Journal of Inequalities in Pure and
Applied Mathematics
POLYNOMIALS AND CONVEX SEQUENCE INEQUALITIES
ROSIHAN M. ALI, M. HUSSAIN KHAN, V. RAVICHANDRAN,
AND K.G. SUBRAMANIAN
School of Mathematical Sciences
Universiti Sains Malaysia
11800 USM, Penang, Malaysia.
EMail: rosihan@cs.usm.my
Department of Mathematics
Islamiah College
Vaniambadi 635 751, India.
EMail: khanhussaff@yahoo.co.in
School of Mathematical Sciences
Universiti Sains Malaysia
11800 USM, Penang, Malaysia.
EMail: vravi@cs.usm.my
Department of Mathematics
Madras Christian College
Tambaram, Chennai- 600 059, India.
EMail: kgsmani@vsnl.net
volume 6, issue 1, article 22,
2005.
Received 12 December, 2004;
accepted 27 January, 2005.
Communicated by: H. Silverman
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
237-04
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Abstract
For a given p-valent analytic function g with positive
in the open unit
Pcoefficients
n , a ≥ 0 satisfying
disk ∆, we study a class of functions f(z) = z p + ∞
a
z
n
n
n=m
1 z(f ∗ g)0 (z)
m+p
<
<α
z ∈ ∆; 1 < α <
.
p
(f ∗ g)(z)
2p
Coefficient inequalities, distortion and covering theorems, as well as closure
theorems are determined. The results obtained extend several known results
as special cases.
2000 Mathematics Subject Classification: 30C45
Key words: Starlike function, Ruscheweyh derivative, Convolution, Positive coefficients, Coefficient inequalities, Growth and distortion theorems.
The authors R. M. Ali and V. Ravichandran respectively acknowledged support from
an IRPA grant 09-02-05-00020 EAR and a post-doctoral research fellowship from
Universiti Sains Malaysia
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Coefficient Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Growth and Distortion Theorems . . . . . . . . . . . . . . . . . . . . . . . 9
4
Closure Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5
Order and Radius Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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1.
Introduction
Let A denote the class of all analytic functions f (z) in the unit disk ∆ := {z ∈
C : |z| < 1} with f (0) = 0 = f 0 (0) − 1. The class M (α) defined by
0 zf (z)
3
M (α) := f ∈ A : <
<α
1<α< ; z∈∆
f (z)
2
was investigated by Uralegaddi et al. [6]. A subclass of M (α) was recently investigated by Owa and Srivastava [3]. Motivated by M (α), we introduce a more
general class P Mg (p, m, α) of analytic functions with positive coefficients. For
two analytic functions
f (z) = z p +
∞
X
an z n
and
g(z) = z p +
n=m
∞
X
bn z n ,
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
n=m
the convolution (or Hadamard product) of f and g, denoted by f ∗g or (f ∗g)(z),
is defined by
∞
X
p
(f ∗ g)(z) := z +
an bn z n .
n=m
P
n
Let T (p, m) be the class of all analytic p-valent functions f (z) = z p − ∞
n=m an z
(an ≥ 0), defined on the unit disk ∆ and let T := T (1, 2). A function
f (z) ∈ T (p, m) is called a function with negative coefficients. The subclass
of T consisting of starlike functions of order α, denoted by T S ∗ (α), was studied by Silverman [5]. Several other classes of starlike functions with negative
coefficients were studied; for e.g. see [2].
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Let P (p, m) be the class of all analytic functions
(1.1)
p
f (z) = z +
∞
X
an z n
(an ≥ 0)
bn z n
(bn > 0)
n=m
and P := P (1, 2).
Definition 1.1. Let
(1.2)
p
g(z) = z +
∞
X
n=m
be a fixed analytic function in ∆. Define the class P Mg (p, m, α) by
z(f ∗ g)0 (z)
1
P Mg (p, m, α) := f ∈ P (p, m) : <
< α,
p
(f ∗ g)(z)
m+p
1<α<
;z ∈ ∆
.
2p
When g(z) = z/(1 − z), p = 1 and m = 2, the class P Mg (p, m, α) reduces
to the subclass P M (α) := P ∩ M (α). When g(z) = z/(1 − z)λ+1 , p = 1 and
m = 2, the class P Mg (p, m, α) reduces to the class:
3
z(Dλ f (z))0
< α,
λ > −1, 1 < α < ; z ∈ ∆
,
Pλ (α) = f ∈ P : <
Dλ f (z)
2
where Dλ denotes the Ruscheweyh derivative of order λ. When
g(z) = z +
∞
X
n=2
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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l n
nz ,
Page 4 of 19
the class of functions P Mg (1, 2, α) reduces to the class P Ml (α) where
3
z(Dl f (z))0
P Ml (α) = f ∈ P : <
< α,
1 < α < ; l ≥ 0; z ∈ ∆
,
Dl f (z)
2
where Dl denotes the Salagean derivative of order l. Also we have
P M (α) ≡ P0 (α) ≡ P M0 (α).
A function f ∈ A(p, m) is in P P C(p, m, α, β) if
!
(1 − β)zf 0 (z) + βp z(zf 0 )0 (z)
1
m+p
<α
β ≥ 0; 0 ≤ α <
<
2p
p
(1 − β)f (z) + βp zf 0 (z)
This class is similar to the class of β-Pascu convex functions of order α and it
unifies the class of P M (α) and the corresponding convex class.
For the newly defined class P Mg (p, m, α), we obtain coefficient inequalities,
distortion and covering theorems, as well as closure theorems. As special cases,
we obtain results for the classes Pλ (α), and P Ml (α). Similar results for the
class P P C(p, m, α, β) also follow from our results, the details of which are
omitted here.
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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2.
Coefficient Inequalities
Throughout the paper, we assume that the function f (z) is given by the equation
(1.1) and g(z) is given by by (1.2). We first prove a necessary and sufficient
condition for functions to be in the class P Mg (p, m, α) in the following:
Theorem 2.1. A function f ∈ P Mg (p, m, α) if and only if
∞
X
m+p
(2.1)
(n − pα)an bn ≤ p(α − 1)
1<α<
.
2p
n=m
Proof. If f ∈ P Mg (p, m, α), then (2.1) follows from
z(f ∗ g)0 (z)
1
<
<α
p
(f ∗ g)(z)
by letting z → 1− through real values. To prove the converse, assume that (2.1)
holds. Then by making use of (2.1), we obtain
z(f ∗ g)0 (z) − p(f ∗ g)(z)
z(f ∗ g)0 (z) − (2α − 1)p(f ∗ g)(z) P∞
n=m (n − p)an bn
P
≤
≤1
2(α − 1)p − ∞
n=m [n − (2α − 1)p]an bn
or equivalently f ∈ P Mg (p, m, α).
Corollary 2.2. A function f ∈ Pλ (α) if and only if
∞
X
3
(n − α)an Bn (λ) ≤ α − 1
1<α<
,
2
n=2
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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where
(2.2)
Bn (λ) =
(λ + 1)(λ + 2) · · · (λ + n − 1)
.
(n − 1)!
Corollary 2.3. A function f ∈ P Mm (α) if and only if
∞
X
3
m
(n − α)an n ≤ α − 1
1<α<
.
2
n=2
Our next theorem gives an estimate for the coefficient of functions in the
class P Mg (p, m, α).
Theorem 2.4. If f ∈ P Mg (p, m, α), then
an ≤
p(α − 1)
(n − pα)bn
with equality only for functions of the form
fn (z) = z p +
p(α − 1) n
z .
(n − pα)bn
Proof. Let f ∈ P Mg (p, m, α). By making use of the inequality (2.1), we have
(n − pα)an bn ≤
∞
X
(n − pα)an bn ≤ p(α − 1)
n=m
or
p(α − 1)
.
an ≤
(n − pα)bn
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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Clearly for
p(α − 1) n
z ∈ P Mg (p, m, α),
(n − pα)bn
fn (z) = z p +
we have
an =
p(α − 1)
.
(n − pα)bn
Corollary 2.5. If f ∈ Pλ (α), then
an ≤
α−1
(n − α)Bn (λ)
with equality only for functions of the form
fn (z) = z +
α−1
zn,
(n − α)Bn (λ)
where Bn (λ) is given by (2.2).
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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Corollary 2.6. If f ∈ P Mm (α), then
an ≤
α−1
(n − α)nm
with equality only for functions of the form
fn (z) = z +
α−1
zn.
(n − α)nm
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3.
Growth and Distortion Theorems
We now prove the growth theorem for the functions in the class P Mg (p, m, α).
Theorem 3.1. If f ∈ P Mg (p, m, α), then
rp −
p(α − 1) m
p(α − 1) m
r ≤ |f (z)| ≤ rp +
r ,
(m − pα)bm
(m − pα)bm
|z| = r < 1,
provided bn ≥ bm ≥ 1. The result is sharp for
(3.1)
f (z) = z p +
p(α − 1) m
z .
(m − pα)bm
Proof. By making use of the inequality (2.1) for f ∈ P Mg (p, m, α) together
with
(m − pα)bm ≤ (n − pα)bn ,
we obtain
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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bm (m − pα)
∞
X
an ≤
n=m
∞
X
(n − pα)an bn ≤ p(α − 1)
n=m
or
(3.2)
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∞
X
n=m
an ≤
p(α − 1)
.
(m − pα)bm
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By using (3.2) for the function f (z) = z p +
have for |z| = r,
p
|f (z)| ≤ r +
∞
X
P∞
n=m
an z n ∈ P Mg (p, m, α), we
an r n
n=m
p
≤r +r
m
∞
X
an
n=m
≤ rp +
p(α − 1) m
r ,
(m − pα)bm
|f (z)| ≥ rp −
p(α − 1) m
r .
(m − pα)bm
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
and similarly,
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
Theorem 3.1 also shows that f (∆) for every f ∈ P Mg (p, m, α) contains the
p(α−1)
disk of radius 1 − (m−pα)b
.
m
α−1
α−1
r2 ≤ |f (z)| ≤ r +
r2
(2 − α)(λ + 1)
(2 − α)(λ + 1)
(|z| = r).
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The result is sharp for
(3.3)
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Corollary 3.2. If f ∈ Pλ (α), then
r−
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f (z) = z +
α−1
z2.
(2 − α)(λ + 1)
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Page 10 of 19
Corollary 3.3. If f ∈ P Mm (α), then
r−
α−1
α−1
r2 ≤ |f (z)| ≤ r +
r2
m
(2 − α)2
(2 − α)2m
(|z| = r).
The result is sharp for
(3.4)
f (z) = z +
α−1
z2.
m
(2 − α)2
The distortion estimates for the functions in the class P Mg (p, m, α) is given
in the following:
Theorem 3.4. If f ∈ P Mg (p, m, α), then
mp(α − 1) m−1
mp(α − 1) m−1
prp−1 −
r
≤ |f 0 (z)| ≤ prp−1 +
r
,
(m − pα)bm
(m − pα)bm
|z| = r < 1,
provided bn ≥ bm . The result is sharp for the function given by (3.1).
Proof. By making use of the inequality (2.1) for f ∈ P Mg (p, m, α), we obtain
∞
X
n=m
an b n ≤
p(α − 1)
(m − pα)
and therefore, again using the inequality (2.1), we get
∞
X
mp(α − 1)
nan ≤
.
(m − pα)bm
n=m
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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For the function f (z) = z p +
P∞
n=m
|f 0 (z)| ≤ prp−1 +
an z n ∈ P Mg (p, m, α), we now have
∞
X
nan rn−1
(|z| = r)
n=m
≤ pr
p−1
+r
m−1
∞
X
nan
n=m
≤ prp−1 +
mp(α − 1) m−1
r
(m − pα)bm
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
and similarly we have
|f 0 (z)| ≥ prp−1 −
mp(α − 1) m−1
r
.
(m − pα)bm
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
Corollary 3.5. If f ∈ Pλ (α), then
1−
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2(α − 1)
2(α − 1)
r ≤ |f 0 (z)| ≤ 1 +
r
(2 − α)(λ + 1)
(2 − α)(λ + 1)
(|z| = r).
The result is sharp for the function given by (3.3)
Corollary 3.6. If f ∈ P Mm (α), then
1−
2(α − 1)
2(α − 1)
r ≤ |f 0 (z)| ≤ 1 +
r
m
(2 − α)2
(2 − α)2m
The result is sharp for the function given by (3.4)
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(|z| = r).
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4.
Closure Theorems
We shall now prove the following closure theorems for the class P Mg (p, m, α).
Let the functions Fk (z) be given by
p
Fk (z) = z +
(4.1)
∞
X
fn,k z n ,
(k = 1, 2, . . . , M ).
n=m
P
Theorem 4.1. Let λk ≥ 0 for k = 1, 2, . . . , M and M
k=1 λk ≤ 1. Let the
function Fk (z) defined by (4.1) be in the class P Mg (p, m, α) for every k =
1, 2, . . . , M . Then the function f (z) defined by
!
∞
M
X
X
f (z) = z p +
λk fn,k z n
n=m
k=1
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
belongs to the class P Mg (p, m, α).
Proof. Since Fk (z) ∈ P Mg (p, m, α), it follows from Theorem 2.1 that
∞
X
(4.2)
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(n − pα)bn fn,k ≤ p(α − 1)
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n=m
for every k = 1, 2, . . . , M. Hence
∞
X
(n − pα)bn
n=m
M
X
!
λk fn,k
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=
k=1
≤
M
X
k=1
M
X
k=1
λk
∞
X
!
(n − pα)bn fn,k
n=m
λk p(α − 1) ≤ p(α − 1).
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By Theorem 2.1, it follows that f (z) ∈ P Mg (p, m, α).
Corollary 4.2. The class P Mg (p, m, α) is closed under convex linear combinations.
Theorem 4.3. Let
Fp (z) = z p and Fn (z) = z p +
p(α − 1) n
z
(n − pα)bn
for n = m, m + 1, . . .. Then f (z) ∈ P Mg (p, m, α) if and only if f (z) can be
expressed in the form
p
f (z) = λp z +
(4.3)
∞
X
λn Fn (z),
n=m
where each λj ≥ 0 and λp +
P∞
n=m
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∞
X
λn p(α − 1) n
z
(n − pα)bn
n=m
and therefore
∞
X
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
λn = 1.
Proof. Let f (z) be of the form (4.3). Then
f (z) = z p +
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∞
X
λn p(α − 1) (n − pα)bn
=
λn = 1 − λp ≤ 1.
(n
−
pα)b
p(α
−
1)
n
n=m
n=m
By Theorem 2.1, we have f (z) ∈ P Mg (p, m, α).
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Conversely, let f (z) ∈ P Mg (p, m, α). From Theorem 2.4, we have
an ≤
p(α − 1)
(n − pα)bn
n = m, m + 1, . . . .
for
Therefore we may take
λn =
(n − pα)bn an
p(α − 1)
for
and
λp = 1 −
∞
X
n = m, m + 1, . . .
λn .
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
n=m
Then
p
f (z) = λp z +
∞
X
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
λn Fn (z).
n=m
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We now prove that the class P Mg (p, m, α) is closed under convolution with
certain functions and give an application of this result to show that the class
P Mg (p, m, α) is closed under the familiar Bernardi integral operator.
P
n
Theorem 4.4. Let h(z) = z p + ∞
n=m hn z be analytic in ∆ with 0 ≤ hn ≤ 1.
If f (z) ∈ P Mg (p, m, α), then (f ∗ h)(z) ∈ P Mg (p, m, α).
Proof. The result follows directly from Theorem 2.1.
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The generalized Bernardi integral operator is defined by the following integral:
Z
c + p z c−1
(4.4)
F (z) =
t f (t)dt (c > −1; z ∈ ∆).
zc 0
Since
F (z) = f (z) ∗
!
∞
X
c
+
p
zp +
zn ,
c
+
n
n=m
we have the following:
Corollary 4.5. If f (z) ∈ P Mg (p, m, α), then F (z) given by (4.4) is also in
P Mg (p, m, α).
A Class of Multivalent
Functions with Positive
Coefficients Defined by
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Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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5.
Order and Radius Results
Let P Sh∗ (p, m, β) be the subclass of P (m, p) consisting of functions f for which
f ∗ h is starlike of order β.
P
n
Theorem 5.1. Let h(z) = z p + ∞
n=m hn z with hn > 0. Let (α − 1)nhn ≤
(n − pα)bn . If f ∈ P Mg (p, m, α), then f ∈ P Sh∗ (p, m, β), where
(n − pα)bn − (α − 1)nhn
β := inf
.
n≥m (n − pα)bn − (α − 1)phn
Proof. Let us first note that the condition (α − 1)nhn ≤ (n − pα)bn implies
f ∈ P Sh∗ (p, m, 0). From the definition of β, it follows that
(n − pα)bn − (α − 1)nhn
β≤
(n − pα)bn − (α − 1)phn
or
(n − pβ)hn
(n − pα)bn
≤
1−β
α−1
and therefore, in view of (2.1),
∞
∞
X
X
(n − pα)
(n − pβ)
an hn ≤
an bn ≤ 1.
p(1
−
β)
p(α
−
1)
n=m
n=m
Thus
P∞
1 z(f ∗ h)0 (z)
n=m (n/p − 1)an hn
·
p (f ∗ h)(z) − 1 ≤ 1 − P∞ an hn ≤ 1 − β
n=m
and therefore f ∈ P Sh∗ (p, m, β).
A Class of Multivalent
Functions with Positive
Coefficients Defined by
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Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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Similarly we can prove the following:
Theorem 5.2. If f ∈ P Mg (p, m, α), then f ∈ P Mh (p, m, β) in |z| < r(α, β)
where
(
1 )
(n − pα) (β − 1) bn n−p
.
r(α, β) := min 1; inf
n≥m (n − pα) (α − 1) hn
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
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References
[1] R.M. ALI, M. HUSSAIN KHAN, V. RAVICHANDRAN AND K.G. SUBRAMANIAN, A class of multivalent functions with negative coefficients
defined by convolution, preprint.
[2] O.P. AHUJA, Hadamard products of analytic functions defined by
Ruscheweyh derivatives, in Current Topics in Analytic Function Theory,
13–28, World Sci. Publishing, River Edge, NJ.
[3] S. OWA AND H.M. SRIVASTAVA, Some generalized convolution properties associated with certain subclasses of analytic functions, J. Inequal. Pure
Appl. Math., 3(3) (2002), Article 42, 13 pp. [ONLINE: http://jipam.
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[4] V. RAVICHANDRAN, On starlike functions with negative coefficients, Far
East J. Math. Sci., 8(3) (2003), 359–364.
A Class of Multivalent
Functions with Positive
Coefficients Defined by
Convolution
Rosihan M. Ali, M. Hussain Khan,
V. Ravichandran and
K.G. Subramanian
Title Page
[5] H. SILVERMAN, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc., 51 (1975), 109–116.
[6] B.A. URALEGADDI, M.D. GANIGI AND S.M. SARANGI, Univalent
functions with positive coefficients, Tamkang J. Math., 25(3) (1994), 225–
230.
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