< 1.

n(p+n)!

a z n+1 
(n+1)! p+n
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O N A C ERTAIN C LASS O F p−VALENT F UNCTIONS W ITH N EGATIVE C OEFFICIENTS
3
Choosing values of z on the real axis and letting z → 1− through real values, we obtain
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(2.3)
ap+n ≤ (A − B)(p − α)p!,
(n
+
1)!
n=1
which obviously is required assertion (2.1). Finally, sharpness follows if we take
(A − B)(p − α)p!(n + 1)!
(2.4)
f (z) = z p −
z p+n .
(p + n)! [n(1 − B) + (A − B)(p − α)]
Corollary 2.2. If f ∈ A∗o (p, A, B, α), then
(A − B)(p − α)p!(n + 1)!
(2.5)
ap+n ≤
.
(p + n)! [n(1 − B) + (A − B)(p − α)]
The equality in (2.5) is attained for the function f given by (2.4).
3. D ISTORTION P ROPERTIES
Theorem 3.1. If f ∈ A∗o (p, A, B, α), then for |z| = r < 1
(3.1)
rp −
2(A − B)(p − α)
rp+1
(p + 1) [(1 − B) + (A − B)(p − α)]
2(A − B)(p − α)
rp+1
≤ |f (z)| ≤ rp +
(p + 1) [(1 − B) + (A − B)(p − α)]
and
(3.2)
prp−1 −
2(A − B)(p − α)
rp
(1 − B) + (A − B)(p − α)
≤ |f 0 (z)| ≤ prp−1 +
2(A − B)(p − α)
rp .
(1 − B) + (A − B)(p − α)
All the inequalities are sharp.
Proof. Let
p
f (z) = z −
∞
X
ap+n z p+n , ap+n > 0.
n=1
From Theorem 2.1, we have
∞
(p + 1)! [(1 − B) + (A − B)(p − α)] X
ap+n
2
n=1
≤
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(n + 1)!
n=1
ap+n
≤ (A − B)(p − α)p!
which
(3.3)
∞
X
ap+n ≤
n=1
2(A − B)(p − α)
(p + 1) [(1 − B) + (A − B)(p − α)]
and
(3.4)
∞
X
(p + n)ap+n ≤
n=1
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
2(A − B)(p − α)
.
(1 − B) + (A − B)(p − α)
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4
H.Ö. G ÜNEY AND S. S ÜMER E KER
Consequently, for |z| = r < 1, we obtain
p
|f (z)| ≤ r + r
p+1
∞
X
ap+n ≤ rp +
2(A − B)(p − α)
rp+1
(p + 1) [(1 − B) + (A − B)(p − α)]
ap+n ≥ rp −
2(A − B)(p − α)
rp+1
(p + 1) [(1 − B) + (A − B)(p − α)]
n=1
and
|f (z)| ≥ rp − rp+1
∞
X
n=1
which prove that the assertion (3.1) of Theorem 3.1 holds.
The inequalities in (3.2) can be proved in a similar manner and we omit the details.
The bounds in (3.1) and (3.2) are attained for the function f given by
f (z) = z p −
(3.5)
2(A − B)(p − α)
z p+1 .
(p + 1) [(1 − B) + (A − B)(p − α)]
Letting r → 1− in the left hand side of (3.1), we have the following:
Corollary 3.2. If f ∈ A∗o (p, A, B, α), then the disc |z| < 1 is mapped by f onto a domain that
contains the disc
(p + 1)(1 − B) + (A − B)(p − α)(p − 1)
|w| <
.
(p + 1) [(1 − B) + (A − B)(p − α)]
The result is sharp with the extremal function f being given by (3.5).
Putting α = 0 in Theorem 3.1 and Corollary 3.2, we get
Corollary 3.3. If f ∈ A∗o (p, A, B, 0), then for |z| = r
rp −
2p(A − B)
rp+1
(p + 1) [(1 − B) + p(A − B)]
≤ |f (z)| ≤ rp +
2p(A − B)
rp+1
(p + 1) [(1 − B) + p(A − B)]
and
prp−1 −
2p(A − B)
2p(A − B)
rp ≤ |f 0 (z)| ≤ prp−1 +
rp .
(1 − B) + p(A − B)
(1 − B) + p(A − B)
The result is sharp with the extremal function
(3.6)
f (z) = z p −
2p(A − B)
z p+1 ;
(p + 1) [(1 − B) + p(A − B)]
z = ∓r.
Corollary 3.4. If f ∈ A∗o (p, A, B, 0), then the disc |z| < 1 is mapped by f onto a domain that
contains the disc
(p + 1)(1 − B) + p(p − 1)(A − B)
|w| <
.
(p + 1) [(1 − B) + p(A − B)]
The result is sharp with the extremal function f being given by (3.6).
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
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O N A C ERTAIN C LASS O F p−VALENT F UNCTIONS W ITH N EGATIVE C OEFFICIENTS
4. R ADII
OF
5
C LOSE -T O -C ONVEXITY, S TARLIKENESS AND C ONVEXITY
Theorem 4.1. Let f ∈ A∗o (p, A, B, α). Then f is p−valent close-to-convex of order δ (0 ≤ δ < p)
in |z| < R1 , where
(
n1 )
(p + n)![n(1 − B) + (A − B)(p − α)] p − δ
.
(4.1)
R1 = inf
n
(A − B)(p − α)(n + 1)p!
p+n
Theorem 4.2. If f ∈ A∗o (p, A, B, α), then f is p−valent starlike of order δ (0 ≤ δ < p) in
|z| < R2 , where
(
n1 )
p−δ
(p + n)![n(1 − B) + (A − B)(p − α)]
(4.2)
R2 = inf
.
n
(A − B)(p − α)(n + 1)!p!
p+n−δ
Theorem 4.3. If f ∈ A∗o (p, A, B, α), then f is a p−valent convex function of order δ (0 ≤ δ < p)
in |z| < R3 , where
(
n1 )
[n(1 − B) + (A − B)(p − α)](p + n − 1)!
p−δ
.
(4.3)
R3 = inf
n
(A − B)(p − α)(n + 1)!(p − 1)!
p+n−δ
In order to establish the required results in Theorems 4.1, 4.2 and 4.3, it is sufficient to show
that
0
f (z)
≤ p − δ for |z| < R1 ,
−
p
z p−1
0
zf (z)
f (z) − p ≤ p − δ for |z| < R2 and
00
zf
(z)
≤ p − δ for |z| < R3 ,
1+
−
p
f 0 (z)
respectively.
Remark 4.4. The results in Theorems 4.1, 4.2 and 4.3 are sharp with the extremal function f
given by (2.4). Furthermore, taking δ = 0 in Theorems 4.1, 4.2 and 4.3, we obtain radius of
close-to-convexity, starlikeness and convexity, respectively.
5. I NTEGRAL O PERATORS
Theorem 5.1. Let c be a real number such that c > −p. If f ∈ A∗o (p, A, B, α), then the function
F defined by
Z
c + p z c−1
(5.1)
F (z) =
t f (t)dt
zc 0
also belongs to A∗o (p, A, B, α).
Proof. Let
f (z) = z p −
∞
X
ap+n z p+n .
n=1
Then from the representation of F , it follows that
p
F (z) = z −
∞
X
bp+n z p+n ,
n=1
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
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6
H.Ö. G ÜNEY AND S. S ÜMER E KER
where bp+n =
c+p
c+p+n
ap+n . Therefore using Theorem 2.1 for the coefficients of F , we have
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
(n + 1)!
bp+n
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
c+p
=
ap+n
(n
+
1)!
c
+
p
+
n
n=1
≤ (A − B)(p − α)p!
since
c+p
c+p+n
< 1 and f ∈ A∗o (p, A, B, α). Hence F ∈ A∗o (p, A, B, α).
Theorem 5.2. Let c be a real number such that c > −p. If F ∈ A∗o (p, A, B, α), then the
function f defined by (5.1) is p−valent in |z| < R∗ , where
(
n1 )
c
+
p
(p
+
n)!
[n(1
−
B)
+
(A
−
B)(p
−
α)]
p
(5.2) R∗ = inf
.
n
c+p+n
(n + 1)!(A − B)(p − α)p!
p+n
The result is sharp. Sharpness follows if we take
c+p+n
(n + 1)!(A − B)(p − α)p!
p
f (z) = z −
z p+n .
c+p
(p + n)! [n(1 − B) + (A − B)(p − α)]
6. C LOSURE P ROPERTIES
In this section we show that the class A∗o (p, A, B, α) is closed under “arithmetic mean” and
“convex linear combinations”.
Theorem 6.1. Let
p
fj (z) = z −
∞
X
ap+n,j z p+n ,
j = 1, 2, ...
n=1
and
p
h(z) = z −
∞
X
bp+n z p+n ,
n=1
where
bp+n =
∞
X
λj ap+n,j ,
λj > 0
j=1
and
P∞
j=1
λj = 1. If fj ∈ A∗o (p, A, B, α) for each j = 1, 2, ..., then h ∈ A∗o (p, A, B, α).
Proof. If fj ∈ A∗o (p, A, B, α), then we have from Theorem 2.1 that
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
ap+n,j ≤ (A − B)(p − α)p!,
(n + 1)!
n=1
Therefore
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
(n + 1)!
j = 1, 2, ....
bp+n
"
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
=
(n + 1)!
n=1
∞
X
!#
λj ap+n,j
j=1
≤ (A − B)(p − α)p!.
Hence, by Theorem 2.1, h ∈ A∗o (p, A, B, α).
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
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O N A C ERTAIN C LASS O F p−VALENT F UNCTIONS W ITH N EGATIVE C OEFFICIENTS
7
Theorem 6.2. The class A∗o (p, A, B, α) is closed under convex linear combinations.
Theorem 6.3. Let fp (z) = z p and
(A − B)(p − α)(n + 1)!p!
z p+n
(p + n)! [n(1 − B) + (A − B)(p − α)]
fp+n = z p −
(n ≥ 1).
Then f ∈ A∗o (p, A, B, α) if and only if it can be expressed in the form
f (z) = λp fp (z) +
∞
X
λn fp+n (z),
z ∈ U,
n=1
where λn ≥ 0 and λp = 1 −
P∞
n=1
λn .
Proof. Let us assume that
f (z) = λp fp (z) +
∞
X
λn fp+n (z)
n=1
p
=z −
∞
X
n=1
(A − B)(p − α)(n + 1)!p!
λn z p+n .
(p + n)! [n(1 − B) + (A − B)(p − α)]
Then from Theorem 2.1 we have
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(n + 1)!
n=1
×
(A − B)(p − α)(n + 1)!p!
λn
(p + n)! [n(1 − B) + (A − B)(p − α)]
≤ (A − B)(p − α)p!.
Hence f ∈ A∗o (p, A, B, α). Conversely, let f ∈ A∗o (p, A, B, α). It follows from Corollary 2.2
that
(A − B)(p − α)(n + 1)!p!
ap+n ≤
.
(p + n)! [n(1 − B) + (A − B)(p − α)]
Setting
and λp = 1 −
λn =
(p + n)! [n(1 − B) + (A − B)(p − α)]
ap+n ,
(A − B)(p − α)(n + 1)!p!
P∞
λn , we have
n=1
p
∞
X
p
∞
X
f (z) = z −
n = 1, 2, . . .
ap+n z p+n
n=1
=z −
p
λn z +
n=1
= λp fp (z) +
∞
X
p
λn z −
n=1
∞
X
∞
X
n=1
λn
(A − B)(p − α)(n + 1)!p!
z p+n
(p + n)! [n(1 − B) + (A − B)(p − α)]
λn fp+n (z).
n=1
This completes the proof of Theorem 6.3.
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
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8
H.Ö. G ÜNEY AND S. S ÜMER E KER
7. D EFINITIONS AND A PPLICATIONS OF F RACTIONAL C ALCULUS
In this section, we shall prove several distortion theorems for functions to general class
Each of these theorems would involve certain operators of fractional calculus we find it to be convenient to recall here the following definition which were used recently
by Owa [2] (and more recently, by Owa and Srivastava [3], and Srivastava and Owa [4] ; see
also Srivastava et al. [5]).
A∗o (p, A, B, α).
Definition 7.1. The fractional integral of order λ is defined, for a function f , by
Z z
1
f (ζ)
−λ
(7.1)
Dz f (z) =
dζ (λ > 0),
Γ(λ) 0 (z − ζ)1−λ
where f is an analytic function in a simply – connected region of the z -plane containing the
origin, and the multiplicity of (z − ζ)λ−1 is removed by requiring log(z − ζ) to be real when
z − ζ > 0.
Definition 7.2. The fractional derivative of order λ is defined, for a function f , by
Z z
1
d
f (ζ)
λ
dζ (0 ≤ λ < 1),
(7.2)
Dz f (z) =
Γ(1 − λ) dz 0 (z − ζ)λ
where f is constrained, and the multiplicity of (z − ζ)−λ is removed, as in Definition 7.1.
Definition 7.3. Under the hypotheses of Definition 7.2, the fractional derivative of order (n+λ)
is defined by
dn
(7.3)
Dzn+λ f (z) = n Dzλ f (z) (0 ≤ λ < 1),
dz
S
where 0 ≤ λ < 1 and n ∈ N0 = N {0}. From Definition 7.2, we have
Dz0 f (z) = f (z)
(7.4)
which, in view of Definition 7.3 yields,
Dzn+0 f (z) =
(7.5)
dn 0
D f (z) = f n (z).
dz n z
Thus, it follows from (7.4) and (7.5) that
lim Dz−λ f (z) = f (z) and
λ→0
lim Dz1−λ f (z) = f 0 (z).
λ→0
Theorem 7.1. Let the function f defined by (1.2) be in the class A∗o (p, A, B, α). Then for z ∈ U
and λ > 0,
−λ
Γ(p + 1)
p+λ
Dz f (z) ≥ |z|
Γ(λ + p + 1)
2(A − B)(p − α)Γ(p + 1)
−
|z|
(λ + p + 1)Γ(λ + p + 1) [(1 − B) + (A − B)(p − α)]
and
−λ
Γ(p + 1)
Dz f (z) ≤ |z|p+λ
Γ(λ + p + 1)
2(A − B)(p − α)Γ(p + 1)
+
|z| .
(λ + p + 1)Γ(λ + p + 1) [(1 − B) + (A − B)(p − α)]
The result is sharp.
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
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O N A C ERTAIN C LASS O F p−VALENT F UNCTIONS W ITH N EGATIVE C OEFFICIENTS
9
Proof. Let
Γ(p + 1 + λ) −λ −λ
z Dz f (z)
Γ(p + 1)
∞
X
Γ(p + n + 1)Γ(p + λ + 1)
p
=z −
ap+n z p+n
Γ(p
+
1)Γ(p
+
n
+
λ
+
1)
n=1
F (z) =
= zp −
∞
X
ϕ(n)ap+n z p+n ,
n=1
where
Γ(p + n + 1)Γ(p + λ + 1)
,
Γ(p + 1)Γ(p + n + λ + 1)
ϕ(n) =
Then by using 0 < ϕ(n) ≤ ϕ(1) =
p+1
p+λ+1
(λ > 0, n ∈ N).
and Theorem 2.1, we observe that
∞
(p + 1)! [(1 − B) + (A − B)(p − α)] X
ap+n
2!
n=1
≤
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(n + 1)!
n=1
ap+n
≤ (A − B)(p − α)p!,
which shows that F (z) ∈ A∗o (p, A, B, α). Consequently, with the aid of Theorem 3.1, we have
p+1
p
|F (z)| ≥ |z | − ϕ(1) |z|
∞
X
ap+n
n=1
≥ |z|p −
2(A − B)(p − α)
|z|p+1
(p + λ + 1)[(1 − B) + (A − B)(p − α)]
and
p+1
p
|F (z)| ≤ |z | + ϕ(1) |z|
∞
X
ap+n
n=1
≤ |z|p +
2(A − B)(p − α)
|z|p+1
(p + λ + 1)[(1 − B) + (A − B)(p − α)]
which completes the proof of Theorem 7.1.By letting λ → 0, Theorem 7.1 reduces at once to
Theorem 3.1.
Corollary 7.2. Under the hypotheses of Theorem 7.1, Dz−λ f (z) is included in a disk with its
center at the origin and radius R1−λ given by
Γ(p + 1)
2(A − B)(p − α)
−λ
R1 =
1+
.
Γ(λ + p + 1)
(p + λ + 1)[(1 − B) + (A − B)(p − α)]
Theorem 7.3. Let the function f defined by (1.2) be in the class A∗o (p, A, B, α). Then,
λ
Γ(p + 1)
p−λ
Dz f (z) ≥ |z|
Γ(p − λ + 1)
2(A − B)(p − α)Γ(2 − λ)Γ(p + 1)
−
|z|
Γ(p − λ + 1)Γ(p − λ + 2)[(1 − B) + (A − B)(p − α)]
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
http://jipam.vu.edu.au/
10
H.Ö. G ÜNEY AND S. S ÜMER E KER
and
λ
Γ(p + 1)
Dz f (z) ≤ |z|p−λ
Γ(p − λ + 1)
2(A − B)(p − α)Γ(2 − λ)Γ(p + 1)
+
|z|
Γ(p − λ + 1)Γ(p − λ + 2)[(1 − B) + (A − B)(p − α)]
for 0 ≤ λ < 1.
Proof. Using similar arguments as given by Theorem 7.1, we can get the result.
Corollary 7.4. Under the hypotheses of Theorem 7.3, Dzλ f (z) is included in the disk with its
center at the origin and radius R2λ given by
Γ(p + 1)
2(A − B)(p − α)Γ(2 − λ)
λ
R2 =
1+
.
Γ(λ + p + 1)
Γ(p − λ + 1)[(1 − B) + (A − B)(p − α)]
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[1] M. NUNOKAWA, On the multivalent functions, Indian J. of Pure and Appl.Math., 20(6) (1989),
577–582.
[2] S. OWA, On the distortion theorems, I., Kyunpook Math.J., 18 (1978 ), 53–59.
[3] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized hypergeometric functions,
Canad. J. Math., 39 (1987), 1057–1077.
[4] H.M. SRIVASTAVA AND S. OWA, Some characterization and distortion theorems involving fractional calculus, linear operators and certain subclasses of analytic functions, Nagoya Mathematics
J., 106 (1987), 1–28.
[5] 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.
J. Inequal. Pure and Appl. Math., 6(4) Art. 97, 2005
http://jipam.vu.edu.au/