General Mathematics Vol. 14, No. 3 (2006), 61–78
Integral Means and fractional calculus
operators for comprehensive family of
univalent functions with negative
coefficients 1
B. A. Frasin, G. Murugusundaramoorthy and N.
Mageshand
Abstract
In this paper, we obtain the integral means inequality for the function f (z) belongs to the class U T (Φ, Ψ, γ, k) of analytic and univalent
functions with negative coefficients defined in [3] with the extremal
functions of this class. And also we derive some distortion theorems
using fractional calculus techniques for the class U T (Φ, Ψ, γ, k).
2000 Mathematics Subject Classification: 30C45.
Keywords:Univalent, starlike, convex, uniformly convex, uniformly
starlike, Hadamard product, integral means, Fractional calculus
1
Received 28 August, 2006
Accepted for publication (in revised form) 19 September, 2006
61
62
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
1
Introduction and definitions
Let A denote the class of functions of the form
∞
X
(1.1)
f (z) = z +
an z n
n=2
which are analytic and univalent in the open disc U = {z : z ∈ C, |z| < 1}.
Also denote by T the subclass of A consisting of functions of the form
(1.2)
f (z) = z −
∞
X
an z n , an ≥ 0, z ∈ U
n=2
introduced and studied by Silverman [16].
Following Gooodman [5, 6], Rønning [12, 13] introduced and studied the
following subclasses
(i) A function f ∈ A is said to be in the class Sp (γ, k), k−uniformly starlike
functions of order γ, if it satisfies the condition
¯ ′
¯
½ ′
¾
¯ zf (z)
¯
zf (z)
(1.3)
Re
− γ > k ¯¯
− 1¯¯ , z ∈ U,
f (z)
f (z)
0 ≤ γ < 1 and k ≥ 0.
(ii) A function f ∈ A is said to be in the class U CV (γ, k), k−uniformly
convex functions of order γ, if it satisfies the condition
¯ ′′ ¯
½
¾
¯ zf (z) ¯
zf ′′ (z)
¯ , z ∈ U,
(1.4)
Re 1 + ′
− γ > k ¯¯ ′
f (z)
f (z) ¯
0 ≤ γ < 1 and k ≥ 0.
Indeed it follows from (1.3) and (1.4) that
(1.5)
f ∈ U CV (γ, k) ⇔ zf ′ ∈ Sp (γ, k).
Definition 1.1 ([3]). Given γ(−1 ≤ γ < 1), k(k ≥ 0) and functions
Φ(z) = z +
∞
X
n=2
n
λn z and Ψ(z) = z +
∞
X
n=2
µn z n
Integral Means and fractional calculus operators...
63
analytic in U, such that λn ≥ 0, µn ≥ 0 and λn ≥ µn for n ≥ 2, we let f ∈ A
is in U (Φ, Ψ, α, β) if (f ∗ Ψ)(z) 6= 0 and
¯
¯
¾
½
¯ (f ∗ Φ)(z)
¯
(f ∗ Φ)(z)
¯
−γ ≥ k¯
− 1¯¯ , ∀ z ∈ U.
Re
(f ∗ Ψ)(z)
(f ∗ Ψ)(z)
where (*) stands for the Hadamard product.
Further let U T (Φ, Ψ, α, β) = U (Φ, Ψ, α, β) ∩ T.
We note that, by taking suitable choice of Φ, Ψ, α and β we obtain the
following subclasses studied in literature.
1. U T
³
z
z
, 1−z
,
(1−z)2
´
γ, 1 = T Sp (γ) (Subrmanian et al., [22])
2. U T
³
z
z
, 1−z
,
(1−z)2
´
γ, k = Sp T (γ, k) (Bharati et al., [1])
3. U T
³
z+z 2
z
, (1−z)
2,
(1−z)3
´
0, 1 = U CT (Subrmanian et al., [21])
4. U T
³
z+z 2
z
, (1−z)
2,
(1−z)3
´
0, k = U CT (k) (Subrmanian et al., [21])
5. U T
³
z+z 2
z
, (1−z)
2,
(1−z)3
´
γ, 1 = U CT (γ) (Bharati et al., [1])
6. U T
³
z
z+z 2
, (1−z)
2,
(1−z)3
´
γ, k = U CT (γ, k) (Bharati et al., [1])
7. U T
³
z
z
, 1−z
,
(1−z)2
´
γ, 0 = ST∗ (γ) (Silverman [16])
64
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
8. U T
³
z
z+z 2
, (1−z)
2,
(1−z)3
´
γ, 0 = KT (γ) (Silverman [16])
9. U T (Φ, Ψ, γ, 0) = ET (Φ, Ψ, γ) (Juneja et al.[7]).
10. U T (Φ, Ψ, 1+β−2α
, 0) = BT (Φ, Ψ, α, β) (Frasin [4]).
2(1−α)
In fact many subclasses of T are defined and studied to investigate coefficient estimates, extreme points, convolution properties and closure properties etc. suitably choosing Φ, Ψ, γ and k.
In this paper, we obtain integral means inequalities for functions f (z) ∈
U T (Φ, Ψ, γ, k) and also we state integral means results for the classes studied in [21, 1, 22, 16, 4] as corollaries.
For analytic functions g(z) and h(z) with g(0) = h(0), g(z) is said to be
subordinate to h(z) if there exists an analytic function w(z) so that w(0) =
0, |w(z)| < 1 (z ∈ U) and g(z) = h(w(z)), we denote this subordination by
g(z) ≺ h(z).
To prove our main results, we need the following lemmas.
Lemma 1.1 ([3]). A function f (z) ∈ U T (Φ, Ψ, γ, k) for γ(−1 ≤ γ < 1)
and k(k ≥ 0) if and only if
(1.1)
∞
X
[(1 + k)λn − (γ + k)µn ]an ≤ 1 − γ.
n=2
The result is sharp with the extremal functions
(1.2)
fn (z) = z −
1−γ n
z , n≥2
σ(γ, k, n)
Integral Means and fractional calculus operators...
65
where σ(γ, k, n) = (1 + k)λn − (γ + k)µn , γ(−1 ≤ γ < 1), k(k ≥ 0) and
n ≥ 2.
Lemma 1.2 ([8]). If the functions f (z) and g(z) are analytic in U with
g(z) ≺ f (z) then
(1.3)
Z2π
0
2
¯
¯
¯g(reiθ )¯ ηdθ ≤
Z2π
0
¯
¯
¯f (reiθ )¯ ηdθ η > 0, z = reiθ and 0 < r < 1.
Integral mean
Applying Lemma 1.1 and Lemma 1.2, we prove the following theorem.
Theorem 2.1. Let η > 0. If f (z) ∈ U T (Φ, Ψ, γ, k), −1 ≤ γ < 1, k ≥ 0 and
iθ
{σ(γ, k, n)}∞
n=2 is non-decreasing sequence, then for z = re and 0 < r < 1,
we have
Z2π
(2.1)
0
where f2 (z) = z −
¯
¯
¯f (reiθ )¯ ηdθ ≤
Z2π
0
¯
¯
¯f2 (reiθ )¯ ηdθ
1−γ
z2.
σ(γ,k,2)
(1−γ) 2
Proof. Let f (z) of the form (1.2) and f2 (z) = z − σ(γ,k,2)
z , then we must
show that
¯
¯
Z2π ¯
Z2π ¯¯
∞
¯
X
¯
¯
(1
−
γ)
¯
n−1 ¯
an z ¯ ηdθ ≤ ¯¯1 −
z ¯¯ ηdθ.
¯1 −
¯
¯
σ(γ, k, 2)
n=2
0
0
By Lemma 1.2, it suffices to show that
1−
∞
X
n=2
an z n−1 ≺ 1 −
1−γ
z
σ(γ, k, 2)
66
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
Setting
(2.2)
1−
∞
X
an z n−1 = 1 −
n=2
1−γ
w(z).
σ(γ, k, 2)
From (2.2) and (1.1), we obtain
¯
¯∞
¯
¯X σ(γ, k, 2)
¯
¯
an z n−1 ¯
|w(z)| = ¯
¯
¯ n=2 1 − γ
≤ |z|
∞
X
σ(γ, k, n)
1−γ
n=2
an
≤ |z| < 1.
This completes the proof of the Theorem 2.1.
By taking different choices of Φ, Ψ, γ and k in the above theorem, we
can state the following integral means results for various subclasses studied
earlier [21, 1, 22, 16, 4].
Corollary 2.2. Let η > 0. If f (z) ∈ U T
then for z = reiθ ; 0 < r < 1, we have
Z2π
(2.3)
0
where g2 (z) = z −
z2
6
¯
¯
¯f (reiθ )¯ ηdθ ≤
Z2π
0
³
z+z 2
z
, (1−z)
2,
(1−z)3
¯
¯
¯g2 (reiθ )¯ ηdθ
.
Corollary 2.3. Let η > 0. If f (z) ∈ U T
³
z
z+z 2
, (1−z)
2,
(1−z)3
and k ≥ 0, then for z = reiθ ; 0 < r < 1, we have
(2.4)
Z2π
0
where g2 (z) = z −
´
0, 1 = U CT,
¯
¯
¯f (reiθ )¯ ηdθ ≤
z2
2(k+2)
.
Z2π
0
¯
¯
¯g2 (reiθ )¯ ηdθ
´
0, k = U CT (k)
Integral Means and fractional calculus operators...
Corollary 2.4. Let η > 0. If f (z) ∈ U T
³
67
z
z+z 2
, (1−z)
2,
(1−z)3
and −1 ≤ γ < 1, then for z = reiθ ; 0 < r < 1, we have
(2.5)
Z2π
0
where g2 (z) = z −
¯
¯
¯f (reiθ )¯ ηdθ ≤
(1−γ) 2
z
2(3−γ)
Z2π
0
´
γ, 1 = U CT (γ)
¯
¯
¯g2 (reiθ )¯ ηdθ
.
Corollary 2.5. Let η > 0. If f (z) ∈ U T
³
z+z 2
z
, (1−z)
2,
(1−z)3
´
γ, k = U CT (γ, k),
−1 ≤ γ < 1 and k ≥ 0, then for z = reiθ ; 0 < r < 1, we have
(2.6)
Z2π
0
where g2 (z) = z −
¯
¯
¯f (reiθ )¯ ηdθ ≤
(1−γ)
z2
2(2−γ+k)
Z2π
0
¯
¯
¯g2 (reiθ )¯ ηdθ
.
Corollary 2.6. Let η > 0. If f (z) ∈ U T
³
z
z
, 1−z
,
(1−z)2
−1 ≤ γ < 1, then for z = reiθ ; 0 < r < 1, we have
(2.7)
Z2π
0
where g2 (z) = z −
¯
¯
¯f (reiθ )¯ ηdθ ≤
(1−γ) 2
z
(3−γ)
Z2π
0
´
γ, 1 = T Sp (γ) and
¯
¯
¯g2 (reiθ )¯ ηdθ
.
Corollary 2.7. Let η > 0. If f (z) ∈ U T
³
z
z
, 1−z
,
(1−z)2
´
γ, k = Sp T (γ, k),
−1 ≤ γ < 1 and k ≥ 0, then for z = reiθ ; 0 < r < 1, we have
(2.8)
Z2π
0
where g2 (z) = z −
By taking γ =
¯
¯
¯f (reiθ )¯ ηdθ ≤
(1−γ)
z2 .
(2−γ+k)
1+β−2α
and
2(1−α)
result of Frasin and Darus [4].
Z2π
0
¯
¯
¯g2 (reiθ )¯ ηdθ
k = 0 in Theorem 2.1 we get the following
68
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
, 0) = BT (Φ, Ψ, α, β),
Corollary 2.8. Let η > 0. If f (z) ∈ U T (Φ, Ψ, 1+β−2α
2(1−α)
0 ≤ β < 1 and 0 ≤ α < 1, then for z = reiθ ; 0 < r < 1, we have
Z2π
(2.9)
0
¯
¯
¯f (reiθ )¯ ηdθ ≤
Z2π
0
¯
¯
¯g2 (reiθ )¯ ηdθ
(1−β) 2
z and ψ(α, β, 2) = 2(1 − α)λ2 − (1 + β − 2α)µ2 .
where g2 (z) = z − ψ(α,β,2)
³
´
z
z
Corollary 2.9.Let η > 0. If f (z) ∈ U T (1−z)2 , 1−z , γ, 0 = ST∗ (γ) and
γ ≥ 0, then for z = reiθ ; 0 < r < 1, we have
(2.10)
Z2π
0
where g2 (z) = z −
¯
¯
¯f (reiθ )¯ ηdθ ≤
1−γ 2
z
2−γ
Z2π
0
¯
¯
¯g2 (reiθ )¯ ηdθ
.
Corollary 2.10.Let η > 0. If f (z) ∈ U T
³
z
z+z 2
, (1−z)
2,
(1−z)3
and γ ≥ 0, then for z = reiθ ; 0 < r < 1, we have
(2.11)
Z2π
0
where g2 (z) = z −
Remark 2.11.
¯
¯
¯f (reiθ )¯ ηdθ ≤
1−γ
z2
2(2−γ)
Z2π
0
´
γ, 0 = KT (γ),
¯
¯
¯g2 (reiθ )¯ ηdθ
.
If we take γ = 0 in ST∗ (γ) of Corollary 2.9 and KT (γ)
of Corollary 2.10, we get the integral means results obtained by Silverman
[17].
3
Fractional Calculus
Many essentially equivalent definitions of fractional calculus (that is fractional derivatives and fractional integrals) have been given in the literature
(cf., e.g., [2],[9],[11], [14], [15], [18]and[19]). We find it to be convenient to
Integral Means and fractional calculus operators...
69
recall here the following definitions which are used earlier by Owa [10](and,
subsequently, by Srivastava and Owa [19]).
Definition 3.1. The fractional integral of order ξ is defined, for a function
f (z), by
Dz−ξ f (z)
(3.1)
1
=
Γ(ξ)
z
Z
0
f (ζ)
dζ
(z − ζ)1−ξ
(ξ > 0),
where the function f (z) is analytic in a simply-connected region of the zplane containing the origin and the multiplicity of the the function (z −
ζ)ξ−1 is removed by requiring the function log(z−ζ) to be real when z−ζ > 0.
Definition 3.2. The fractional derivative of order ξ is defined, for a function f (z), by
(3.2)
d
1
Dz ξf (z) =
Γ(−ξ) dz
Z
0
z
f (ζ)
dζ
(z − ζ)1−ξ
(0 ≤ ξ < 1),
where the function f (z) is constrained, and the multiplicity of the the function (z − ζ)−ξ is removed as in Definition 3.1
Definition 3.3. Under the hypotheses of Definition 3.2, the fractional
derivative of order n + λ is defined by
(3.3)
Dzm+ξ f (z) =
dm
D ξf (z)
dz m z
(0 ≤ ξ < 1; m ∈ N0 ).
Remark 3.4. From Definition 3.2, we have Dz0 f (z) = f (z), which in
view of Definition 3.3 yields Dzm+0 f (z) =
dm
D0 f (z)
dz m z
= f (m) (z). Thus,
limDz−ξ f (z) = f (z) and limDz1−ξ f (z) = f ′ (z) .
ξ→0
ξ→0
We need the following definition of fractional integral operator given by
Srivastava, Saigo and Owa[20].
70
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
Definition 3.5 For real number η > 0, µ and δ, the fractional integral
η,µ,δ
operator I0,z
is defined by
(3.4)
η,µ,δ
I0,z
f (z)
z −η−µ
=
Γ(η)
Z
z
(z − t)η−1 F (η + µ, −δ; η; 1 − t/z)f (t)dt,
0
where a function f (z) is analytic in a simply-connected region of the z-plane
containing the origin with the order
f (z) = O(|z| ε)
(z → 0),
with ε > max{0, µ − δ} − 1. Here F (a, b; c; z) is the Gauss hypergeometric
function defined by
(3.5)
F (a, b; c; z) =
∞
X
(a)n (b)n
n=0
(c)n (1)n
zn,
where (ν)n is the Pochhammer symbol defined by
(3.6)

Γ(ν + n)  1
(ν)n =
=
 ν(ν + 1)(ν + 2) · · · (ν + n − 1)
Γ(ν)
(n = 0)
(n ∈ N)
and the multiplicity of (z − t)η−1 is removed by requiring log (z − t) to be
real when z − t > 0.
Remark 3.4. For µ = −η, we note that
η,−η,δ
I0,z
f (z) = Dz−η f (z).
In order to prove our result for the fractional integral operator, we have
to recall here the following lemma due to Srivastava, Saigo and Owa [20].
Integral Means and fractional calculus operators...
71
Lemma 3.7. If η > 0 and n > µ − δ − 1, then
(3.7)
η,µ,δ n
I0,z
z =
Γ(n + 1)Γ(n − µ + δ + 1) n−µ
z .
Γ(n − µ + 1)Γ(n + η + δ + 1)
With aid of Lemma 3.7, we prove
Theorem 3.8. Let η > 0, µ < 2, η + δ > −2, µ − δ < 2, µ(η + δ) ≤ 3η.
Let the function f (z) defined by (1.2) be in the class U T (Φ, Ψ, γ, k). If
{σ(γ, k, n)}∞
n=2 is a non-decreasing sequence. Then we have
(3.8)
µ
¶
¯
¯
Γ(2 − µ + δ) |z|1−µ
2(1 − γ)(2 − µ + δ)
¯
¯ η,µ,δ
1−
|z|
¯I0,z f (z)¯ ≥
Γ(2 − µ)Γ(2 + η + δ)
(2 − µ)(2 + η + δ)σ(γ, k, 2)
and
(3.9)
¶
µ
¯
¯
Γ(2 − µ + δ) |z|1−µ
2(1 − γ)(2 − µ + δ)
¯ η,µ,δ
¯
|z|
1+
¯I0,z f (z)¯ ≤
Γ(2 − µ)Γ(2 + η + δ)
(2 − µ)(2 + η + δ)σ(γ, k, 2)
for z ∈ U0 , where
(3.10)

 U
U0 =
 U−{0}
(µ ≤ 1),
(µ > 1).
The equalities in (3.8) and(3.9) are attained for the function f (z) given by
(3.11)
f (z) = z −
2(1 − γ)(2 − µ + δ)
z2.
(2 − µ)(2 + η + δ)σ(γ, k, 2)
Proof. By using Lemma 3.7, we have
72
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
η,µ,δ
I0,z
f (z) =
(3.12)
Γ(2 − µ + δ)
z 1−µ
Γ(2 − µ)Γ(2 + η + δ)
−
∞
X
n=2
Γ(n + 1)Γ(n − µ + δ + 1)
an z n−µ
Γ(n − µ + 1)Γ(n + η + δ + 1)
(z ∈ U0 ).
Letting
G(z) =
(3.13)
Γ(2 − µ)Γ(2 + η + δ)
η,µ,δ
zµI0,z
f (z)
Γ(2 − µ + δ)
=z−
∞
X
g(n)an z n ,
n=2
where
(3.14)
g(n) =
(2 − µ + δ)n−1 (1)n
(2 − µ)n−1 (2 + η + δ)n−1
(n ≥ 2),
we can see that the function g(k) is non-increasing for integers n(n ≥ 2),and
thus we have
(3.15)
0 < g(n) ≤ g(2) =
2(2 − µ + δ)
.
(2 − µ)(2 + η + δ)
From Lemma 1.1, we obtain
(3.16)
σ(γ, k, 2)
∞
X
n=2
an ≤
∞
X
n=2
σ(γ, k, n)an ≤ 1 − γ
Integral Means and fractional calculus operators...
73
Hence, using (3.15) and (3.16), we have
(3.17)
|G(z)| ≥ |z| − g(2) |z|
2
∞
X
an ≥ |z| −
2(1 − γ)(2 − µ + δ)
|z|2 ,
(2 − µ)(2 + η + δ)σ(γ, k, 2)
∞
X
an ≤ |z| +
2(1 − γ)(2 − µ + δ)
|z|2 ,
(2 − µ)(2 + η + δ)σ(γ, k, 2)
n=2
and
(3.18)
|G(z)| ≤ |z| + g(2) |z|
2
n=2
for z ∈ U0 , where U0 is defined by (3.10). This completes the proof Theorem
3.8.
By using the same proof as in Theorem 3.8, we can prove
Theorem 3.9. Let the function f (z) be defined by (1.2) be in the class
U T (Φ, Ψ, γ, k) . If {σ(γ, k, n)/n}∞
n=2 is a non-decreasing sequence. Then
we have
(3.19)
¶
µ
¯
¯
Γ(2 − µ + δ) |z|1−µ
4(1 − γ)(2 − µ + δ)
¯
¯ η,µ,δ
|z|
1−
¯I0,z f (z)¯ ≥
Γ(2 − µ)Γ(2 + η + δ)
(2 − µ)(2 + η + δ)σ(γ, k, 2)
and
(3.20)
µ
¶
¯
¯
Γ(2 − µ + δ) |z|1−µ
4(1 − γ)(2 − µ + δ)
¯
¯ η,µ,δ
1+
|z|
¯I0,z f (z)¯ ≤
Γ(2 − µ)Γ(2 + η + δ)
(2 − µ)(2 + η + δ)σ(γ, k, 2)
for z ∈ U0 , where U0 is defined by (3.10). The equalities in (3.19) and (3.20)
are attained for the function f (z) given by (3.11).
Taking µ = −η = −ξ Theorem 3.8, we get
74
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
Corollary 3.10. Let the function f (z) defined by (1.2) be in the class
U T (Φ, Ψ, γ, k). If {σ(γ, k, n)}∞
n=2 is a non-decreasing sequence. Then we
have
(3.21)
and
(3.22)
1+ξ
¯
¯ −ξ
¯Dz f (z)¯ ≥ |z|
Γ(2 + ξ)
µ
1+ξ
¯
¯ −ξ
¯Dz f (z)¯ ≤ |z|
Γ(2 + ξ)
µ
1+
¶
2(1 − γ)
|z|
1−
(2 + ξ)σ(γ, k, 2)
2(1 − γ)
|z|
(2 + ξ)σ(γ, k, 2)
¶
for ξ > 0, z ∈ U. The result is sharp for the function
(3.23)
Dz−ξ f (z)
|z|1+ξ
=
Γ(2 + ξ)
µ
¶
2(1 − γ)
|z| .
1−
(2 + ξ)σ(γ, k, 2)
Taking µ = −η = ξ in Theorem 3.9, we get
Corollary 3.11.Let the function
f (z) defined by (1.2) be in the class
U T (Φ, Ψ, γ, k). If {σ(γ, k, n)/n}∞
n=2 is a non-decreasing sequence. Then
we have
(3.24)
|z|1−ξ
|Dz ξf (z)| ≥
Γ(2 − ξ)
µ
|z|1−ξ
|Dz ξf (z)| ≤
Γ(2 − ξ)
µ
4(1 − γ)
|z|
1−
(2 − ξ)σ(γ, k, 2)
¶
and
(3.25)
¶
4(1 − γ)
|z|
1+
(2 − ξ)σ(γ, k, 2)
for 0 ≤ ξ < 1, z ∈ U. The result is sharp for the function given by (3.23).
Integral Means and fractional calculus operators...
75
Letting ξ = 0 in Corollary 3.10, we have
Corollary 3.12 ([3]). Let the function f (z) defined by (1.2) be in the
class U T (Φ, Ψ, γ, k). If {σ(γ, k, n)}∞
n=2 is a non-decreasing sequence. Then
we have
1−γ
1−γ
|z| ≤ |f (z)| ≤ 1 +
|z|
σ(γ, k, 2)
σ(γ, k, 2)
for ξ > 0, z ∈ U. The result is sharp for the function
(3.26)
1−
1−γ 2
z .
σ(γ, k, 2)
Letting ξ → 1 in Corollary 3.11,we have
(3.27)
f (z) = z −
Corollary 2.7 ([3]).Let the function f (z) defined by (1.2) be in the class
U T (Φ, Ψ, γ, k). If {σ(γ, k, n)/n}∞
n=2 is a non-decreasing sequence. Then we
have
2(1 − γ)
2(1 − γ)
|z| ≤ |f ′ (z)| ≤ 1 +
|z|
σ(γ, k, 2)
σ(γ, k, 2)
for 0 ≤ ξ < 1, z ∈ U. The result is sharp for the function given by (3.27).
(3.28)
1−
References
[1] R.Bharati, R.Parvatham and A.Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math., 26 1 (1997), 17 - 32.
[2] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricemi, “ Tables
of integral Transforms,” voll. II, McGraw-Hill Book Co., NewYork,
Toronto and London,1954.
76
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
[3] B.A.Frasin, Comprehensive
family
of
uniformly
analytic
func-
tions, Tamkang J. Math., 36 3 (2005), 243 - 254.
[4] B.A.Frasin, Generalization to classes for analytic functions with negative coefficients, Jour. of Inst. of Math. & Comp. Sci.(Math. Ser.)
Vol.12 No. 1(1999), 75-80.
[5] A.W.
Goodman, On
uniformly
convex
functions, Ann.
polon.
Math., 56 (1991), 87-92.
[6] A.W. Goodman, On uniformly starlike functions, J. Math. Anal. &
Appl., 155(1991),364-370.
[7] O.P. Juneja, T.R.Reddy and M.L. Mogra, A convolution approach
for analytic functions with negative coefficients, Soochow J. Math. 11
(1985), 69-81.
[8] J.E.Littlewood, On inequalities in theory of functions, Proc. London
Math. Soc., 23 (1925), 481 - 519.
[9] K.B. Oldham and T. Spanier, The Fractional Calculus: Theory and
Applications of Differentiation and Integral to Arbitrary Order, Academic Press, NewYork and London, 1974.
[10] S. Owa, On the distortion theorems I, Kyungpook Math. J. 18 (1978),
53-59.
[11] S. Owa, M. Saigo and H.M.Srivastava, Some characterization theorems
for starlike and convex functions involving a certain fractional integral
operators, J. Math. Anal. 140 (1989), 419-426.
Integral Means and fractional calculus operators...
77
[12] F.Rønning, Uniformly convex functions and a corresponding class of
starlike functions, Proc. Amer. Math. Soc., 118,(1993),189-196.
[13] F.Rønning, Integral
representations
for
bounded
starlike
func-
tions, Annal. Polon. Math., 60,(1995),289-297.
[14] M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. College General Ed. Kyushu Univ. 11
(1978),135-143.
[15] S.G. Samko, A.A. Kilbas and O.I. Marchev, “Integrals and Derivatives of Fractional Order and Some of Their Applications, ” (Russian),
Nauka i Teknika, Minsk, 1987.
[16] H. Silverman, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc. , 51, (1975), 109 - 116.
[17] H. Silverman, Integral means for univalent functions with negative coefficients, Houston J. Math., 23 (1997), 169- 174.
[18] H.M.Srivastava and R.G. Buchman, “Convolution Integral Equations
with Special Functions Kernels,” John Wiely and Sons, NewYork, London, Sydney and Toronto, 1977.
[19] H.M.Srivastava and S. Owa (Eds.),“ Univalent functions, Fractional
Calculus, and Their Applications,”Halsted Press (Ellis Horworod Limited, Chichester ), John Wiely and Sons, NewYork, Chichester, Brisbane and Toronto, 1989.
78
B. A. Frasin, G.Murugusundaramoorthy and N.Mageshand
[20] 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. .
[21] K.G.Subramanian, G.Murugusundaramoorthy, P.Balasubrahmanyam
and H.Silverman, Subclasses of uniformly convex and uniformly starlike functions. Math. Japonica., 42 N0.3,(1995),517 - 522.
[22] K.G.Subramanian,
and
H.Silverman,
T.V.Sudharsan,
Classes
of
P.Balasubrahmanyam
uniformly
tions. Publ.Math.Debrecen., 53 3-4,(1998),309 -315 .
Department of Mathematics,
Al al-Bayt University,
P.O. Box: 130095 Mafraq, Jordan.
E-mail address: bafrasin@yahoo.com.
School of Science and Humanities,
Vellore Institute of Technology, Deemed University,
Vellore - 632014, India.
E-mail address:gmsmoorthy@yahoo.com
Department of Mathematics,
Adhiyamaan College of Engineering,
Hosur - 635109,India.
E-mail address: nmagi 2000@yahoo.co.in
starlike
func-