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 functions𝑓 of the form
𝑓(𝑧) = 1 𝑧+
∞
∑
𝑛=1
𝑎𝑛𝑧𝑛, (1.1)
defined on the punctured unit disk
Δ∗:={𝑧∈ℂ: 0<∣𝑧∣<1}.
The Hadamard product or convolution of two functions𝑓(𝑧) given by (1.1) and
𝑔(𝑧) = 1 𝑧 +
∞
∑
𝑛=1
𝑔𝑛𝑧𝑛 (1.2)
is defined by
(𝑓∗𝑔)(𝑧) := 1 𝑧 +
∞
∑
𝑛=1
𝑎𝑛𝑔𝑛𝑧𝑛.
A function𝑓 ∈Σ ismeromorphic starlike of order𝛼(0≤𝛼 <1) if
−ℜ
(𝑧𝑓′(𝑧)
𝑓(𝑧) )
> 𝛼 (𝑧∈Δ := Δ∗∪ {0}).
The class of all such functions is denoted by Σ∗(𝛼). Similarly the class of convex
functions of order𝛼is defined. Let Σ𝑃 be the class of functions𝑓 ∈Σ with𝑎𝑛 ≥0.
The subclass of Σ𝑃 consisting of starlike functions of order𝛼is denoted by Σ∗𝑃(𝛼).
Now, we define a new class of functions in Definition 1.
2000Mathematics Subject Classification. 30C50.
Key words and phrases. Meromorphic functions, starlike function, convolution, positive coef-ficients, coefficient inequalities, integral operator.
c
⃝2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted September 14, 2009. Published September 3, 2010 .
Definition 1. Let 0≤𝛼 <1. Further, let𝑓(𝑧)∈Σ𝑝 be given by (1.1), 0≤𝜆 <1
The class𝑀𝑃(𝛼, 𝜆)is defined by
𝑀𝑃(𝛼, 𝜆) = {
𝑓 ∈Σ𝑃 :ℜ
( 𝑧𝑓′(𝑧)
(𝜆−1)𝑓(𝑧) +𝜆𝑧𝑓′(𝑧)
) > 𝛼
} .
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 in-equalities, as well as closure results for the class𝑀𝑃(𝛼, 𝜆). Properties of a certain
integral operator and its inverse defined on the new class 𝑀𝑃(𝛼, 𝜆) are also
dis-cussed.
2. Coefficients Inequalities
Our first theorem gives a necessary and sufficient condition for a function𝑓 to be in the class𝑀𝑃(𝛼, 𝜆).
Theorem 2.1. Let 𝑓(𝑧)∈Σ𝑃 be given by (1.1). Then𝑓 ∈𝑀𝑃(𝛼, 𝜆)if and only if
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)} 𝑎𝑛≤1−𝛼. (2.1)
Proof. If𝑓 ∈𝑀𝑝(𝛼, 𝜆),then
ℜ
( 𝑧𝑓′(𝑧)
(𝜆−1)𝑓(𝑧) +𝜆𝑧𝑓′(𝑧)
) =ℜ
{ −1 +∑∞
𝑛=1𝑛𝑎𝑛𝑧𝑛+1
−1 +∑∞
𝑛=1(𝜆−1 +𝜆𝑛)𝑎𝑛𝑧𝑛+1 }
> 𝛼.
By letting𝑧→1−, we have
{ −1 +∑∞ 𝑛=1𝑛𝑎𝑛
−1 +∑∞
𝑛=1(𝜆−1 +𝜆𝑛)𝑎𝑛 }
> 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).
Using (2.1), we see that
𝑧𝑓′(𝑧)− {(𝜆−1)𝑓(𝑧) +𝜆𝑧𝑓′(𝑧)}
𝑧𝑓′(𝑧) + (1−2𝛼){(𝜆−1)𝑓(𝑧) +𝜆𝑧𝑓′(𝑧)}
=
∑∞
𝑛=1(1−𝜆)(𝑛+ 1)𝑎𝑛𝑧 𝑛+1
−2(1−𝛼) +∑∞
𝑛=1[{1 + (1−2𝛼)𝜆}𝑛+ (1−2𝛼)(𝜆−1)]𝑎𝑛𝑧𝑛+1
≤
∑∞
𝑛=1(1−𝜆)(𝑛+ 1)𝑎𝑛
2(1−𝛼)−∑∞
𝑛=1[{1 + (1−2𝛼)𝜆}𝑛+ (1−2𝛼)(𝜆−1)]𝑎𝑛
≤1.
Thus we have𝑓 ∈𝑀𝑝(𝛼, 𝜆). □
Remark 2.2. Let𝑓(𝑧)∈Σ𝑃 be given by (1.1). Then𝑓 ∈Σ∗𝑃(𝛼) if and only if ∞
∑
𝑛=1
(𝑛+𝛼)𝑎𝑛≤1−𝛼.
Our next result gives the coefficient estimates for functions in𝑀𝑃(𝛼, 𝜆).
Theorem 2.3. If 𝑓 ∈𝑀𝑃(𝛼, 𝜆), then
𝑎𝑛≤
1−𝛼
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}, 𝑛= 1,2,3, . . . .
The result is sharp for the functions 𝐹𝑛(𝑧)given by
𝐹𝑛(𝑧) =
1 𝑧 +
1−𝛼
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑧
𝑛, 𝑛= 1,2,3, . . . .
Proof. If𝑓 ∈𝑀𝑃(𝛼, 𝜆), then we have, for each𝑛,
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑎𝑛≤ ∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑎𝑛≤1−𝛼.
Therefore we have
𝑎𝑛≤
1−𝛼
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}.
Since
𝐹𝑛(𝑧) =
1 𝑧 +
1−𝛼
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑧
𝑛
satisfies the conditions of Theorem 2.1, 𝐹𝑛(𝑧)∈𝑀𝑃(𝛼, 𝜆) and the equality is
at-tained for this function. □
For𝜆= 0,we have the following corollary.
Remark 2.4. If𝑓 ∈Σ∗𝑃(𝛼), then
𝑎𝑛 ≤
1−𝛼
𝑛+𝛼, 𝑛= 1,2,3, . . . .
Theorem 2.5. If 𝑓 ∈𝑀𝑃(𝛼, 𝜆), then
1 𝑟 −
1−𝛼
1 +𝛼−2𝛼𝜆𝑟≤ ∣𝑓(𝑧)∣ ≤ 1 𝑟 +
1−𝛼
1 +𝛼−2𝛼𝜆𝑟 (∣𝑧∣=𝑟).
The result is sharp for
𝑓(𝑧) =1 𝑧 +
1−𝛼
{1 +𝛼−2𝛼𝜆}𝑧. (2.2)
Proof. Since𝑓(𝑧) = 1𝑧+∑∞
𝑛=1𝑎𝑛𝑧𝑛, we have
∣𝑓(𝑧)∣ ≤ 1
𝑟 +
∞
∑
𝑛=1
𝑎𝑛𝑟𝑛≤
1 𝑟+𝑟
∞
∑
𝑛=1
𝑎𝑛.
Since,
∞
∑
𝑛=1
𝑎𝑛≤
1−𝛼 1 +𝛼−2𝛼𝜆.
Using this, we have
∣𝑓(𝑧)∣ ≤ 1
𝑟+
Similarly
∣𝑓(𝑧)∣ ≥ 1
𝑟−
1−𝛼 1 +𝛼−2𝛼𝜆𝑟.
The result is sharp for𝑓(𝑧) = 1𝑧+1+𝛼1−−𝛼2𝛼𝜆𝑧. □
Similarly we have the following:
Theorem 2.6. If 𝑓 ∈𝑀𝑃(𝛼, 𝜆), then
1 𝑟2 −
1−𝛼
1 +𝛼−2𝛼𝜆 ≤ ∣𝑓
′(𝑧)∣ ≤ 1
𝑟2 +
1−𝛼
1 +𝛼−2𝛼𝜆 (∣𝑧∣=𝑟).
The result is sharp for the function given by (2.2).
3. Neighborhoods for the class 𝑀𝑝(𝛾)(𝛼, 𝜆)
In this section, we determine the neighborhood for the class 𝑀𝑝(𝛾)(𝛼, 𝜆), which
we define as follows:
Definition 2. A function𝑓 ∈Σ𝑝is said to be in the class𝑀 (𝛾)
𝑝 (𝛼, 𝜆)if there exists
a function 𝑔∈𝑀𝑝(𝛼, 𝜆)such that
𝑓(𝑧) 𝑔(𝑧) −1
<1−𝛾, (𝑧∈Δ,0≤𝛾 <1). (3.1)
Following the earlier works on neighborhoods of analytic functions by Goodman [6] and Ruscheweyh [14], we define the𝛿-neighborhood of a function𝑓 ∈Σ𝑝 by
𝑁𝛿(𝑓) := {
𝑔∈Σ𝑝 : 𝑔(𝑧) =
1 𝑧+
∞
∑
𝑛=1
𝑏𝑛𝑧𝑛 and ∞
∑
𝑛=1
𝑛∣𝑎𝑛−𝑏𝑛∣ ≤𝛿 }
. (3.2)
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. Let𝑓 ∈𝑁𝛿(𝑔). Then we find from (3.2) that
∞
∑
𝑛=1
𝑛∣𝑎𝑛−𝑏𝑛∣ ≤𝛿, (3.4)
which implies the coefficient inequality
∞
∑
𝑛=1
∣𝑎𝑛−𝑏𝑛∣ ≤𝛿, (𝑛∈ℕ). (3.5)
Since𝑔∈𝑀𝑝(𝛼, 𝜆), we have [cf. equation (2.1)]
∞
∑
𝑛=1
𝑏𝑛 ≤
1−𝛼
so that
𝑓(𝑧) 𝑔(𝑧)−1
< ∑∞
𝑛=1∣𝑎𝑛−𝑏𝑛∣
1−∑∞ 𝑛=1𝑏𝑛
= 𝛿(1 +𝛼−2𝛼𝜆) 2𝛼−2𝛼𝜆 = 1−𝛾,
provided 𝛾 is given by (3.3). Hence, by definition, 𝑓 ∈𝑀𝑝(𝛾)(𝛼, 𝜆) for 𝛾 given by
(3.3), which completes the proof. □
4. Closure Theorems
Let the functions𝐹𝑘(𝑧) be given by
𝐹𝑘(𝑧) =
1 𝑧+
∞
∑
𝑛=1
𝑓𝑛,𝑘𝑧𝑛, 𝑘= 1,2, ..., 𝑚. (4.1)
We shall prove the following closure theorems for the class𝑀𝑃(𝛼, 𝜆).
Theorem 4.1. Let the function𝐹𝑘(𝑧)defined by (4.1) be in the class𝑀𝑃(𝛼, 𝜆)for
every 𝑘= 1,2, ..., 𝑚. Then the function𝑓(𝑧)defined by
𝑓(𝑧) = 1 𝑧+
∞
∑
𝑛=1
𝑎𝑛𝑧𝑛 (𝑎𝑛≥0)
belongs to the class 𝑀𝑃(𝛼, 𝜆),where 𝑎𝑛=𝑚1 ∑
𝑚
𝑘=1𝑓𝑛,𝑘 (𝑛= 1,2, ..)
Proof. Since𝐹𝑛(𝑧)∈𝑀𝑃(𝛼, 𝜆), it follows from Theorem 2.1 that
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑓𝑛,𝑘 ≤1−𝛼 (4.2)
for every𝑘= 1,2, .., 𝑚.Hence
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑎𝑛 = ∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}
( 1 𝑚
𝑚 ∑
𝑘=1
𝑓𝑛,𝑘 )
= 1 𝑚
𝑚 ∑
𝑘=1 (∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑓𝑛,𝑘 )
≤ 1−𝛼.
By Theorem 2.1, it follows that𝑓(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 class𝑀𝑃(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 class𝑀𝑃(𝛼, 𝜆). Since for 0≤𝜆≤1,
𝐻(𝑧) = 1 𝑧+
∞
∑
𝑛=1
we observe that
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}[𝜆𝑓𝑛,1+ (1−𝜆)𝑓𝑛,2]
= 𝜆
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑓𝑛,1+ (1−𝜆) ∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑓𝑛,2
≤ 1−𝛼.
By Theorem 2.1, we have𝐻(𝑧)∈𝑀𝑃(𝛼, 𝜆). □
Theorem 4.3. Let 𝐹0(𝑧) =1𝑧 and𝐹𝑛(𝑧) =1𝑧 +{𝑛+𝛼−1𝛼𝜆(1+𝑛)−𝛼 }𝑧𝑛 for𝑛= 1,2, . . .. Then 𝑓(𝑧) ∈ 𝑀𝑃(𝛼, 𝜆) if and only if 𝑓(𝑧) can be expressed in the form 𝑓(𝑧) = ∑∞
𝑛=0𝜆𝑛𝐹𝑛(𝑧)where𝜆𝑛≥0 and∑∞𝑛=0𝜆𝑛= 1.
Proof. Let
𝑓(𝑧) =
∞
∑
𝑛=0
𝜆𝑛𝐹𝑛(𝑧)
=1 𝑧 +
∞
∑
𝑛=1
𝜆𝑛(1−𝛼)
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑧
𝑛.
Then
∞
∑
𝑛=1
𝜆𝑛
1−𝛼
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}
(1−𝛼) =
∞
∑
𝑛=1
𝜆𝑛= 1−𝜆0≤1.
By Theorem 2.1, we have𝑓(𝑧)∈𝑀𝑃(𝛼, 𝜆).
Conversely, let𝑓(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)}
1−𝛼 𝑎𝑛 for 𝑛= 1,2, ...
and
𝜆0= 1− ∞
∑
𝑛=1
𝜆𝑛.
Then
𝑓(𝑧) =
∞
∑
𝑛=0
𝜆𝑛𝐹𝑛(𝑧).
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
𝑓(𝑧) = 1 𝑧+
∞
∑
𝑛=1
𝑎𝑛𝑧𝑛, (5.1)
to its sequence of partial sums
𝑓1(𝑧) =
1
𝑧 and𝑓𝑘(𝑧) = 1 𝑧 +
𝑘 ∑
𝑛=1
𝑎𝑛𝑧𝑛 (5.2)
when the coefficients are sufficiently small to satisfy the condition analogous to
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)} 𝑎𝑛≤1−𝛼.
For the sake of brevity we rewrite it as
∞
∑
𝑛=1
𝑑𝑛∣𝑎𝑛∣ ≤1−𝛼, (5.3)
where
𝑑𝑛:=𝑛+𝛼−𝛼𝜆(1 +𝑛) (5.4)
More precisely we will determine sharp lower bounds forℜ{𝑓(𝑧)/𝑓𝑘(𝑧)}andℜ{𝑓𝑘(𝑧)/𝑓(𝑧)}.
In this connection we make use of the well known results thatℜ{1+𝑤(𝑧)1−𝑤(𝑧)}>0 (𝑧∈
Δ) if and only if𝜔(𝑧) =
∞
∑ 𝑛=1
𝑐𝑛𝑧𝑛 satisfies the inequality∣𝜔(𝑧)∣ ≤ ∣𝑧∣.Unless
other-wise stated, we will assume that 𝑓 is of the form (1.1) and its sequence of partial sums is denoted by𝑓𝑘(𝑧) =1𝑧 +∑
𝑘
𝑛=1𝑎𝑛𝑧𝑛.
Theorem 5.1. Let 𝑓(𝑧)∈𝑀𝑃(𝛼, 𝜆)be given by (5.1)satisfies condition, (2.1)
Re {𝑓(𝑧)
𝑓𝑘(𝑧) }
≥𝑑𝑘+1(𝜆, 𝛼)−1 +𝛼
𝑑𝑘+1(𝜆, 𝛼)
(𝑧∈𝑈) (5.5)
where
𝑑𝑛(𝜆, 𝛼)≥ {
1−𝛼, 𝑖𝑓 𝑛= 1,2,3, . . . , 𝑘 𝑑𝑘+1(𝜆, 𝛼), 𝑖𝑓 𝑛=𝑘+ 1, 𝑘+ 2, . . . .
(5.6)
The result (5.5) is sharp with the function given by
𝑓(𝑧) = 1 𝑧+
1−𝛼 𝑑𝑘+1(𝜆, 𝛼)
𝑧𝑘+1. (5.7)
Proof. Define the function𝑤(𝑧) by
1 +𝑤(𝑧) 1−𝑤(𝑧) =
𝑑𝑘+1(𝜆, 𝛼)
1−𝛼
[𝑓(𝑧) 𝑓𝑘(𝑧)
−𝑑𝑘+1(𝜆, 𝛼)−1 +𝛼
= 1 +
𝑘 ∑ 𝑛=1
𝑎𝑛𝑧𝑛+1+ (𝑑
𝑘+1(𝜆,𝛼)
1−𝛼
) ∞
∑ 𝑛=𝑘+1
𝑎𝑛𝑧𝑛+1
1 +
𝑘 ∑ 𝑛=1
𝑎𝑛𝑧𝑛+1
(5.8)
It suffices to show that∣𝑤(𝑧)∣ ≤1.Now, from (5.8) we can write
𝑤(𝑧) =
(𝑑 𝑘+1(𝜆,𝛼)
1−𝛼
) ∞
∑ 𝑛=𝑘+1
𝑎𝑛𝑧𝑛+1
2 + 2
𝑘 ∑ 𝑛=1
𝑎𝑛𝑧𝑛+1+ (𝑑
𝑘+1(𝜆,𝛼)
1−𝛼
) ∞
∑ 𝑘=𝑛+1
𝑎𝑛𝑧𝑛+1
.
Hence we obtain
∣𝑤(𝑧)∣ ≤
(𝑑 𝑘+1(𝜆,𝛼)
1−𝛼
) ∞
∑ 𝑘=𝑛+1
∣𝑎𝑛∣
2−2
𝑘 ∑ 𝑛=1
∣𝑎𝑛∣ − (𝑑
𝑘+1(𝜆,𝛼)
1−𝛼
) ∞ ∑ 𝑛=𝑘+1
∣𝑎𝑛∣
Now∣𝑤(𝑧)∣ ≤1 if
2 (𝑑
𝑘+1(𝜆, 𝛼)
1−𝛼
) ∞
∑
𝑛=𝑘+1
∣𝑎𝑛∣ ≤2−2 𝑘 ∑ 𝑛=1 ∣𝑎𝑛∣ or, equivalently, 𝑘 ∑ 𝑛=1 ∣𝑎𝑛∣+
𝑑𝑘+1(𝜆, 𝛼)
1−𝛼
∞
∑
𝑛=𝑘+1
∣𝑎𝑛∣ ≤1.
From the condition (2.1), it is sufficient to show that
𝑘 ∑
𝑛=1
∣𝑎𝑛∣+
𝑑𝑘+1(𝜆, 𝛼)
1−𝛼
∞ ∑ 𝑛=𝑘+1 ∣𝑎𝑛∣ ≤ ∞ ∑ 𝑛=1 𝑑𝑛(𝜆, 𝛼)
1−𝛼 ∣𝑎𝑛∣
which is equivalent to
𝑘 ∑
𝑛=1 (𝑑
𝑛(𝜆, 𝛼)−1 +𝛼
1−𝛼 ) ∣𝑎𝑛∣ + ∞ ∑ 𝑛=𝑘+1 (
𝑑𝑛(𝜆, 𝛼)−𝑑𝑘+1(𝜆, 𝛼)
1−𝛼
)
∣𝑎𝑛∣
≥ 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) 𝑓𝑘(𝑧)
= 1 + 1−𝛼 𝑑𝑘+1(𝜆, 𝛼)
𝑧𝑛→1− 1−𝛼
𝑑𝑘+1(𝜆, 𝛼)
= 𝑑𝑘+1(𝜆, 𝛼)−1 +𝛼 𝑑𝑘+1(𝜆, 𝛼)
which shows the bound (5.5) is the best possible for each𝑘∈ℕ. □
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)
Re {
𝑓𝑘(𝑧)
𝑓(𝑧) }
≥ 𝑑𝑘+1(𝜆, 𝛼)
𝑑𝑘+1(𝜆, 𝛼) + 1−𝛼
(𝑧∈𝑈) (5.10)
where
𝑑𝑘+1(𝜆, 𝛼)≥1−𝛼
𝑑𝑛(𝜆, 𝛼)≥ {
1−𝛼, 𝑖𝑓 𝑛= 1,2,3, . . . , 𝑘 𝑑𝑘+1(𝜆, 𝛼), 𝑖𝑓 𝑛=𝑘+ 1, 𝑘+ 2, . . . .
(5.11)
The result (5.10) is sharp with the function given by
𝑓(𝑧) = 1 𝑧+
1−𝛼 𝑑𝑘+1(𝜆, 𝛼)
𝑧𝑘+1. (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 class𝑀𝑃(𝛼, 𝜆).
Theorem 6.1. Let the function 𝑓 be in the class 𝑀𝑃(𝛼, 𝜆). Then𝑓 is
meromor-phically starlike of order𝜌(0≤𝜌 <1),in ∣𝑧∣< 𝑟1(𝛼, 𝜆, 𝜌),where
𝑟1(𝛼, 𝜆, 𝜌) = inf 𝑛≥1
[ (1−𝜌)(1−𝛼)
(𝑛+ 2−𝜌){𝑛+𝛼−𝛼𝜆(1 +𝑛)}
]𝑛+11
, (6.1)
Proof. Since,
𝑓(𝑧) = 1 𝑧+
∞
∑
𝑛=1
𝑎𝑛𝑧𝑛,
we get
𝑓′(𝑧) =−1
𝑧2+ ∞
∑
𝑛=1
𝑛𝑎𝑛𝑧𝑛−1.
It is sufficient to show that
−𝑧𝑓
′(𝑧)
𝑓(𝑧) −1
≤1−𝜌 (6.2)
or equivalently
𝑧𝑓′(𝑧) 𝑓(𝑧) + 1
=
∞
∑
𝑛=1
(𝑛+ 1)𝑎𝑛𝑧𝑛
1 𝑧−
∞
∑
𝑛=1
𝑎𝑛𝑧𝑛
≤1−𝜌
or
∞
∑
𝑛=1
(𝑛+ 2−𝜌 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−𝜌
)
∣𝑧∣𝑛+1≤ 1−𝛼
or, if
∣𝑧∣ ≤
[ (1−𝜌)(1−𝛼)
(𝑛+ 2−𝜌){𝑛+𝛼−𝛼𝜆(1 +𝑛)}
]𝑛1+1
, 𝑛≥1. (6.3)
This completes the proof of Theorem 6.1. □
Theorem 6.2. Let the function𝑓 in the class𝑀𝑃(𝛼, 𝜆).Then𝑓 is meromorphically
convex of order𝜌, (0≤𝜌 <1),in∣𝑧∣< 𝑟2(𝛼, 𝜆, 𝜌),where
𝑟2(𝛼, 𝜆, 𝜌) = inf 𝑛≥1
[
(1−𝜌)(1−𝛼)
𝑛(𝑛+ 2−𝜌){𝑛+𝛼−𝛼𝜆(1 +𝑛)}
]𝑛+11
, 𝑛≧1, (6.4)
Proof. Since,
𝑓(𝑧) = 1 𝑧−
∞
∑
𝑛=1
𝑎𝑛𝑧𝑛,
we get
𝑓′(𝑧) =−1
𝑧2− ∞
∑
𝑛=1
𝑛𝑎𝑛𝑧𝑛−1.
It is sufficient to show that
−1−𝑧𝑓
′′(𝑧)
𝑓′(𝑧) −1
≤1−𝜌or equivalently (6.5)
𝑧𝑓′′(𝑧) 𝑓′(𝑧) + 2
=
∞
∑
𝑛=1
𝑛(𝑛+ 1)𝑎𝑛𝑧𝑛−1
−1
𝑧2 − ∞
∑
𝑛=1
𝑛𝑎𝑛𝑧𝑛−1
≤1−𝜌or
∞
∑
𝑛=1 (
𝑛(𝑛+ 2−𝜌) 1−𝜌
)
𝑎𝑛∣𝑧∣𝑛+1≤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−𝜌
)
∣𝑧∣𝑛+1≤ (1−𝛼)
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}
or, if
∣𝑧∣ ≤
[
(1−𝜌)(1−𝛼)
𝑛(𝑛+ 2−𝜌){𝑛+𝛼−𝛼𝜆(1 +𝑛)}
]𝑛+11
, 𝑛≥1. (6.6)
This completes the proof of Theorem 6.2. □
7. Integral Operators
In this section, we consider integral transforms of functions in the class𝑀𝑝(𝛼, 𝜆).
Theorem 7.1. Let the function𝑓(𝑧)given by (1) be in𝑀𝑝(𝛼, 𝜆).Then the integral operator
𝐹(𝑧) =𝑐 ∫ 1
0
𝑢𝑐𝑓(𝑢𝑧)𝑑𝑢 (0< 𝑢≤1,0< 𝑐 <∞)
is in𝑀𝑝(𝛿, 𝜆), where
The result is sharp for the function𝑓(𝑧) =1𝑧 +{1+𝛼1−−𝛼2𝛼𝜆}𝑧.
Proof. Let𝑓(𝑧)∈𝑀𝑝(𝛼, 𝜆). Then
𝐹(𝑧) = 𝑐 ∫ 1
0
𝑢𝑐𝑓(𝑢𝑧)𝑑𝑢
= 𝑐 ∫ 1
0 (
𝑢𝑐−1 𝑧 +
∞
∑
𝑛=1
𝑓𝑛𝑢𝑛+𝑐𝑧𝑛 )
𝑑𝑢
= 1 𝑧+
∞
∑
𝑛=1
𝑐
𝑐+𝑛+ 1𝑓𝑛𝑧
𝑛.
It is sufficient to show that
∞
∑
𝑛=1
𝑐{𝑛+𝛿−𝛿𝜆(1 +𝑛)}
(𝑐+𝑛+ 1)(1−𝛿) 𝑎𝑛≤1. (7.1)
Since𝑓 ∈𝑀𝑝(𝛼, 𝜆), we have
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}
(1−𝛼) 𝑎𝑛≤1.
Note that (7.1) is satisfied if
𝑐{𝑛+𝛿−𝛿𝜆(1 +𝑛)}
(𝑐+𝑛+ 1)(1−𝛿) ≤
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}
(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 for𝛿, we have
𝛿≤ (𝑐+𝑛+ 1){𝑛+𝛼−𝛼𝜆(1 +𝑛)} −𝑐𝑛(1−𝛼)
𝑐(1−𝛼){1−𝜆(1 +𝑛)}+{(𝑛+𝛼−𝛼𝜆(1 +𝑛)}(𝑐+𝑛+ 1) =𝐹(𝑛).
A simple computation will show that 𝐹(𝑛) is increasing and 𝐹(𝑛)≥𝐹(1). Using
this, the results follows. □
For the choice of𝜆= 0, we have the following result of Uralegaddi and Ganigi [15].
Remark 7.2. Let the function𝑓(𝑧)defined by (1) be inΣ∗𝑝(𝛼). Then the integral operator
𝐹(𝑧) =𝑐 ∫ 1
0
𝑢𝑐𝑓(𝑢𝑧)𝑑𝑢 (0< 𝑢≤1,0< 𝑐 <∞)
is inΣ∗𝑝(𝛿), where𝛿= 1+𝛼+𝑐𝛼1+𝛼+𝑐 .The result is sharp for the function
𝑓(𝑧) = 1 𝑧+
1−𝛼 1 +𝛼𝑧.
Theorem 7.3. Let 𝑓(𝑧), given by (1), be in 𝑀𝑝(𝛼, 𝜆),
𝐹(𝑧) =1
𝑐[(𝑐+ 1)𝑓(𝑧) +𝑧𝑓
′(𝑧)] = 1
𝑧+
∞
∑
𝑛=1
𝑐+𝑛+ 1 𝑐 𝑓𝑛𝑧
𝑛, 𝑐 >0. (7.2)
Then𝐹(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)}
)1/(𝑛+1)
, 𝑛= 1,2,3, . . . .
The result is sharp for the function𝑓𝑛(𝑧) =1𝑧+{𝑛+𝛼−1𝛼𝜆(1+𝑛)−𝛼 }𝑧𝑛, 𝑛= 1,2,3, . . . .
Proof. Let𝑤=(𝜆−1)𝑓(𝑧)+𝜆𝑧𝑓𝑧𝑓′(𝑧) ′(𝑧). Then it is sufficient to show that
𝑤−1 𝑤+ 1−2𝛽
<1.
A computation shows that this is satisfied if
∞
∑
𝑛=1
{𝑛+𝛽−𝛽𝜆(1 +𝑛)}(𝑐+𝑛+ 1) (1−𝛽)𝑐 𝑎𝑛∣𝑧∣
𝑛+1≤1. (7.3)
Since𝑓 ∈𝑀𝑝(𝛼, 𝜆), by Theorem 2.1, we have
∞
∑
𝑛=1
{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑎𝑛 ≤1−𝛼.
The equation (7.3) is satisfied if
{𝑛+𝛽−𝛽𝜆(1 +𝑛)}(𝑐+𝑛+ 1) (1−𝛽)𝑐 𝑎𝑛∣𝑧∣
𝑛+1≤{𝑛+𝛼−𝛼𝜆(1 +𝑛)}𝑎𝑛
1−𝛼 .
Solving for∣𝑧∣, we get the result. □
For the choice of𝜆= 0, we have the following result of Uralegaddi and Ganigi [15].
Remark 7.4. Let the function𝑓(𝑧)defined by (1) be inΣ∗𝑝(𝛼)and𝐹(𝑧)given by
(7.2). Then𝐹(𝑧)is inΣ∗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)
)1/(𝑛+1)
, 𝑛= 1,2,3, . . . .
The result is sharp for the function𝑓𝑛(𝑧) = 1𝑧+𝑛+𝛼1−𝛼𝑧𝑛, 𝑛= 1,2,3, . . . .
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S.Kavitha
Department of Mathematics, Madras Christian College East Tambaram, Chennai-600 059, India
E-mail address:[email protected]
S. Sivasubramanian
Department of Mathematics, University College of Engineering, Tindivanam Anna University Chennai, Saram-604 307,
India
E-mail address:[email protected]
K. Muthunagai
Department of Mathematics, Sri Sairam Engineering College Tambaram, Chennai-44, India
