Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 3, Article 68, 2005
NEW SUBCLASSES OF MEROMORPHIC p−VALENT FUNCTIONS
B.A. FRASIN AND G. MURUGUSUNDARAMOORTHY
D EPARTMENT OF M ATHEMATICS
A L AL -BAYT U NIVERSITY
P.O. B OX : 130095
M AFRAQ , J ORDAN .
bafrasin@yahoo.com
D EPARTMENT OF M ATHEMATICS ,
V ELLORE I NSTITUTE OF T ECHNOLOGY, D EEMED U NIVERSITY,
V ELLORE , TN-632 014 I NDIA
gmsmoorthy@yahoo.com
Received 20 October, 2004; accepted 02 June, 2005
Communicated by A. Sofo
A BSTRACT. In this paper, we introduce two subclasses Ω∗p (α) and Λ∗p (α) of meromorphic pvalent 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.
Key words and phrases: Meromorphic p−valent functions, Meromorphically starlike and convex functions.
2000 Mathematics Subject Classification. 30C45, 30C50.
1. I NTRODUCTION AND D EFINITIONS
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)
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
202-04
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B.A. F RASIN AND G. M URUGUSUNDARAMOORTHY
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.
2. C OEFFICIENT I NEQUALITIES
Theorem 2.1. Let σn (p, k, α) = (p + n + k − 1) + |p + n + 2α − k − 1|. If f (z) ∈ Σp satisfies
∞
X
(2.1)
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) ∈ 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)
(z ∈ D).
zf 0 (z)
<1
+ (2α − k) f (z)
We note that
P
zf 0 (z)
2p+n−1
+
k
k−p+ ∞
(p
+
n
+
k
−
1)a
z
f (z)
p+n−1
Pn=1
zf 0 (z)
=
∞
2α − k − p + n=1 (p + n + 2α − k − 1)ap+n−1 z 2p+n−1 +
(2α
−
k)
f (z)
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|
P∞
k − p + n=1 (p + n + k − 1) |ap+n−1 |
P
<
.
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
∞
X
1
4(p − α)
(2.2)
f (z) = p +
z p+n−1
z
n(n
+
1)σ
(p,
k,
α)
n
n=1
(p ∈ N)
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
∞
X
(2.3)
(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).
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N EW S UBCLASSES OF M EROMORPHIC p−VALENT F UNCTIONS
3
Example 2.2. The function f (z) given by
∞
X
4(p − α)
1
f (z) = p +
z p+n−1
z
n(n + 1)(p + n − 1)σn (p, k, α)
n=1
(2.4)
belongs to the class Λp (α).
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) = Σ∗K (α) the class of meromorphically convex functions of order α in D.
3. D ISTORTION T HEOREMS
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
1
2(p − α)
−
rp ≤ |f (z)|
p
r
p + k + |p + 2α − k|
1
2(p − α)
≤ p+
rp ,
r
p + k + |p + 2α − k|
(3.1)
and
(3.2)
p
rp+1
−
2p(p − α)
rp−1 ≤ |f 0 (z)|
p + k + |p + 2α − k|
p
2p(p − α)
≤ 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) =
1
2(p − α)
+
zp
p
z
p + k + |p + 2α − k|
(p ∈ N; z ∈ D).
Proof. Since f ∈ Ω∗p (α), from the inequality (2.1), we have
(3.4)
∞
X
|ap+n−1 | ≤
n=1
J. Inequal. Pure and Appl. Math., 6(3) Art. 68, 2005
2(p − α)
.
p + k + |p + 2α − k|
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B.A. F RASIN AND G. M URUGUSUNDARAMOORTHY
Thus, for 0 < |z| = r < 1, and making use of (3.4) we have
∞
1 X
(3.5)
|f (z)| ≤ p +
|ap+n−1 | |z|p+n−1
z
n=1
∞
X
1
p
≤ p +r
|ap+n−1 |
r
n=1
≤
1
2(p − α)
+
rp
p
r
p + k + |p + 2α − k|
and
∞
1 X
|f (z)| ≥ p −
|ap+n−1 | |z|p+n−1
z
(3.6)
≥
1
−r
rp
n=1
∞
X
p
|ap+n−1 |
n=1
1
2(p − α)
≥ p−
rp .
r
p + k + |p + 2α − k|
We also observe that
∞
(3.7)
p + k + |p + 2α − k| X
(p + n − 1) |ap+n−1 | ≤ 2(p − α)
p
n=1
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
0
≤
≤
p
rp+1
p
rp+1
+ rp−1
∞
X
(p + n − 1) |ap+n−1 |
n=1
+
2p(p − α)
rp−1
p + k + |p + 2α − k|
and
∞
(3.9)
|f 0 (z)| ≥
≥
≥
X
p
−
(p + n − 1) |ap+n−1 | |z|p+n−2
p+1
|z|
n=1
p
rp+1
p
rp+1
−r
p−1
∞
X
(p + n − 1) |ap+n−1 |
n=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
J. Inequal. Pure and Appl. Math., 6(3) Art. 68, 2005
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N EW S UBCLASSES OF M EROMORPHIC p−VALENT F UNCTIONS
5
Theorem 3.2. If the function f (z) defined by (1.1) is in the class Λ∗p (α), then for 0 < |z| = r <
1, we have
(3.10)
1
2(p − α)
−
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
(3.11)
p
rp+1
−
2(p − α)
rp−1 ≤ |f 0 (z)|
p + k + |p + 2α − k|
p
2(p − α)
≤ p+1 +
rp−1 .
r
p + k + |p + 2α − k|
The bounds in (3.10) and (3.11) are attained for the functions f (z) given by
(3.12)
g(z) =
2(p − α)
1
+
zp
p
z
p[p + k − 1 + |p + 2α − k|]
(p ∈ N; z ∈ D).
4. R ADII OF S TARLIKENESS AND C ONVEXITY
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)
(p ∈ N).
(4.1)
r1 = 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| < r2 ,
where
1
2p+n−1
p(p − δ)σn (p, k, α)
(4.2)
r2 = inf
(p ∈ N).
n≥1
2[(p + n − 1)[3p + n − 1 − δ](p − α)
The results (4.1) and (4.2) are sharp for the function f (z) given by
f (z) =
(4.3)
1
2(p − α) p+n−1
+
z
p
z
σn (p, k, α)
(p ∈ N; z ∈ D).
Proof. It suffices to prove that
0
zf (z)
≤ p − δ,
+
p
f (z)
(4.4)
for |z| ≤ r1 . We have
0
P∞
zf (z)
n=1 (2p + n − 1)ap+n−1 z p+n−1 P∞
(4.5)
1
f (z) + p = p+n−1
+
a
z
p+n−1
p
n=1
z
P∞
(2p + n − 1) |ap+n−1 | |z|2p+n−1
≤ n=1 P∞
.
1 − n=1 |ap+n−1 | |z|2p+n−1
Hence (4.5) holds true if
(4.6)
∞
X
(2p + n − 1) |ap+n−1 | |z|2p+n−1 ≤ (p − δ) 1 −
n=1
J. Inequal. Pure and Appl. Math., 6(3) Art. 68, 2005
∞
X
!
|ap+n−1 | |z|2p+n−1
,
n=1
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6
B.A. F RASIN AND G. M URUGUSUNDARAMOORTHY
or
∞
X
3p + n − 1 − δ
(4.7)
(p − δ)
n=1
|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)
(n ≥ 1).
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).
In precisely the same manner, we can find the radius of convexity asserted by (4.2), by requiring
that
00
zf (z)
≤ p − δ,
+
p
+
1
(4.10)
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
(4.11)
r3 = 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| < r4 ,
where
1
2p+n−1
p(p − δ)(p + n − 1)σn (p, k, α)
(4.12)
r4 = inf
(p ∈ N).
n≥1
2[(p + n − 1)[3p + n − 1 − δ](p − α)
The results (4.11) and (4.12) are sharp for the function g(z) given by
(4.13)
g(z) =
1
2(p − α)
+
z p+n−1
z p (p + n − 1)σn (p, k, α)
(p ∈ N; z ∈ D).
5. C LOSURE T HEOREMS
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),
j=1
J. Inequal. Pure and Appl. Math., 6(3) Art. 68, 2005
bj ≥ 0 and
m
X
bj = 1).
j=1
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N EW S UBCLASSES OF M EROMORPHIC p−VALENT F UNCTIONS
7
Proof. From (5.2), we can write h(z) as
∞
h(z) =
(5.3)
X
1
+
cp+n−1 z p+n−1 ,
z p n=1
where
cp+n−1 =
(5.4)
m
X
j ∈ {1, 2, . . . , m}.
bj ap+n−1,j ,
j=1
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
bj
j=1
m
X
∞
X
σn (p, k, α)
2(p − α)
n=1
!
|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
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).
Theorem 5.3. Let
fp−1 (z) =
(5.5)
1
zp
(z ∈ D)
and
(5.6)
fp+n−1 (z) =
1
2(p − α) p+n−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
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
∞
X
(5.8)
f (z) =
λp+n−1 fn+p−1 (z)
n=0
=
Since
1
2(p − α)
+
λp+n−1 z p+n−1 .
p
z
σn (p, k, α)
∞
∞
X
X
σn (p, k, α) 2(p − α)
.
λp+n−1 =
λp+n−1 = 1 − λp−1 ≤ 1,
2(p − α) σn (p, k, α)
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).
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8
B.A. F RASIN AND G. M URUGUSUNDARAMOORTHY
Conversely, let us suppose that f (z) ∈ Ω∗p (α). Since
2(p − α)
σn (p, k, α)
|ap+n−1 | ≤
(n ≥ 1),
setting
λp+n−1 =
σn (p, k, α)
|ap+n−1 | ,
2(p − α)
and
λp−1 = 1 −
∞
X
(n ≥ 1)
λp+n−1,
n=1
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
gp−1 (z) =
(5.9)
1
zp
(z ∈ D)
and
gp+n−1 (z) =
(5.10)
1
2(p − α)
+
z p+n−1
p
z
(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
g(z) =
(5.11)
∞
X
λp+n−1 gp+n−1 (z)
n=0
where λp+n−1 ≥ 0, (n ∈ N0 ) and
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.
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
h(z) = tf (z) + (1 − t)g(z),
(5.12)
∗
is also in the class Ω (α) (or in
(0 ≤ t ≤ 1)
Λ∗p (α)).
6. C ONVOLUTION P ROPERTIES
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,
∞
X
1
(6.2)
(f1 ∗ f2 )(z) = p +
ap+n−1,1 ap+n−1,2 z p+n−1 .
z
n=1
Finally, we prove the following.
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N EW S UBCLASSES OF M EROMORPHIC p−VALENT F UNCTIONS
9
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
1
p ([p + k + |p + 2α − k|]2 − 4(p − α)2 )
(k + 1 − p − n) ≤ η =
, (k ≥ p; p, n ∈ N).
2
4(p − α)2 + [p + k + |p + 2α − k|]2
The result is sharp.
(6.3)
Proof. For fj (z) ∈ Ω∗ (α) (j = 1, 2), we need to find the largest η such that
∞
X
σn (p, k, η)
(6.4)
n=1
2(p − η)
|ap+n−1,1 | |ap+n−1,2 | ≤ 1.
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.
Therefore, by the Cauchy-Schwarz inequality, we have
∞
q
X
σn (p, k, α)
|ap+n−1,1 | |ap+n−1,2 | ≤ 1.
(6.7)
2(p
−
α)
n=1
Thus it is sufficient to show that
q
σn (p, k, η)
σn (p, k, α)
(6.8)
|ap+n−1,1 | |ap+n−1,2 | ≤
|ap+n−1,1 | |ap+n−1,2 |,
2(p − η)
2(p − α)
that is, that
q
(p − η)σn (p, k, α)
(6.9)
|ap+n−1,1 | |ap+n−1,2 | ≤
,
(n ≥ 1).
(p − α)σn (p, k, η)
From (6.7), we have
q
2(p − α)
|ap+n−1,1 | |ap+n−1,2 | ≤
.
σn (p, k, α)
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 ≥
(6.11)
1
2
(n ≥ 1)
(n ≥ 1).
(k + 1 − p − n), where k ≥ p and p, n ∈ N. It follows from (6.10) that
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
p ([p + k + |p + 2α − k|]2 − 4(p − α)2 )
,
4(p − α)2 + [p + k + |p + 2α − k|]2
which proves the main assertion of Theorem 6.1.
Finally, by taking the functions
(6.12)
(6.13)
η ≤ Ψ(1) =
fj (z) =
1
2(p − α) p+n−1
+
z
,
p
z
σn (p, k, α)
J. Inequal. Pure and Appl. Math., 6(3) Art. 68, 2005
(j = 1, 2)
http://jipam.vu.edu.au/
10
B.A. F RASIN AND G. M URUGUSUNDARAMOORTHY
we can see the result is sharp.
Similarly, and as the above proof, we can prove the following.
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
p (p[p + k + |p + 2α − k|]2 − 4(p − α)2 )
1
(k +1−p−n) ≤ ξ =
, (k ≥ p; p, n ∈ N).
2
4(p − α)2 + p[p + k + |p + 2α − k|]2
The result is sharp for the functions
2(p − α)
1
(6.15)
fj (z) = p +
z p+n−1 ,
(j = 1, 2).
z
(p + n − 1)σn (p, k, α)
(6.14)
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http://jipam.vu.edu.au/