Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 3(2010), Pages 109-121.
A NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH
POSITIVE COEFFICIENTS
S. KAVITHA, S. SIVASUBRAMANIAN, K. MUTHUNAGAI
Abstract. In the present investigation, the authors define a new class of
meromorphic functions defined in the punctured unit disk Δ∗ := {𝑧 ∈ ℂ : 0 <
∣𝑧∣ < 1}. Coefficient inequalities, growth and distortion inequalities, as well
as closure results are obtained. We also prove a Property using an integral
operator and its inverse defined on the new class.
1. Introduction
Let Σ denote the class of normalized meromorphic functions 𝑓 of the form
∞
𝑓 (𝑧) =
1 ∑
+
𝑎𝑛 𝑧 𝑛 ,
𝑧 𝑛=1
(1.1)
defined on the punctured unit disk
Δ∗ := {𝑧 ∈ ℂ : 0 < ∣𝑧∣ < 1}.
The Hadamard product or convolution of two functions 𝑓 (𝑧) given by (1.1) and
∞
𝑔(𝑧) =
1 ∑
+
𝑔𝑛 𝑧 𝑛
𝑧 𝑛=1
(1.2)
is defined by
∞
(𝑓 ∗ 𝑔)(𝑧) :=
1 ∑
𝑎𝑛 𝑔𝑛 𝑧 𝑛 .
+
𝑧 𝑛=1
A function 𝑓 ∈ Σ is meromorphic starlike of order 𝛼 (0 ≤ 𝛼 < 1) if
( ′ )
𝑧𝑓 (𝑧)
−ℜ
> 𝛼 (𝑧 ∈ Δ := Δ∗ ∪ {0}).
𝑓 (𝑧)
The class of all such functions is denoted by Σ∗ (𝛼). Similarly the class of convex
functions of order 𝛼 is defined. Let Σ𝑃 be the class of functions 𝑓 ∈ Σ with 𝑎𝑛 ≥ 0.
The subclass of Σ𝑃 consisting of starlike functions of order 𝛼 is denoted by Σ∗𝑃 (𝛼).
Now, we define a new class of functions in Definition 1.
2000 Mathematics Subject Classification. 30C50.
Key words and phrases. Meromorphic functions, starlike function, convolution, positive coefficients, coefficient inequalities, integral operator.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted September 14, 2009. Published September 3, 2010 .
109
110
KAVITHA ET AL.
Definition 1. Let 0 ≤ 𝛼 < 1. Further, let 𝑓 (𝑧) ∈ Σ𝑝 be given by (1.1), 0 ≤ 𝜆 < 1
The class 𝑀𝑃 (𝛼, 𝜆) is defined by
{
(
)
}
𝑧𝑓 ′ (𝑧)
𝑀𝑃 (𝛼, 𝜆) = 𝑓 ∈ Σ𝑃 : ℜ
>𝛼 .
(𝜆 − 1)𝑓 (𝑧) + 𝜆𝑧𝑓 ′ (𝑧)
Clearly, 𝑀𝑃 (𝛼, 0) reduces to the class Σ∗𝑃 (𝛼).
The class Σ∗𝑃 (𝛼) and various other subclasses of Σ have been studied rather
extensively by Clunie [4], Nehari and Netanyahu [8], Pommerenke ([9], [10]), Royster
[11], and others (cf., e.g., Bajpai [2] , Mogra et al. [7], Uralegaddi and Ganigi [16],
Cho et al. [3], Aouf [3], and Uralegaddi and Somanatha [15]; see also Duren [[5],
pages 29 and 137], and Srivastava and Owa [[13], pages 86 and 429]) (see also [1]).
In this paper, we obtain the coefficient inequalities, growth and distortion inequalities, as well as closure results for the class 𝑀𝑃 (𝛼, 𝜆). Properties of a certain
integral operator and its inverse defined on the new class 𝑀𝑃 (𝛼, 𝜆) are also discussed.
2. Coefficients Inequalities
Our first theorem gives a necessary and sufficient condition for a function 𝑓 to
be in the class 𝑀𝑃 (𝛼, 𝜆).
Theorem 2.1. Let 𝑓 (𝑧) ∈ Σ𝑃 be given by (1.1). Then 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆) if and only if
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑎𝑛 ≤ 1 − 𝛼.
(2.1)
𝑛=1
Proof. If 𝑓 ∈ 𝑀𝑝 (𝛼, 𝜆), then
∑∞
)
{
}
(
−1 + 𝑛=1 𝑛𝑎𝑛 𝑧 𝑛+1
𝑧𝑓 ′ (𝑧)
∑∞
=ℜ
> 𝛼.
ℜ
(𝜆 − 1)𝑓 (𝑧) + 𝜆𝑧𝑓 ′ (𝑧)
−1 + 𝑛=1 (𝜆 − 1 + 𝜆𝑛)𝑎𝑛 𝑧 𝑛+1
By letting 𝑧 → 1− , we have
{
−1 +
−1 +
∑∞
∑∞
𝑛𝑎𝑛
(𝜆
−
1
+ 𝜆𝑛)𝑎𝑛
𝑛=1
𝑛=1
}
> 𝛼.
This shows that (2.1) holds.
Conversely assume that (2.1) holds. It is sufficient to show that
𝑧𝑓 ′ (𝑧) − {(𝜆 − 1)𝑓 (𝑧) + 𝜆𝑧𝑓 ′ (𝑧)}
𝑧𝑓 ′ (𝑧) + (1 − 2𝛼) {(𝜆 − 1)𝑓 (𝑧) + 𝜆𝑧𝑓 ′ (𝑧)} < 1 (𝑧 ∈ Δ).
we see that
Using (2.1),
𝑧𝑓 ′ (𝑧) − {(𝜆 − 1)𝑓 (𝑧) + 𝜆𝑧𝑓 ′ (𝑧)}
𝑧𝑓 ′ (𝑧) + (1 − 2𝛼) {(𝜆 − 1)𝑓 (𝑧) + 𝜆𝑧𝑓 ′ (𝑧)} ∑∞
(1 − 𝜆)(𝑛 + 1)𝑎𝑛 𝑧 𝑛+1
𝑛=1
∑∞
= −2(1 − 𝛼) + 𝑛=1 [{1 + (1 − 2𝛼)𝜆}𝑛 + (1 − 2𝛼)(𝜆 − 1)] 𝑎𝑛 𝑧 𝑛+1 ∑∞
𝑛=1 (1 − 𝜆)(𝑛 + 1)𝑎𝑛
∑∞
≤
≤ 1.
2(1 − 𝛼) − 𝑛=1 [{1 + (1 − 2𝛼)𝜆}𝑛 + (1 − 2𝛼)(𝜆 − 1)] 𝑎𝑛
Thus we have 𝑓 ∈ 𝑀𝑝 (𝛼, 𝜆).
For the choice of 𝜆 = 0, we get the following.
□
A NEW SUBCLASS OF MEROMORPHIC FUNCTION...
111
Remark 2.2. Let 𝑓 (𝑧) ∈ Σ𝑃 be given by (1.1). Then 𝑓 ∈ Σ∗𝑃 (𝛼) if and only if
∞
∑
(𝑛 + 𝛼)𝑎𝑛 ≤ 1 − 𝛼.
𝑛=1
Our next result gives the coefficient estimates for functions in 𝑀𝑃 (𝛼, 𝜆).
Theorem 2.3. If 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆), then
1−𝛼
𝑎𝑛 ≤
,
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
𝑛 = 1, 2, 3, . . . .
The result is sharp for the functions 𝐹𝑛 (𝑧) given by
1
1−𝛼
𝐹𝑛 (𝑧) = +
𝑧𝑛,
𝑧
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
𝑛 = 1, 2, 3, . . . .
Proof. If 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆), then we have, for each 𝑛,
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑎𝑛 ≤
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑎𝑛 ≤ 1 − 𝛼.
𝑛=1
Therefore we have
𝑎𝑛 ≤
1−𝛼
.
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
Since
1−𝛼
1
+
𝑧𝑛
𝑧
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
satisfies the conditions of Theorem 2.1, 𝐹𝑛 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆) and the equality is attained for this function.
□
𝐹𝑛 (𝑧) =
For 𝜆 = 0, we have the following corollary.
Remark 2.4. If 𝑓 ∈ Σ∗𝑃 (𝛼), then
1−𝛼
𝑎𝑛 ≤
,
𝑛+𝛼
𝑛 = 1, 2, 3, . . . .
Theorem 2.5. If 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆), then
1
1−𝛼
1
1−𝛼
−
𝑟 ≤ ∣𝑓 (𝑧)∣ ≤ +
𝑟
𝑟
1 + 𝛼 − 2𝛼𝜆
𝑟
1 + 𝛼 − 2𝛼𝜆
The result is sharp for
1
1−𝛼
𝑓 (𝑧) = +
𝑧.
𝑧
{1 + 𝛼 − 2𝛼𝜆}
∑∞
Proof. Since 𝑓 (𝑧) = 𝑧1 + 𝑛=1 𝑎𝑛 𝑧 𝑛 , we have
∣𝑓 (𝑧)∣ ≤
∞
∞
∑
1 ∑
1
+
𝑎𝑛 𝑟 𝑛 ≤ + 𝑟
𝑎𝑛 .
𝑟 𝑛=1
𝑟
𝑛=1
Since,
∞
∑
𝑎𝑛 ≤
𝑛=1
1−𝛼
.
1 + 𝛼 − 2𝛼𝜆
Using this, we have
∣𝑓 (𝑧)∣ ≤
1−𝛼
1
+
𝑟.
𝑟
1 + 𝛼 − 2𝛼𝜆
(∣𝑧∣ = 𝑟).
(2.2)
112
KAVITHA ET AL.
Similarly
1
1−𝛼
−
𝑟.
𝑟
1 + 𝛼 − 2𝛼𝜆
∣𝑓 (𝑧)∣ ≥
The result is sharp for 𝑓 (𝑧) =
1
𝑧
+
1−𝛼
1+𝛼−2𝛼𝜆 𝑧.
□
Similarly we have the following:
Theorem 2.6. If 𝑓 ∈ 𝑀𝑃 (𝛼, 𝜆), then
1
1−𝛼
1−𝛼
1
−
≤ ∣𝑓 ′ (𝑧)∣ ≤ 2 +
𝑟2
1 + 𝛼 − 2𝛼𝜆
𝑟
1 + 𝛼 − 2𝛼𝜆
(∣𝑧∣ = 𝑟).
The result is sharp for the function given by (2.2).
(𝛾)
3. Neighborhoods for the class 𝑀𝑝 (𝛼, 𝜆)
(𝛾)
In this section, we determine the neighborhood for the class 𝑀𝑝 (𝛼, 𝜆), which
we define as follows:
(𝛾)
Definition 2. A function 𝑓 ∈ Σ𝑝 is said to be in the class 𝑀𝑝 (𝛼, 𝜆) if there exists
a function 𝑔 ∈ 𝑀𝑝 (𝛼, 𝜆) such that
𝑓 (𝑧)
(𝑧 ∈ Δ, 0 ≤ 𝛾 < 1).
(3.1)
𝑔(𝑧) − 1 < 1 − 𝛾,
Following the earlier works on neighborhoods of analytic functions by Goodman
[6] and Ruscheweyh [14], we define the 𝛿-neighborhood of a function 𝑓 ∈ Σ𝑝 by
{
}
∞
∞
∑
1 ∑
𝑛
𝑁𝛿 (𝑓 ) := 𝑔 ∈ Σ𝑝 : 𝑔(𝑧) = +
𝑏𝑛 𝑧 and
𝑛∣𝑎𝑛 − 𝑏𝑛 ∣ ≤ 𝛿 .
(3.2)
𝑧 𝑛=1
𝑛=1
Theorem 3.1. If 𝑔 ∈ 𝑀𝑝 (𝛼, 𝜆) and
𝛾 =1−
𝛿(1 + 𝛼 − 2𝛼𝜆)
,
2𝛼 − 2𝛼𝜆
(3.3)
then
𝑁𝛿 (𝑔) ⊂ 𝑀𝑝(𝛾) (𝛼, 𝜆).
Proof. Let 𝑓 ∈ 𝑁𝛿 (𝑔). Then we find from (3.2) that
∞
∑
𝑛∣𝑎𝑛 − 𝑏𝑛 ∣ ≤ 𝛿,
(3.4)
𝑛=1
which implies the coefficient inequality
∞
∑
∣𝑎𝑛 − 𝑏𝑛 ∣ ≤ 𝛿,
(𝑛 ∈ ℕ).
(3.5)
𝑛=1
Since 𝑔 ∈ 𝑀𝑝 (𝛼, 𝜆), we have [cf. equation (2.1)]
∞
∑
𝑛=1
𝑏𝑛 ≤
1−𝛼
,
1 + 𝛼 − 2𝛼𝜆
(3.6)
A NEW SUBCLASS OF MEROMORPHIC FUNCTION...
113
so that
∑∞
𝑓 (𝑧)
<
−
1
𝑔(𝑧)
∣𝑎𝑛 − 𝑏𝑛 ∣
𝑛=1∑
∞
1 − 𝑛=1 𝑏𝑛
𝛿(1 + 𝛼 − 2𝛼𝜆)
2𝛼 − 2𝛼𝜆
1 − 𝛾,
=
=
(𝛾)
provided 𝛾 is given by (3.3). Hence, by definition, 𝑓 ∈ 𝑀𝑝 (𝛼, 𝜆) for 𝛾 given by
(3.3), which completes the proof.
□
4. Closure Theorems
Let the functions 𝐹𝑘 (𝑧) be given by
∞
𝐹𝑘 (𝑧) =
1 ∑
𝑓𝑛,𝑘 𝑧 𝑛 ,
+
𝑧 𝑛=1
𝑘 = 1, 2, ..., 𝑚.
(4.1)
We shall prove the following closure theorems for the class 𝑀𝑃 (𝛼, 𝜆).
Theorem 4.1. Let the function 𝐹𝑘 (𝑧) defined by (4.1) be in the class 𝑀𝑃 (𝛼, 𝜆) for
every 𝑘 = 1, 2, ..., 𝑚. Then the function 𝑓 (𝑧) defined by
∞
𝑓 (𝑧) =
1 ∑
𝑎𝑛 𝑧 𝑛
+
𝑧 𝑛=1
belongs to the class 𝑀𝑃 (𝛼, 𝜆), where 𝑎𝑛 =
1
𝑚
(𝑎𝑛 ≥ 0)
∑𝑚
𝑘=1
𝑓𝑛,𝑘 (𝑛 = 1, 2, ..)
Proof. Since 𝐹𝑛 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆), it follows from Theorem 2.1 that
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑓𝑛,𝑘 ≤ 1 − 𝛼
(4.2)
𝑛=1
for every 𝑘 = 1, 2, .., 𝑚. Hence
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑎𝑛
=
𝑛=1
∞
∑
(
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
𝑛=1
=
𝑚
1 ∑
𝑓𝑛,𝑘
𝑚
)
𝑘=1
(∞
)
𝑚
1 ∑ ∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑓𝑛,𝑘
𝑚
𝑛=1
𝑘=1
≤ 1 − 𝛼.
By Theorem 2.1, it follows that 𝑓 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆).
□
Theorem 4.2. The class 𝑀𝑃 (𝛼, 𝜆) is closed under convex linear combination.
Proof. Let the function 𝐹𝑘 (𝑧) given by (4.1) be in the class 𝑀𝑃 (𝛼, 𝜆). Then it is
enough to show that the function
𝐻(𝑧) = 𝜆𝐹1 (𝑧) + (1 − 𝜆)𝐹2 (𝑧) (0 ≤ 𝜆 ≤ 1)
is also in the class 𝑀𝑃 (𝛼, 𝜆). Since for 0 ≤ 𝜆 ≤ 1,
∞
𝐻(𝑧) =
1 ∑
+
[𝜆𝑓𝑛,1 + (1 − 𝜆)𝑓𝑛,2 ]𝑧 𝑛 ,
𝑧 𝑛=1
114
KAVITHA ET AL.
we observe that
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} [𝜆𝑓𝑛,1 + (1 − 𝜆)𝑓𝑛,2 ]
𝑛=1
= 𝜆
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑓𝑛,1 + (1 − 𝜆)
𝑛=1
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑓𝑛,2
𝑛=1
≤ 1 − 𝛼.
By Theorem 2.1, we have 𝐻(𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆).
□
1−𝛼
𝑧 𝑛 for 𝑛 = 1, 2, . . ..
Theorem 4.3. Let 𝐹0 (𝑧) = 𝑧1 and 𝐹𝑛 (𝑧) = 𝑧1 + {𝑛+𝛼−𝛼𝜆(1+𝑛)}
Then
if 𝑓 (𝑧) can be expressed in the form 𝑓 (𝑧) =
∑∞ 𝑓 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆) if and only
∑∞
𝜆
𝐹
(𝑧)
where
𝜆
≥
0
and
𝑛
𝑛=0 𝑛 𝑛
𝑛=0 𝜆𝑛 = 1.
Proof. Let
𝑓 (𝑧) =
∞
∑
𝜆𝑛 𝐹𝑛 (𝑧)
𝑛=0
∞
=
1 ∑
𝜆𝑛 (1 − 𝛼)
+
𝑧𝑛.
𝑧 𝑛=1 {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
Then
∞
∑
𝑛=1
𝜆𝑛
∞
1−𝛼
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} ∑
=
𝜆𝑛 = 1 − 𝜆0 ≤ 1.
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
(1 − 𝛼)
𝑛=1
By Theorem 2.1, we have 𝑓 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆).
Conversely, let 𝑓 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆). From Theorem 2.3, we have
𝑎𝑛 ≤
1−𝛼
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
for
𝑛 = 1, 2, ..
we may take
𝜆𝑛 =
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
𝑎𝑛
1−𝛼
for
𝑛 = 1, 2, ...
and
𝜆0 = 1 −
∞
∑
𝜆𝑛 .
𝑛=1
Then
𝑓 (𝑧) =
∞
∑
𝜆𝑛 𝐹𝑛 (𝑧).
𝑛=0
□
A NEW SUBCLASS OF MEROMORPHIC FUNCTION...
115
5. Partial Sums
Silverman [12] determined sharp lower bounds on the real part of the quotients
between the normalized starlike or convex functions and their sequences of partial
sums. As a natural extension, one is interested to search results analogous to
those of Silverman for meromorphic univalent functions. In this section, motivated
essentially by the work of silverman [12] and Cho and Owa [3] we will investigate
the ratio of a function of the form
∞
1 ∑
𝑓 (𝑧) = +
𝑎𝑛 𝑧 𝑛 ,
(5.1)
𝑧 𝑛=1
to its sequence of partial sums
𝑘
1 ∑
1
and 𝑓𝑘 (𝑧) = +
𝑎𝑛 𝑧 𝑛
𝑓1 (𝑧) =
𝑧
𝑧 𝑛=1
(5.2)
when the coefficients are sufficiently small to satisfy the condition analogous to
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑎𝑛 ≤ 1 − 𝛼.
𝑛=1
For the sake of brevity we rewrite it as
∞
∑
𝑑𝑛 ∣𝑎𝑛 ∣ ≤ 1 − 𝛼,
(5.3)
𝑛=1
where
𝑑𝑛 := 𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)
(5.4)
More precisely we will determine sharp lower bounds for ℜ{𝑓 (𝑧)/𝑓
} ℜ{𝑓𝑘 (𝑧)/𝑓 (𝑧)}.
{ 𝑘 (𝑧)} and
1+𝑤(𝑧)
> 0 (𝑧 ∈
In this connection we make use of the well known results thatℜ 1−𝑤(𝑧)
∞
∑
Δ) if and only if 𝜔(𝑧) =
𝑐𝑛 𝑧 𝑛 satisfies the inequality ∣𝜔(𝑧)∣ ≤ ∣𝑧∣. Unless other𝑛=1
wise stated, we will assume that 𝑓 is of the form (1.1) and its sequence of partial
∑𝑘
sums is denoted by 𝑓𝑘 (𝑧) = 𝑧1 + 𝑛=1 𝑎𝑛 𝑧 𝑛 .
Theorem 5.1. Let 𝑓 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆) be given by (5.1)satisfies condition, (2.1)
}
{
𝑑𝑘+1 (𝜆, 𝛼) − 1 + 𝛼
𝑓 (𝑧)
≥
(𝑧 ∈ 𝑈 )
(5.5)
Re
𝑓𝑘 (𝑧)
𝑑𝑘+1 (𝜆, 𝛼)
where
{
𝑑𝑛 (𝜆, 𝛼) ≥
1 − 𝛼,
𝑑𝑘+1 (𝜆, 𝛼),
𝑖𝑓 𝑛 = 1, 2, 3, . . . , 𝑘
𝑖𝑓 𝑛 = 𝑘 + 1, 𝑘 + 2, . . . .
(5.6)
The result (5.5) is sharp with the function given by
𝑓 (𝑧) =
1
1−𝛼
+
𝑧 𝑘+1 .
𝑧
𝑑𝑘+1 (𝜆, 𝛼)
Proof. Define the function 𝑤(𝑧) by
[
]
1 + 𝑤(𝑧)
𝑑𝑘+1 (𝜆, 𝛼) 𝑓 (𝑧)
𝑑𝑘+1 (𝜆, 𝛼) − 1 + 𝛼
=
−
1 − 𝑤(𝑧)
1−𝛼
𝑓𝑘 (𝑧)
𝑑𝑘+1 (𝜆, 𝛼)
(5.7)
116
KAVITHA ET AL.
1+
𝑘
∑
𝑎𝑛 𝑧 𝑛+1 +
𝑛=1
=
1+
(
𝑑𝑘+1 (𝜆,𝛼)
1−𝛼
𝑘
∑
𝑎𝑛
) ∑
∞
𝑎𝑛 𝑧 𝑛+1
𝑛=𝑘+1
(5.8)
𝑧 𝑛+1
𝑛=1
It suffices to show that ∣𝑤(𝑧)∣ ≤ 1. Now, from (5.8) we can write
(
𝑤(𝑧) =
2+2
𝑘
∑
𝑑𝑘+1 (𝜆,𝛼)
1−𝛼
𝑎𝑛
𝑧 𝑛+1
+
) ∑
∞
𝑎𝑛 𝑧 𝑛+1
𝑛=𝑘+1
(
𝑛=1
𝑑𝑘+1 (𝜆,𝛼)
1−𝛼
.
) ∑
∞
𝑎𝑛
𝑧 𝑛+1
𝑘=𝑛+1
Hence we obtain
(
∣𝑤(𝑧)∣ ≤
2−2
𝑘
∑
𝑑𝑘+1 (𝜆,𝛼)
1−𝛼
∣𝑎𝑛 ∣ −
(
𝑛=1
) ∑
∞
∣𝑎𝑛 ∣
𝑘=𝑛+1
𝑑𝑘+1 (𝜆,𝛼)
1−𝛼
) ∑
∞
∣𝑎𝑛 ∣
𝑛=𝑘+1
Now ∣𝑤(𝑧)∣ ≤ 1 if
(
2
𝑑𝑘+1 (𝜆, 𝛼)
1−𝛼
) ∑
∞
∣𝑎𝑛 ∣ ≤ 2 − 2
𝑘
∑
∣𝑎𝑛 ∣
𝑛=1
𝑛=𝑘+1
or, equivalently,
𝑘
∑
∣𝑎𝑛 ∣ +
𝑛=1
∞
𝑑𝑘+1 (𝜆, 𝛼) ∑
∣𝑎𝑛 ∣ ≤ 1.
1−𝛼
𝑛=𝑘+1
From the condition (2.1), it is sufficient to show that
∞
∞
∑
𝑑𝑘+1 (𝜆, 𝛼) ∑
𝑑𝑛 (𝜆, 𝛼)
∣𝑎𝑛 ∣ +
∣𝑎𝑛 ∣ ≤
∣𝑎𝑛 ∣
1
−
𝛼
1−𝛼
𝑛=1
𝑛=1
𝑘
∑
𝑛=𝑘+1
which is equivalent to
)
𝑘 (
∑
𝑑𝑛 (𝜆, 𝛼) − 1 + 𝛼
𝑛=1
+
1−𝛼
∣𝑎𝑛 ∣
)
∞ (
∑
𝑑𝑛 (𝜆, 𝛼) − 𝑑𝑘+1 (𝜆, 𝛼)
∣𝑎𝑛 ∣
1−𝛼
𝑛=𝑘+1
≥ 0
(5.9)
To see that the function given by (5.7) gives the sharp result, we observe that for
𝑧 = 𝑟𝑒𝑖𝜋/𝑘
𝑓 (𝑧)
1−𝛼
1−𝛼
= 1+
𝑧𝑛 → 1 −
𝑓𝑘 (𝑧)
𝑑𝑘+1 (𝜆, 𝛼)
𝑑𝑘+1 (𝜆, 𝛼)
𝑑𝑘+1 (𝜆, 𝛼) − 1 + 𝛼
when 𝑟 → 1− .
=
𝑑𝑘+1 (𝜆, 𝛼)
A NEW SUBCLASS OF MEROMORPHIC FUNCTION...
which shows the bound (5.5) is the best possible for each 𝑘 ∈ ℕ.
117
□
The proof of the next theorem is much akin to that of the earlier theorem and
hence we state the theorem without proof.
Theorem 5.2. Let 𝑓 (𝑧) ∈ 𝑀𝑃 (𝛼, 𝜆) be given by (5.1)satisfies condition, (2.1)
{
}
𝑓𝑘 (𝑧)
𝑑𝑘+1 (𝜆, 𝛼)
Re
≥
(𝑧 ∈ 𝑈 )
(5.10)
𝑓 (𝑧)
𝑑𝑘+1 (𝜆, 𝛼) + 1 − 𝛼
where
𝑑𝑘+1 (𝜆, 𝛼) ≥ 1 − 𝛼
{
1 − 𝛼,
𝑖𝑓 𝑛 = 1, 2, 3, . . . , 𝑘
𝑑𝑛 (𝜆, 𝛼) ≥
(5.11)
𝑑𝑘+1 (𝜆, 𝛼),
𝑖𝑓 𝑛 = 𝑘 + 1, 𝑘 + 2, . . . .
The result (5.10) is sharp with the function given by
𝑓 (𝑧) =
1−𝛼
1
+
𝑧 𝑘+1 .
𝑧
𝑑𝑘+1 (𝜆, 𝛼)
(5.12)
6. Radius of meromorphic starlikeness and meromorphic convexity
The radii of starlikeness and convexity for the class are given by the following
theorems for the class 𝑀𝑃 (𝛼, 𝜆).
Theorem 6.1. Let the function 𝑓 be in the class 𝑀𝑃 (𝛼, 𝜆). Then 𝑓 is meromorphically starlike of order 𝜌(0 ≤ 𝜌 < 1), in ∣𝑧∣ < 𝑟1 (𝛼, 𝜆, 𝜌), where
1
[
] 𝑛+1
(1 − 𝜌)(1 − 𝛼)
𝑟1 (𝛼, 𝜆, 𝜌) = inf
,
(6.1)
𝑛≥1 (𝑛 + 2 − 𝜌) {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
Proof. Since,
∞
𝑓 (𝑧) =
1 ∑
+
𝑎𝑛 𝑧 𝑛 ,
𝑧 𝑛=1
we get
∞
𝑓 ′ (𝑧) = −
∑
1
𝑛𝑎𝑛 𝑧 𝑛−1 .
+
2
𝑧
𝑛=1
It is sufficient to show that
𝑧𝑓 ′ (𝑧)
−
𝑓 (𝑧) − 1 ≤ 1 − 𝜌
or equivalently
or
∞
∑
𝑛
(𝑛 + 1)𝑎𝑛 𝑧 ′
𝑧𝑓 (𝑧)
𝑛=1
≤1−𝜌
∞
𝑓 (𝑧) + 1 = 1 ∑
−
𝑎𝑛 𝑧 𝑛 𝑧
𝑛=1
)
∞ (
∑
𝑛+2−𝜌
𝑛=1
1−𝜌
𝑎𝑛 ∣𝑧∣𝑛+1 ≤ 1,
for 0 ≤ 𝜌 < 1, and ∣𝑧∣ < 𝑟1 (𝛼, 𝜆, 𝜌). By Theorem 2.1, (6.2) will be true if
(
)
𝑛+2−𝜌
1−𝛼
∣𝑧∣𝑛+1 ≤
1−𝜌
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
(6.2)
118
KAVITHA ET AL.
or, if
[
(1 − 𝜌)(1 − 𝛼)
(𝑛 + 2 − 𝜌) {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
This completes the proof of Theorem 6.1.
1
] 𝑛+1
∣𝑧∣ ≤
,
𝑛 ≥ 1.
(6.3)
□
Theorem 6.2. Let the function 𝑓 in the class 𝑀𝑃 (𝛼, 𝜆). Then 𝑓 is meromorphically
convex of order 𝜌, (0 ≤ 𝜌 < 1), in ∣𝑧∣ < 𝑟2 (𝛼, 𝜆, 𝜌), where
1
[
] 𝑛+1
(1 − 𝜌)(1 − 𝛼)
,
𝑛 ≧ 1, (6.4)
𝑟2 (𝛼, 𝜆, 𝜌) = inf
𝑛≥1 𝑛 (𝑛 + 2 − 𝜌) {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
Proof. Since,
∞
𝑓 (𝑧) =
we get
1 ∑
−
𝑎𝑛 𝑧 𝑛 ,
𝑧 𝑛=1
∞
𝑓 ′ (𝑧) = −
∑
1
−
𝑛𝑎𝑛 𝑧 𝑛−1 .
2
𝑧
𝑛=1
It is sufficient to show that
′′
−1 − 𝑧𝑓 (𝑧) − 1 ≤ 1 − 𝜌 or equivalently
′
𝑓 (𝑧)
∞
∑
𝑛−1 𝑛(𝑛 + 1)𝑎𝑛 𝑧
′′
𝑧𝑓 (𝑧)
𝑛=1
≤ 1 − 𝜌 or
∞
𝑓 ′ (𝑧) + 2 = ∑
1
𝑛−1 𝑛𝑎𝑛 𝑧
− 2 −
𝑧
𝑛=1
(
)
∞
∑
𝑛(𝑛 + 2 − 𝜌)
𝑎𝑛 ∣𝑧∣𝑛+1 ≤ 1,
1
−
𝜌
𝑛=1
for 0 ≤ 𝜌 < 1, and ∣𝑧∣ < 𝑟2 (𝛼, 𝜆, 𝜌). By Theorem 2.1, (6.5) will be true if
(
)
𝑛(𝑛 + 2 − 𝜌)
(1 − 𝛼)
∣𝑧∣𝑛+1 ≤
1−𝜌
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
or, if
1
[
] 𝑛+1
(1 − 𝜌)(1 − 𝛼)
∣𝑧∣ ≤
, 𝑛 ≥ 1.
𝑛(𝑛 + 2 − 𝜌) {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
This completes the proof of Theorem 6.2.
(6.5)
(6.6)
□
7. Integral Operators
In this section, we consider integral transforms of functions in the class 𝑀𝑝 (𝛼, 𝜆).
Theorem 7.1. Let the function 𝑓 (𝑧) given by (1) be in 𝑀𝑝 (𝛼, 𝜆). Then the integral
operator
∫ 1
𝐹 (𝑧) = 𝑐
𝑢𝑐 𝑓 (𝑢𝑧)𝑑𝑢
(0 < 𝑢 ≤ 1, 0 < 𝑐 < ∞)
0
is in 𝑀𝑝 (𝛿, 𝜆), where
𝛿=
(𝑐 + 2) {1 + 𝛼 − 2𝛼𝜆} − 𝑐(1 − 𝛼)
.
𝑐(1 − 𝛼) {1 − 2𝜆} + (1 + 𝛼) {1 − 2𝜆} (𝑐 + 2)
A NEW SUBCLASS OF MEROMORPHIC FUNCTION...
The result is sharp for the function 𝑓 (𝑧) =
1
𝑧
+
119
1−𝛼
{1+𝛼−2𝛼𝜆} 𝑧.
Proof. Let 𝑓 (𝑧) ∈ 𝑀𝑝 (𝛼, 𝜆). Then
∫ 1
𝑢𝑐 𝑓 (𝑢𝑧)𝑑𝑢
𝐹 (𝑧) = 𝑐
0
)
∫ 1 ( 𝑐−1 ∑
∞
𝑢
= 𝑐
+
𝑓𝑛 𝑢𝑛+𝑐 𝑧 𝑛 𝑑𝑢
𝑧
0
𝑛=1
∞
=
1 ∑
𝑐
+
𝑓𝑛 𝑧 𝑛 .
𝑧 𝑛=1 𝑐 + 𝑛 + 1
It is sufficient to show that
∞
∑
𝑐 {𝑛 + 𝛿 − 𝛿𝜆(1 + 𝑛)}
𝑎𝑛 ≤ 1.
(𝑐 + 𝑛 + 1)(1 − 𝛿)
𝑛=1
(7.1)
Since 𝑓 ∈ 𝑀𝑝 (𝛼, 𝜆), we have
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
𝑎𝑛 ≤ 1.
(1 − 𝛼)
𝑛=1
Note that (7.1) is satisfied if
𝑐 {𝑛 + 𝛿 − 𝛿𝜆(1 + 𝑛)}
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
≤
.
(𝑐 + 𝑛 + 1)(1 − 𝛿)
(1 − 𝛼)
Rewriting the inequality, we have
𝑐 {𝑛 + 𝛿 − 𝛿𝜆(1 + 𝑛)} (1 − 𝛼) ≤ (𝑐 + 𝑛 + 1)(1 − 𝛿) {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} .
Solving for 𝛿, we have
𝛿≤
(𝑐 + 𝑛 + 1) {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} − 𝑐𝑛(1 − 𝛼)
= 𝐹 (𝑛).
𝑐(1 − 𝛼) {1 − 𝜆(1 + 𝑛)} + {(𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} (𝑐 + 𝑛 + 1)
A simple computation will show that 𝐹 (𝑛) is increasing and 𝐹 (𝑛) ≥ 𝐹 (1). Using
this, the results follows.
□
For the choice of 𝜆 = 0, we have the following result of Uralegaddi and Ganigi
[15].
Remark 7.2. Let the function 𝑓 (𝑧) defined by (1) be in Σ∗𝑝 (𝛼). Then the integral
operator
∫ 1
𝐹 (𝑧) = 𝑐
𝑢𝑐 𝑓 (𝑢𝑧)𝑑𝑢
(0 < 𝑢 ≤ 1, 0 < 𝑐 < ∞)
0
is in
Σ∗𝑝 (𝛿),
where 𝛿 =
1+𝛼+𝑐𝛼
1+𝛼+𝑐 .
The result is sharp for the function
𝑓 (𝑧) =
Also we have the following:
1 1−𝛼
+
𝑧.
𝑧
1+𝛼
120
KAVITHA ET AL.
Theorem 7.3. Let 𝑓 (𝑧), given by (1), be in 𝑀𝑝 (𝛼, 𝜆),
∞
𝐹 (𝑧) =
1
1 ∑ 𝑐+𝑛+1
[(𝑐 + 1)𝑓 (𝑧) + 𝑧𝑓 ′ (𝑧)] = +
𝑓𝑛 𝑧 𝑛 ,
𝑐
𝑧 𝑛=1
𝑐
𝑐 > 0.
(7.2)
Then 𝐹 (𝑧) is in 𝑀𝑝 (𝛼, 𝜆) for ∣𝑧∣ ≤ 𝑟(𝛼, 𝜆, 𝛽) where
(
𝑟(𝛼, 𝜆, 𝛽) = inf
𝑛
𝑐(1 − 𝛽) {𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)}
(1 − 𝛼)(𝑐 + 𝑛 + 1) {𝑛 + 𝛽 − 𝛽𝜆(1 + 𝑛)}
The result is sharp for the function 𝑓𝑛 (𝑧) =
Proof. Let 𝑤 =
𝑧𝑓 ′ (𝑧)
(𝜆−1)𝑓 (𝑧)+𝜆𝑧𝑓 ′ (𝑧) .
1
𝑧
)1/(𝑛+1)
,
1−𝛼
+ {𝑛+𝛼−𝛼𝜆(1+𝑛)}
𝑧𝑛,
𝑛 = 1, 2, 3, . . . .
𝑛 = 1, 2, 3, . . . .
Then it is sufficient to show that
𝑤−1 𝑤 + 1 − 2𝛽 < 1.
A computation shows that this is satisfied if
∞
∑
{𝑛 + 𝛽 − 𝛽𝜆(1 + 𝑛)} (𝑐 + 𝑛 + 1)
𝑎𝑛 ∣𝑧∣𝑛+1 ≤ 1.
(1
−
𝛽)𝑐
𝑛=1
(7.3)
Since 𝑓 ∈ 𝑀𝑝 (𝛼, 𝜆), by Theorem 2.1, we have
∞
∑
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑎𝑛 ≤ 1 − 𝛼.
𝑛=1
The equation (7.3) is satisfied if
{𝑛 + 𝛽 − 𝛽𝜆(1 + 𝑛)} (𝑐 + 𝑛 + 1)
{𝑛 + 𝛼 − 𝛼𝜆(1 + 𝑛)} 𝑎𝑛
𝑎𝑛 ∣𝑧∣𝑛+1 ≤
.
(1 − 𝛽)𝑐
1−𝛼
Solving for ∣𝑧∣, we get the result.
□
For the choice of 𝜆 = 0, we have the following result of Uralegaddi and Ganigi
[15].
Remark 7.4. Let the function 𝑓 (𝑧) defined by (1) be in Σ∗𝑝 (𝛼) and 𝐹 (𝑧) given by
(7.2). Then 𝐹 (𝑧) is in Σ∗𝑝 (𝛼) for ∣𝑧∣ ≤ 𝑟(𝛼, 𝛽) where
(
𝑟(𝛼, 𝛽) = inf
𝑛
𝑐(1 − 𝛽)(𝑛 + 𝛼)
(1 − 𝛼)(𝑐 + 𝑛 + 1)(𝑛 + 𝛽)
The result is sharp for the function 𝑓𝑛 (𝑧) =
1
𝑧
)1/(𝑛+1)
+
,
1−𝛼 𝑛
𝑛+𝛼 𝑧 ,
𝑛 = 1, 2, 3, . . . .
𝑛 = 1, 2, 3, . . . .
Acknowledgments. The authors thank Prof.G. Murugusundaramoorthy for his
suggestions and comments on the original manuscript( in particular on the partial
sums chapter). The authors also would like to thank the anonymous referee for
his/her comments.
A NEW SUBCLASS OF MEROMORPHIC FUNCTION...
121
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S.Kavitha
Department of Mathematics, Madras Christian College
East Tambaram, Chennai-600 059, India
E-mail address: kavithass19@rediffmail.com
S. Sivasubramanian
Department of Mathematics, University College of Engineering, Tindivanam
Anna University Chennai, Saram-604 307,
India
E-mail address: sivasaisastha@rediffmail.com
K. Muthunagai
Department of Mathematics, Sri Sairam Engineering College
Tambaram, Chennai-44, India
E-mail address: muthunagaik@yahoo.com