Journal of Inequalities in Pure and
Applied Mathematics
NEW SUBCLASSES OF MEROMORPHIC p−VALENT FUNCTIONS
B.A. FRASIN AND G. MURUGUSUNDARAMOORTHY
Department of Mathematics
Al al-Bayt University
P.O. Box: 130095
Mafraq, Jordan.
EMail: bafrasin@yahoo.com
Vellore Institute of Technology, Deemed University,
Vellore, TN-632 014 India.
EMail: gmsmoorthy@yahoo.com
volume 6, issue 3, article 68,
2005.
Received 20 October, 2004;
accepted 02 June, 2005.
Communicated by: A. Sofo
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
202-04
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Abstract
In this paper, we introduce two subclasses Ω∗p (α) and Λ∗p (α) of meromorphic
p-valent functions in the punctured disk D = {z : 0 < |z| < 1}. Coefficient
inequalities, distortion theorems, the radii of starlikeness and convexity, closure
theorems and Hadamard product ( or convolution) of functions belonging to
these classes are obtained.
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words: Meromorphic p−valent functions, Meromorphically starlike and convex
functions.
Contents
1
Introduction and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Coefficient Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Distortion Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Radii of Starlikeness and Convexity . . . . . . . . . . . . . . . . . . . . . 11
5
Closure Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6
Convolution Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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1.
Introduction and Definitions
Let Σp denote the class of functions of the form:
∞
(1.1)
X
1
f (z) = p +
ap+n−1 z p+n−1
z
n=1
(p ∈ N),
which are analytic and p-valent in the punctured unit disk D = {z : 0 < |z| <
1}. A function f ∈ Σp is said to be in the class Ωp (α) of meromorphic p-valently
starlike functions of order α in D if and only if
zf 0 (z)
(1.2)
Re −
>α
(z ∈ D; 0 ≤ α < p; p ∈ N).
f (z)
Furthermore, a function f ∈ Σp is said to be in the class Λp (α) of meromorphic
p-valently convex functions of order α in D if and only if
zf 00 (z)
(1.3)
Re −1 − 0
>α
(z ∈ D; 0 ≤ α < p; p ∈ N).
f (z)
The classes Ωp (α), Λp (α) and various other subclasses of Σp have been studied rather extensively by Aouf et.al. [1] – [3], Joshi and Srivastava [4], Kulkarni
et. al. [5], Mogra [6], Owa et. al. [7], Srivastava and Owa [8], Uralegaddi and
Somantha [9], and Yang [10].
In the next section we derive sufficient conditions for f (z) to be in the classes
Ωp (α) and Λp (α), which are obtained by using coefficient inequalities.
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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2.
Coefficient Inequalities
Theorem 2.1. Let σn (p, k, α) = (p + n + k − 1) + |p + n + 2α − k − 1|. If
f (z) ∈ Σp satisfies
(2.1)
∞
X
σn (p, k, α) |ap+n−1 | < 2(p − α)
n=1
for some α (0 ≤ α < p) and some k (k ≥ p), then f (z) ∈ Ωp (α).
Proof. Suppose that (2.1) holds true for α (0 ≤ α < p) and k (k ≥ p). For
f (z) ∈ Σp , it suffices to show that
zf 0 (z)
+k
f (z)
zf 0 (z)
<1
+ (2α − k) New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
(z ∈ D).
f (z)
We note that
zf 0 (z)
+
k
f (z)
zf 0 (z)
+ (2α − k) f (z)
P
k−p+ ∞
(p + n + k − 1)ap+n−1 z 2p+n−1
n=1
P∞
=
2α − k − p + n=1 (p + n + 2α − k − 1)ap+n−1 z 2p+n−1 P
2p+n−1
k−p+ ∞
n=1 (p + n + k − 1) |ap+n−1 | |z|
≤
P
2p+n−1
p + k − 2α − ∞
n=1 |p + n + 2α − k − 1| |ap+n−1 | |z|
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P
(p + n + k − 1) |ap+n−1 |
k−p+ ∞
Pn=1
<
.
∞
p + k − 2α − n=1 |p + n + 2α − k − 1| |ap+n−1 |
The last expression is bounded above by 1 if
∞
∞
X
X
k−p+ (p+n+k−1) |ap+n−1 | < p+k−2α−
|p + n + 2α − k − 1| |ap+n−1 |
n=1
n=1
which is equivalent to our condition (2.1) of the theorem.
Example 2.1. The function f (z) given by
New Subclasses of
Meromorphic p−Valent
Functions
∞
(2.2)
X
4(p − α)
1
f (z) = p +
z p+n−1
z
n(n
+
1)σ
(p,
k,
α)
n
n=1
(p ∈ N)
B.A. Frasin and
G. Murugusundaramoorthy
belongs to the class Ωp (α).
Since f (z) ∈ Ωp (α) if and only if zf 0 (z) ∈ Λp (α), we can prove:
Theorem 2.2. If f (z) ∈ Σp satisfies
(2.3)
∞
X
(p + n − 1)σn (p, k, α) |ap+n−1 | < 2(p − α)
n=1
for some α(0 ≤ α < p) and some k(k ≥ p), then f (z) ∈ Λp (α).
Example 2.2. The function f (z) given by
(2.4)
f (z) =
∞
X
1
4(p − α)
+
z p+n−1
p
z
n(n
+
1)(p
+
n
−
1)σ
(p,
k,
α)
n
n=1
belongs to the class Λp (α).
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In view of Theorem 2.1 and Theorem 2.2, we now define the subclasses:
Ω∗p (α) ⊂ Ωp (α) and Λ∗p (α) ⊂ Λp (α),
which consist of functions f (z) ∈ Σp satisfying the conditions (2.1) and (2.3),
respectively.
Letting p = 1, 1 ≤ k ≤ n + 2α, where 0 ≤ α < 1 in Theorem 2.1 and
Theorem 2.2, we have the following corollaries:
Corollary 2.3. If f (z) ∈ Σ1 satisfies
∞
X
(n + α) |an | < 1 − α
n=1
then f (z) ∈ Ω1 (α) = Σ∗ (α) the class of meromorphically starlike functions of
order α in D.
Corollary 2.4. If f (z) ∈ Σ1 satisfies
∞
X
n(n + α) |an | < 1 − α
n=1
then f (z) ∈ Λ1 (α) =
order α in D.
Σ∗K (α)
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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the class of meromorphically convex functions of
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3.
Distortion Theorems
A distortion property for functions in the class Ω∗p (α) is contained in
Theorem 3.1. If the function f (z) defined by (1.1) is in the class Ω∗p (α), then
for 0 < |z| = r < 1,we have
(3.1)
1
2(p − α)
−
rp ≤ |f (z)|
p
r
p + k + |p + 2α − k|
2(p − α)
1
≤ p+
rp ,
r
p + k + |p + 2α − k|
and
p
(3.2)
rp+1
−
2p(p − α)
rp−1
p + k + |p + 2α − k|
≤ |f 0 (z)|
2p(p − α)
p
≤ p+1 +
rp−1 .
r
p + k + |p + 2α − k|
The bounds in (3.1) and (3.2) are attained for the functions f (z) given by
(3.3)
f (z) =
Proof. Since f ∈
(3.4)
1
2(p − α)
+
zp
p
z
p + k + |p + 2α − k|
Ω∗p (α),
∞
X
n=1
(p ∈ N; z ∈ D).
from the inequality (2.1), we have
|ap+n−1 | ≤
2(p − α)
.
p + k + |p + 2α − k|
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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Thus, for 0 < |z| = r < 1, and making use of (3.4) we have
(3.5)
∞
1 X
|f (z)| ≤ p +
|ap+n−1 | |z|p+n−1
z
≤
≤
1
+r
rp
n=1
∞
X
p
|ap+n−1 |
n=1
1
2(p − α)
+
rp
p
r
p + k + |p + 2α − k|
and
(3.6)
X
∞
1
|f (z)| ≥ p −
|ap+n−1 | |z|p+n−1
z
≥
≥
1
−r
rp
n=1
∞
X
p
B.A. Frasin and
G. Murugusundaramoorthy
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|ap+n−1 |
n=1
1
2(p − α)
−
rp .
rp p + k + |p + 2α − k|
We also observe that
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∞
(3.7)
New Subclasses of
Meromorphic p−Valent
Functions
p + k + |p + 2α − k| X
(p + n − 1) |ap+n−1 | ≤ 2(p − α)
p
n=1
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which readily yields the following distortion inequalities:
∞
(3.8)
X
p
|f (z)| ≤ p+1 +
(p + n − 1) |ap+n−1 | |z|p+n−2
|z|
n=1
∞
X
p
≤ p+1 + rp−1
(p + n − 1) |ap+n−1 |
r
n=1
0
≤
p
rp+1
+
2p(p − α)
rp−1
p + k + |p + 2α − k|
and
∞
(3.9)
X
p
|f (z)| ≥ p+1 −
(p + n − 1) |ap+n−1 | |z|p+n−2
|z|
n=1
∞
X
p
(p + n − 1) |ap+n−1 |
≥ p+1 − rp−1
r
n=1
0
≥
p
rp+1
−
2p(p − α)
rp−1 .
p + k + |p + 2α − k|
This completes the proof of Theorem 3.1.
Similarly, for function f (z) ∈ Λ∗p (α), and making use of (2.3), we can prove
Theorem 3.2. If the function f (z) defined by (1.1) is in the class Λ∗p (α), then
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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for 0 < |z| = r < 1, we have
(3.10)
2(p − α)
1
−
rp ≤ |f (z)|
p
r
p[p + k + |p + 2α − k|]
2(p − α)
1
rp ,
≤ p+
r
p[p + k + |p + 2α − k|]
and
p
(3.11)
r
−
p+1
2(p − α)
rp−1
p + k + |p + 2α − k|
≤ |f 0 (z)|
2(p − α)
p
≤ p+1 +
rp−1 .
r
p + k + |p + 2α − k|
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
Title Page
The bounds in (3.10) and (3.11) are attained for the functions f (z) given by
(3.12)
g(z) =
1
2(p − α)
+
zp
p
z
p[p + k − 1 + |p + 2α − k|]
(p ∈ N; z ∈ D).
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4.
Radii of Starlikeness and Convexity
The radii of starlikeness and convexity for the classes Ω∗p (α) is given by
Theorem 4.1. If the function f (z) be defined by (1.1) is in the class Ω∗p (α), then
f (z) is meromorphically p-valently starlike of order δ(0 ≤ δ < p) in |z| < r1 ,
where
1
2p+n−1
(p − δ)σn (p, k, α)
(4.1)
r1 = inf
(p ∈ N).
n≥1
2(3p + n + 1 − δ)(p − α)
Furthermore, f (z) is meromorphically p-valently convex of order δ(0 ≤ δ < p)
in |z| < r2 , where
(4.2) r2 = inf
n≥1
p(p − δ)σn (p, k, α)
2[(p + n − 1)[3p + n − 1 − δ](p − α)
(4.3)
(p ∈ N; z ∈ D).
Proof. It suffices to prove that
0
zf (z)
≤ p − δ,
(4.4)
+
p
f (z)
B.A. Frasin and
G. Murugusundaramoorthy
1
2p+n−1
The results (4.1) and (4.2) are sharp for the function f (z) given by
1
2(p − α) p+n−1
f (z) = p +
z
z
σn (p, k, α)
New Subclasses of
Meromorphic p−Valent
Functions
(p ∈ N).
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for |z| ≤ r1 . We have
0
P∞
n=1 (2p + n − 1)ap+n−1 z p+n−1 zf (z)
P∞
(4.5)
1
f (z) + p = p+n−1
+
a
z
p+n−1
p
n=1
P∞ z
2p+n−1
n=1 (2p + n − 1) |ap+n−1 | |z|
≤
.
P
2p+n−1
1− ∞
n=1 |ap+n−1 | |z|
Hence (4.5) holds true if
(4.6)
∞
X
New Subclasses of
Meromorphic p−Valent
Functions
2p+n−1
(2p + n − 1) |ap+n−1 | |z|
n=1
≤ (p − δ) 1 −
∞
X
!
2p+n−1
|ap+n−1 | |z|
,
B.A. Frasin and
G. Murugusundaramoorthy
n=1
Title Page
or
(4.7)
∞
X
3p + n − 1 − δ
n=1
(p − δ)
|ap+n−1 | |z|2p+n−1 ≤ 1,
with the aid of (2.1), (4.7) is true if
(4.8)
3p + n − 1 − δ 2p+n−1 σn (p, k, α)
|z|
≤
(p − δ)
2(p − α)
Solving (4.8) for |z|, we obtain
1
2p+n−1
(p − δ)σn (p, k, α)
(4.9)
|z| <
2(3p + n + 1 − δ)(p − α)
(n ≥ 1).
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(n ≥ 1).
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In precisely the same manner, we can find the radius of convexity asserted by
(4.2), by requiring that
00
zf (z)
≤ p − δ,
(4.10)
+
p
+
1
f 0 (z)
in view of (2.1). This completes the proof of Theorem 4.1.
Similarly, we can get the radii of starlikeness and convexity for functions in
the class Λ∗p (α).
Theorem 4.2. If the function f (z) be defined by (1.1) is in the class Λ∗p (α), then
f (z) is meromorphically p-valently starlike of order δ(0 ≤ δ < p) in |z| < r3 ,
where
1
(p − δ)(p + n − 1)σn (p, k, α) 2p+n−1
(p ∈ N).
(4.11)
r3 = inf
n≥1
2(3p + n + 1 − δ)(p − α)
Furthermore, f (z) is meromorphically p-valently convex of order δ(0 ≤ δ < p)
in |z| < r4 , where
(4.12) r4 = inf
n≥1
p(p − δ)(p + n − 1)σn (p, k, α)
2[(p + n − 1)[3p + n − 1 − δ](p − α)
1
2p+n−1
(p ∈ N).
The results (4.11) and (4.12) are sharp for the function g(z) given by
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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(4.13)
1
2(p − α)
g(z) = p +
z p+n−1
z
(p + n − 1)σn (p, k, α)
(p ∈ N; z ∈ D).
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5.
Closure Theorems
Let the functions fj (z) be defined, for j ∈ {1, 2, . . . , m},by
∞
(5.1)
X
1
fj (z) = p +
ap+n−1,j z p+n−1 ,
z
n=1
(z ∈ D).
Now, we shall prove the following results for the closure of functions in the
classes Ω∗p (α) and Λ∗p (α).
Theorem 5.1. Let the functions fj (z), j ∈ {1, 2, . . . , m}, defined by (5.1) be in
the class Ω∗p (α). Then the function h(z) ∈ Ω∗p (α) where
(5.2)
h(z) =
m
X
bj fj (z),
bj ≥ 0 and
j=1
m
X
B.A. Frasin and
G. Murugusundaramoorthy
bj = 1).
j=1
Title Page
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Proof. From (5.2), we can write h(z) as
∞
(5.3)
New Subclasses of
Meromorphic p−Valent
Functions
X
1
h(z) = p +
cp+n−1 z p+n−1 ,
z
n=1
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where
(5.4)
Close
cp+n−1 =
m
X
j=1
bj ap+n−1,j ,
j ∈ {1, 2, . . . , m}.
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Since fj (z) ∈ Ω∗p (α), ( j ∈ {1, 2, . . . , m}), from (2.1) , we have
!
X
∞ m
X
σn (p, k, α)
bj |ap+n−1,j |
2(p − α)
n=1
j=1
=
≤
m
X
j=1
m
X
bj
∞
X
σn (p, k, α)
n=1
2(p − α)
!
|ap+n−1,j |
bj = 1,
j=1
which shows that h(z) ∈ Ω∗p (α). This completes the proof of Theorem 5.1.
Using the same technique as in the proof of Theorem 5.1, we have
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
Theorem 5.2. Let the functions fj (z), j ∈ {1, 2, . . . , m}, defined by (5.1) be in
the class Λ∗p (α). Then the function h(z) ∈ Λ∗p (α), where h(z) defined by (5.2).
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Theorem 5.3. Let
Contents
(5.5)
1
fp−1 (z) = p
z
(z ∈ D)
and
(5.6)
fp+n−1 (z) =
2(p − α) p+n−1
1
+
z
,
z p σn (p, k, α)
where n ∈ N0 = N ∪ {0}; z ∈ D. Then f (z) ∈ Ω∗p (α) if and only if it can be
expressed in the form
∞
X
(5.7)
f (z) =
λp+n−1 fp+n−1 (z)
n=0
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where λp+n−1 ≥ 0, (n ∈ N0 ) and
P∞
n=0
λp+n−1 = 1.
Proof. From (5.5), (5.6) and (5.7), it is easily seen that
f (z) =
(5.8)
∞
X
λp+n−1 fn+p−1 (z)
n=0
=
2(p − α)
1
+
λp+n−1 z p+n−1 .
z p σn (p, k, α)
Since
∞
∞
X
X
σn (p, k, α) 2(p − α)
.
λp+n−1 =
λp+n−1 = 1 − λp−1 ≤ 1,
2(p
−
α)
σ
(p,
k,
α)
n
n=1
n=1
it follows from Theorem 2.1 that the function f (z) given by (5.6) is in the class
Ω∗p (α).
Conversely, let us suppose that f (z) ∈ Ω∗p (α). Since
2(p − α)
|ap+n−1 | ≤
σn (p, k, α)
(n ≥ 1),
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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setting
λp+n−1 =
σn (p, k, α)
|ap+n−1 | ,
2(p − α)
and
λp−1 = 1 −
∞
X
n=1
(n ≥ 1)
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λp+n−1,
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it follows that
f (z) =
∞
X
λp+n−1 fp+n−1 (z).
n=0
This completes the proof of the theorem.
Similarly, we can prove the same result for the class Λ∗p (α).
Theorem 5.4. Let
(5.9)
gp−1 (z) =
1
zp
(z ∈ D)
and
(5.10)
B.A. Frasin and
G. Murugusundaramoorthy
gp+n−1 (z) =
1
2(p − α)
+
z p+n−1
z p (p + n − 1)σn (p, k, α)
where n ∈ N0 and z ∈ D. Then g(z) ∈ Λ∗p (α) if and only if it can be expressed
in the form
(5.11)
New Subclasses of
Meromorphic p−Valent
Functions
g(z) =
∞
X
λp+n−1 gp+n−1 (z)
n=0
where λp+n−1 ≥ 0, (n ∈ N0 ) and
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P∞
n=0
λp+n−1 = 1.
Next, we state a theorem which exhibits the fact that the classes Ω∗ (α) and
Λ∗p (α) are closed under convex linear combinations. The proof is fairly straightforward so we omit it.
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Theorem 5.5. Suppose that f (z) and g(z) are in the class Ω∗ (α) (or in Λ∗p (α)).
Then the function h(z) defined by
(5.12)
h(z) = tf (z) + (1 − t)g(z),
(0 ≤ t ≤ 1)
is also in the class Ω∗ (α) (or in Λ∗p (α)).
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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6.
Convolution Properties
For functions
∞
(6.1)
X
1
fj (z) = p +
ap+n−1,j z p+n−1 ,
z
n=1
(j = 1, 2)
belonging to the class Σp , we denote by (f1 ∗ f2 )(z) the Hadamard product (or
convolution) of the functions f1 (z) and f2 (z), that is,
(6.2)
(f1 ∗ f2 )(z) =
New Subclasses of
Meromorphic p−Valent
Functions
∞
X
1
+
ap+n−1,1 ap+n−1,2 z p+n−1 .
z p n=1
B.A. Frasin and
G. Murugusundaramoorthy
Finally, we prove the following.
Theorem 6.1. Let each of the functions fj (z) ( j = 1, 2) defined by (6.1) be in
the class Ω∗ (α). Then (f1 ∗ f2 )(z) ∈ Ω∗ (η), where
2
(6.3)
2
1
p ([p + k + |p + 2α − k|] − 4(p − α) )
(k + 1 − p − n) ≤ η =
,
2
4(p − α)2 + [p + k + |p + 2α − k|]2
(k ≥ p; p, n ∈ N).
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The result is sharp.
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∗
Proof. For fj (z) ∈ Ω (α) (j = 1, 2), we need to find the largest η such that
(6.4)
∞
X
σn (p, k, η)
n=1
2(p − η)
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Page 19 of 24
|ap+n−1,1 | |ap+n−1,2 | ≤ 1.
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From (2.1), we have
∞
X
σn (p, k, α)
(6.5)
n=1
2(p − α)
|ap+n−1,1 | ≤ 1
and
∞
X
σn (p, k, α)
(6.6)
n=1
2(p − α)
|ap+n−1,2 | ≤ 1.
New Subclasses of
Meromorphic p−Valent
Functions
Therefore, by the Cauchy-Schwarz inequality, we have
∞
q
X
σn (p, k, α)
(6.7)
|ap+n−1,1 | |ap+n−1,2 | ≤ 1.
2(p
−
α)
n=1
B.A. Frasin and
G. Murugusundaramoorthy
Thus it is sufficient to show that
σn (p, k, η)
(6.8)
|ap+n−1,1 | |ap+n−1,2 |
2(p − η)
q
σn (p, k, α)
|ap+n−1,1 | |ap+n−1,2 |,
≤
2(p − α)
that is, that
q
(p − η)σn (p, k, α)
,
(6.9)
|ap+n−1,1 | |ap+n−1,2 | ≤
(p − α)σn (p, k, η)
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(n ≥ 1)
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(n ≥ 1).
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From (6.7), we have
2(p − α)
.
|ap+n−1,1 | |ap+n−1,2 | ≤
σn (p, k, α)
q
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Consequently, we need only to prove that
2(p − α)
(p − η)σn (p, k, α)
≤
,
σn (p, k, α)
(p − α)σn (p, k, η)
(6.10)
Let η ≥
that
(6.11)
1
2
(n ≥ 1).
(k + 1 − p − n), where k ≥ p and p, n ∈ N. It follows from (6.10)
η≤
p[σn (p, k, α)]2 − 4(p − α)2 (p + n − 1)
= Ψ(n).
4(p − α)2 + [σn (p, k, α)]2
Since Ψ(k) is an increasing function of n (n ≥ 1), letting n = 1 in (6.11), we
obtain
(6.12)
p ([p + k + |p + 2α − k|]2 − 4(p − α)2 )
η ≤ Ψ(1) =
,
4(p − α)2 + [p + k + |p + 2α − k|]2
fj (z) =
1
2(p − α) p+n−1
+
z
,
p
z
σn (p, k, α)
B.A. Frasin and
G. Murugusundaramoorthy
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which proves the main assertion of Theorem 6.1.
Finally, by taking the functions
(6.13)
New Subclasses of
Meromorphic p−Valent
Functions
(j = 1, 2)
we can see the result is sharp.
Similarly, and as the above proof, we can prove the following.
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Theorem 6.2. Let each of the functions fj (z) (j = 1, 2) defined by (6.1) be in
the class Λ∗p (α). Then (f1 ∗ f2 )(z) ∈ Λ∗p (ξ), where
(6.14)
1
p (p[p + k + |p + 2α − k|]2 − 4(p − α)2 )
(k + 1 − p − n) ≤ ξ =
,
2
4(p − α)2 + p[p + k + |p + 2α − k|]2
(k ≥ p; p, n ∈ N).
The result is sharp for the functions
(6.15)
2(p − α)
1
fj (z) = p +
z p+n−1 ,
z
(p + n − 1)σn (p, k, α)
(j = 1, 2).
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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References
[1] M.K. AOUF, New criteria for multivalent meromorphic starlike functions
of order alpha, Proc. Japan. Acad. Ser. A. Math. Sci., 69 (1993), 66–70.
[2] M.K. AOUF AND H.M. HOSSEN, New criteria for meromorphic p-valent
starlike functions, Tsukuba J. Math., 17 (1993) 481–486.
[3] M.K. AOUF AND H.M. SRIVASTAVA, A new criteria for meromorphic
p-valent convex functions of order alpha, Math. Sci. Res. Hot-line, 1(8)
(1997), 7–12.
New Subclasses of
Meromorphic p−Valent
Functions
[4] S.B. JOSHI AND H.M. SRIVASTAVA, A certain family of meromorphically multivalent functions, Computers Math. Appl., 38 (1999), 201–211.
B.A. Frasin and
G. Murugusundaramoorthy
[5] S.R. KUKARNI, U.H. NAIK AND H.M. SRIVASTAVA, A certain class
of meromorphically p-valent quasi-convex functions, Pan Amer. Math. J.,
8(1) (1998), 57–64.
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[6] M.L. MOGRA, Meromorphic multivalent functions with positive coefficients I and II, Math. Japon., 35 (1990), 1–11 and 1089–1098.
[7] S. OWA, H.E. DARWISH AND M.K. AOUF, Meromorphic multivalent
functions with positive and fixed second coefficients, Math. Japon., 46
(1997), 231–236.
[8] H.M. SRIVASTAVA AND S. OWA (Eds.), Current Topics in Analytic
Function Theory, World Scientific, Singapore/New Jersey/London/Hong
Kong, (1992).
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[9] B.A. URALEGADDI AND C. SOMANATHA, Certain classes of meromorphic multivalent functions, Tamkang J. Math., 23 (1992), 223–231.
[10] D.G. YANG, On new subclasses of meromorphic p-valent functions, J.
Math. Res. Exposition, 15 (1995) 7–13.
New Subclasses of
Meromorphic p−Valent
Functions
B.A. Frasin and
G. Murugusundaramoorthy
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