Bulletin of Mathematical analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 2 Issue 2(2010), Pages 69-76.
HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH
GENERALIZED HYPERGEOMETRIC FUNCTIONS
GANGADHARAN. MURUGUSUNDARAMOORTHY AND KALIEPPAN.UMA
Abstract. The generalized hypergeometric function is used here to introduce
a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc. Among the results presented in
this paper include the coefficient bounds, distortion inequality and covering
property, extreme points and certain inclusion results for this generalized class
of functions.
1. Introduction
A continuous function 𝑓 = 𝑢 + 𝑖𝑣 is a complex- valued harmonic function in
a complex domain Ω if both 𝑢 and 𝑣 are real and harmonic in Ω. In any simplyconnected domain 𝐷 ⊂ Ω, we can write 𝑓 = ℎ + 𝑔, where ℎ and 𝑔 are analytic in
𝐷. We call ℎ the analytic part and 𝑔 the co-analytic part of 𝑓. A necessary and
sufficient condition for 𝑓 to be locally univalent and orientation preserving in 𝐷 is
that ∣ℎ′ (𝑧)∣ > ∣𝑔 ′ (𝑧)∣ in 𝐷 (see [2]).
Denote by ℋ the family of functions
𝑓 =ℎ+𝑔
(1.1)
which are harmonic, univalent and orientation preserving in the open unit disc
𝒰 = {𝑧 : ∣𝑧∣ < 1} so that 𝑓 is normalized by 𝑓 (0) = ℎ(0) = 𝑓𝑧 (0) − 1 = 0. Thus, for
𝑓 = ℎ + 𝑔 ∈ ℋ, the functions ℎ and 𝑔 analytic 𝒰 can be expressed in the following
forms:
∞
∞
∑
∑
ℎ(𝑧) = 𝑧 +
𝑎𝑛 𝑧 𝑛 , 𝑔(𝑧) =
𝑏𝑛 𝑧 𝑛 (∣𝑏1 ∣ < 1),
𝑛=2
𝑛=1
and 𝑓 (𝑧) is then given by
𝑓 (𝑧) = 𝑧 +
∞
∑
𝑛=2
𝑎𝑛 𝑧 𝑛 +
∞
∑
𝑏𝑛 𝑧 𝑛
(∣𝑏1 ∣ < 1).
(1.2)
𝑛=1
We note that the family ℋ of orientation preserving, normalized harmonic univalent
functions reduces to the well known class 𝒮 of normalized univalent functions if the
2000 Mathematics Subject Classification. 30C45, 30C50.
Key words and phrases. Harmonic univalent functions, hypergeometric function , distortion
bounds, extreme points, convolution. inclusion property.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted November, 2009. Published May, 2010.
69
70
GANGADHARAN. MURUGUSUNDARAMOORTHY AND KALIEPPAN.UMA
co-analytic part of 𝑓 is identically zero, i.e. 𝑔 ≡ 0. Also, we denote by ℋ the
subfamily of ℋ consisting of harmonic functions 𝑓 = ℎ + 𝑔 of the form
𝑓 (𝑧) = 𝑧 −
∞
∑
∣𝑎𝑛 ∣𝑧 𝑛 +
∞
∑
∣𝑏𝑛 ∣𝑧 𝑛 , (∣𝑏1 ∣ < 1)
(1.3)
𝑛=1
𝑛=2
introduced and studied by Silverman[12].
The Hadamard product (or convolution) of two power series
𝜙(𝑧) = 𝑧 +
∞
∑
𝜙𝑛 𝑧 𝑛
(1.4)
𝜓𝑛 𝑧 𝑛
(1.5)
𝑛=2
and
𝜓(𝑧) = 𝑧 +
∞
∑
𝑛=2
in 𝑆 is defined (as usual) by
(𝜙 ∗ 𝜓)(𝑧) = 𝜙(𝑧) ∗ 𝜓(𝑧) = 𝑧 +
∞
∑
𝜙𝑛 𝜓𝑛 𝑧 𝑛 .
(1.6)
𝑛=2
For positive real values of 𝛼1 , . . . , 𝛼𝑙 and 𝛽1 , . . . , 𝛽𝑚 (𝛽𝑗 ∕= 0, −1, . . . ; 𝑗 = 1, 2, . . . , 𝑚)
the generalized hypergeometric function 𝑙 𝐹𝑚 (𝑧) is defined by
𝑙 𝐹𝑚 (𝑧) ≡ 𝑙 𝐹𝑚 (𝛼1 , . . . 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ; 𝑧)
(𝑙 ≤ 𝑚 + 1; 𝑙, 𝑚 ∈ 𝑁0
:=
∞
∑
(𝛼1 )𝑛 . . . (𝛼𝑙 )𝑛 𝑧 𝑛
(𝛽1 )𝑛 . . . (𝛽𝑚 )𝑛 𝑛!
𝑛=0
(1.7)
:= 𝑁 ∪ {0}; 𝑧 ∈ 𝑈 ),
where 𝑁 denotes the set of all positive integers and (𝑎)𝑛 is the Pochhammer symbol
defined by
{
1,
𝑛=0
(𝑎)𝑛 =
(1.8)
𝑎(𝑎 + 1)(𝑎 + 2) . . . (𝑎 + 𝑛 − 1), 𝑛 ∈ 𝑁.
The notation 𝑙 𝐹𝑚 is quite useful for representing many well-known functions
such as the exponential, the Binomial, the Bessel and Laguerre polynomial. Let
𝐻[𝛼1 , . . . 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ] : 𝒮 → 𝒮
be a linear operator defined by
𝐻[𝛼1 , . . . 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ]𝜙(𝑧)
:= 𝑧 𝑙 𝐹𝑚 (𝛼1 , 𝛼2 , . . . 𝛼𝑙 ; 𝛽1 , 𝛽2 . . . , 𝛽𝑚 ; 𝑧) ∗ 𝜙(𝑧)
∞
∑
= 𝑧+
𝜔𝑛 (𝛼1 ; 𝑙; 𝑚) 𝜙𝑛 𝑧 𝑛 ,
(1.9)
𝑛=2
where
𝜔𝑛 (𝛼1 ; 𝑙; 𝑚) =
1
(𝛼1 )𝑛−1 . . . (𝛼𝑙 )𝑛−1
.
(𝛽1 )𝑛−1 . . . (𝛽𝑚 )𝑛−1 (𝑛 − 1)!
(1.10)
𝑙
For notational simplicity, we use a shorter notation 𝐻𝑚
[𝛼1 ] for 𝐻[𝛼1 , . . . 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ]
in the sequel.It follows from (1.9) that
𝐻01 [1]𝜙(𝑧) = 𝜙(𝑧), 𝐻01 [2]𝜙(𝑧) = 𝑧𝜙′ (𝑧).
𝑙
The linear operator 𝐻𝑚
[𝛼1 ] is called Dziok-Srivastava operator (see [4])introduced
by Dziok and Srivastava which was subsequently extended by Dziok and Raina [3]
HARMONIC UNIVALENT FUNCTIONS . . .
71
by using the generalized hypergeometric function , recently Srivastava etal.([11])
1
defined the linear operator ℒ𝜏,𝛼
𝜆,𝑙,𝑚 as follows:
ℒ0𝜆,𝛼1 𝜙(𝑧) = 𝜙(𝑧),
𝛼1
𝑙
𝑙
′
1
ℒ1,𝛼
𝜆,𝑙,𝑚 𝜙(𝑧) = (1 − 𝜆)𝐻𝑚 [𝛼1 ]𝜙(𝑧) + 𝜆(𝐻𝑚 [𝛼1 ]𝜙(𝑧)) = ℒ𝜆,𝑙,𝑚 𝜙(𝑧), (𝜆 ≥ 0), (1.11)
1,𝛼1
𝛼1
1
ℒ2,𝛼
𝜆,𝑙,𝑚 𝜙(𝑧) = ℒ𝜆,𝑙,𝑚 (ℒ𝜆,𝑙,𝑚 𝜙(𝑧))
(1.12)
and in general ,
𝜏 −1,𝛼1
𝛼1
1
ℒ𝜏,𝛼
𝜆,𝑙,𝑚 𝜙(𝑧) = ℒ𝜆,𝑙,𝑚 (ℒ𝜆,𝑙,𝑚 𝜙(𝑧)), (𝑙 ≤ 𝑚+1; 𝑙, 𝑚 ∈ 𝑁0 = 𝑁 ∪{0}; 𝑧 ∈ 𝑈 ). (1.13)
If the function 𝜙(𝑧) is given by (1.4), then we see from (1.9), (1.10),(1.11)
and(1.13) that
1
ℒ𝜏,𝛼
𝜆,𝑙,𝑚 𝜙(𝑧)
:= 𝑧 +
∞
∑
𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) 𝜙𝑛 𝑧 𝑛 ,
(1.14)
𝑛=2
where
𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) =
(
(𝛼1 )𝑛−1 . . . (𝛼𝑙 )𝑛−1 [1 + 𝜆(𝑛 − 1)]
(𝛽1 )𝑛−1 . . . (𝛽𝑚 )𝑛−1
(𝑛 − 1)!
)𝜏
, (𝑛 ∈ 𝑁 ∖ {1}, 𝜏 ∈ 𝑁0(1.15)
)
1
unless otherwise stated. We note that when 𝜏 = 1 and 𝜆 = 0 the linear operatorℒ𝜏,𝛼
𝜆,𝑙,𝑚
would reduce to the familiar Dziok-Srivastava linear operator given by (see [4]), includes (as its special cases) various other linear operators introduced and studied
by Carlson and Shaffer [1], Owa [9]and Ruscheweyh [10] .
In view of the relationship (1.15) and the linear operator (1.14) for the harmonic
function 𝑓 = ℎ + 𝑔 given by (1.1), we define the operator
𝜏,𝛼1
𝜏,𝛼1
1
ℒ𝜏,𝛼
𝜆,𝑙,𝑚 𝑓 (𝑧) = ℒ𝜆,𝑙,𝑚 ℎ(𝑧) + ℒ𝜆,𝑙,𝑚 𝑔(𝑧),
(1.16)
and introduce below a new subclass ℒ𝐻 (𝜏 ; 𝜆; 𝛾) of ℋ in terms of the operator given
by (1.16).
Let ℒ𝐻 (𝜏 ; 𝜆; 𝛾) denote a subclass of ℋ consisting of functions of the form 𝑓 =
ℎ + 𝑔 given by (1.2) satisfying the condition that
}
{
𝜏,𝛼1
1
′
′
)
𝑧(ℒ𝜏,𝛼
∂ (
𝜆,𝑙,𝑚 ℎ(𝑧)) − 𝑧(ℒ𝜆,𝑙,𝑚 𝑔(𝑧))
𝜏,𝛼1
𝑎𝑟𝑔ℒ𝜆,𝑙,𝑚 𝑓 (𝑧) > 𝛾 = Re
≥ 𝛾, (1.17)
𝜏,𝛼1
1
∂𝜃
ℒ𝜏,𝛼
𝜆,𝑙,𝑚 ℎ(𝑧) + (ℒ𝜆,𝑙,𝑚 𝑔(𝑧))
(
𝑧 = 𝑟𝑒𝑖𝜃 ; 0 ≤ 𝜃 < 2𝜋; 0 ≤ 𝑟 < 1; 0 ≤ 𝛾 < 1; 𝑧 ∈ 𝕌
)
1
where ℒ𝜏,𝛼
𝜆,𝑙,𝑚 𝑓 (𝑧) is given by (1.16). We also let ℒ𝐻 (𝜏 ; 𝜆; 𝛾) = ℒ𝐻 (𝜏 ; 𝜆; 𝛾) ∩ ℋ.
In this paper, we obtain coefficient conditions for the classes ℒ𝐻 (𝜏 ; 𝜆; 𝛾) and
ℒ𝐻 (𝜏 ; 𝜆; 𝛾). A representation theorem, inclusion properties and distortion bounds
for the class ℒ𝐻 (𝜏 ; 𝜆; 𝛾) are also established.
2. Coefficient bounds
Due to Jahangiri [7], we state the following sufficient coefficient bound for the
harmonic functions 𝑓 ∈ ℋ(𝛾) the class of harmonic starlike functions of order
𝛾, (0 ≤ 𝛾 < 1) a subclass of ℋ consisting of functions of the form 𝑓 = ℎ + 𝑔 given
∂
by (1.2) satisfying the condition that ∂𝜃
(𝑎𝑟𝑔𝑓 (𝑧)) > 𝛾.
72
GANGADHARAN. MURUGUSUNDARAMOORTHY AND KALIEPPAN.UMA
Theorem 2.1. [7]. Let 𝑓 = ℎ + 𝑔 be given by (1.2). If
]
∞ [
∑
𝑛−𝛾
𝑛+𝛾
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ ≤ 2,
1−𝛾
1−𝛾
𝑛=1
(2.1)
where 𝑎1 = 1 and 0 ≤ 𝛾 < 1, then 𝑓 ∈ ℋ(𝛾).
The following result gives a sufficient coefficient condition for the harmonic functions 𝑓 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
Theorem 2.2. Let 𝑓 = ℎ + 𝑔 be given by (1.2). If
]
∞ [
∑
𝑛−𝛾
𝑛+𝛾
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ 𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) ≤ 2,
1
−
𝛾
1
−
𝛾
𝑛=1
(2.2)
where 𝑎1 = 1 and 0 ≤ 𝛾 < 1, then 𝑓 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
𝑛+𝛾 𝜏
𝜏
Proof. Since 𝑛 ≤ 𝑚𝑖𝑛{ 𝑛−𝛾
1−𝛾 𝜔𝑛 (𝛼1 ; 𝜆; 𝑙; 𝑚); 1−𝛾 𝜔𝑛 (𝛼1 ; 𝜆; 𝑙; 𝑚)}, it follows from Theorem2.1 that 𝑓 ∈ ℋ(𝛾) and hence 𝑓 is harmonic, orientation preserving and univalent in 𝕌. Suppose the condition (2.2) holds true. To show that 𝑓 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾),
we show (in view of (1.17)) that
ℜ
⎧ (
)′
(
)′ ⎫
𝜏,𝛼1


1
⎨ 𝑧 ℒ𝜏,𝛼
⎬
ℎ(𝑧)
−
𝑧
ℒ
𝑔(𝑧)
𝜆,𝑙,𝑚
𝜆,𝑙,𝑚

⎩
𝜏,𝛼1
1
ℒ𝜏,𝛼
𝜆,𝑙,𝑚 ℎ(𝑧) + ℒ𝜆,𝑙,𝑚 𝑔(𝑧)
{
=ℜ

⎭
𝐴(𝑧)
𝐵(𝑧)
}
≥ 𝛾 (𝑧 ∈ 𝕌),
where
𝜏,𝛼1
′
1
′
𝐴(𝑧) = 𝑧(ℒ𝜏,𝛼
𝜆,𝑙,𝑚 ℎ(𝑧)) −𝑧(ℒ𝜆,𝑙,𝑚 𝑔(𝑧)) = 𝑧+
∞
∑
𝑛𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑎𝑛 𝑧 𝑛 −
𝑛=2
∞
∑
𝑛𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑏𝑛 𝑧 𝑛
𝑛=1
and
𝐵(𝑧) = 𝑧 +
∞
∑
𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑎𝑛 𝑧 𝑛
𝑛=2
+
∞
∑
𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑏𝑛 𝑧 𝑛 .
𝑛=1
Using the fact that Re {𝑤} ≥ 𝛾 if and only if ∣1 − 𝛾 + 𝑤∣ ≥ ∣1 + 𝛾 − 𝑤∣, it suffices
to show that
∣𝐴(𝑧) + (1 − 𝛾)𝐵(𝑧)∣ − ∣𝐴(𝑧) − (1 + 𝛾)𝐵(𝑧)∣ ≥ 0.
(2.3)
Substituting for 𝐴(𝑧) and 𝐵(𝑧) in (2.3), and performing elementary calculations,
we find that
∣𝐴(𝑧) + (1 − 𝛾)𝐵(𝑧)∣ − ∣𝐴(𝑧) − (1 + 𝛾)𝐵(𝑧)∣
{
}
]
∞ [
∑
𝑛−𝛾
𝑛+𝛾
𝜏
𝑛−1
≥ 2(1 − 𝛾)∣𝑧∣ 2 −
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ 𝜔𝑛 (𝛼1 ; 𝜆; 𝑙; 𝑚)∣𝑧∣
1−𝛾
1−𝛾
𝑛=1
{
}
]
∞ [
∑
𝑛−𝛾
𝑛+𝛾
> 2(1 − 𝛾) 2 −
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ 𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) ≥ 0,
1−𝛾
1−𝛾
𝑛=1
which implies that 𝑓 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
□
HARMONIC UNIVALENT FUNCTIONS . . .
73
The harmonic function
∞
∞
∑
∑
1−𝛾
1−𝛾
𝑛
𝑓 (𝑧) = 𝑧 +
𝑥
𝑧
+
𝑦 (𝑧)𝑛 ,
𝑛
𝜏 (𝛼 ; 𝜆; 𝑙; 𝑚)
𝜏 (𝛼 ; 𝜆; 𝑙; 𝑚) 𝑛
(𝑛
−
𝛾)𝜔
(𝑛
+
𝛾)𝜔
1
1
𝑛
𝑛
𝑛=2
𝑛=1
(2.4)
∞
∞
∑
∑
where
∣𝑥𝑛 ∣ +
∣𝑦𝑛 ∣ = 1 shows that the coefficient bound given by (2.2) is
𝑛=2
𝑛=1
sharp.
The functions of the form (2.4) are in ℒ𝐻 (𝜏 ; 𝜆; 𝛾) because
)
∞ (
∑
(𝑛 + 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
(𝑛 − 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣
1−𝛾
1−𝛾
𝑛=1
=1 +
∞
∑
∣𝑥𝑛 ∣ +
𝑛=2
∞
∑
∣𝑦𝑛 ∣ = 2.
𝑛=1
Our next theorem gives a necessary and sufficient condition for functions of the
form (1.3) to be in the class ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
Theorem 2.3. For 𝑎1 = 1 and 0 ≤ 𝛾 < 1, 𝑓 = ℎ + 𝑔 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾) if and only if
]
∞ [
∑
𝑛−𝛾
𝑛+𝛾
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ 𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) ≤ 2.
(2.5)
1
−
𝛾
1
−
𝛾
𝑛=1
Proof. Since ℒ𝐻 (𝜏 ; 𝜆; 𝛾) ⊂ ℒ𝐻 (𝜏 ; 𝜆; 𝛾), we only need to prove the ”only if” part
of the theorem. To this end, for functions 𝑓 of the form (1.3), we notice that the
condition
{
}
𝜏,𝛼1 ′
1
′
𝑧(ℒ𝜏,𝛼
𝜆,𝑙,𝑚 ℎ (𝑧)) − 𝑧(ℒ𝜆,𝑙,𝑚 𝑔 (𝑧))
ℜ
≥𝛾
𝜏,𝛼1
1
ℒ𝜏,𝛼
𝜆,𝑙,𝑚 ℎ(𝑧) + (ℒ𝜆,𝑙,𝑚 𝑔(𝑧))
implies that
⎫
⎧
∞
∞
∑
∑

𝑛
𝑛
𝜏
𝜏


(𝛼
;
𝜆;
𝑙;
𝑚)𝑏
(𝛼
;
𝜆;
𝑙;
𝑚)𝑎
𝑧
−
(𝑛
+
𝛾)𝜔
(1
−
𝛾)𝑧
−
(𝑛
−
𝛾)𝜔
𝑧
1
1
𝑛
𝑛
⎬
⎨
𝑛
𝑛
𝑛=1
𝑛=2
≥ 0.
ℜ
∞
∞
∑ 𝜏
∑ 𝜏




𝑧−
𝜔𝑛 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑎𝑛 𝑧 𝑛 +
𝜔𝑛 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑏𝑛 𝑧 𝑛
⎭
⎩
𝑛=2
𝑛=1
The above required condition must hold for all values of 𝑧 in 𝕌. Upon choosing
the values of 𝑧 on the positive real axis where 0 ≤ 𝑧 = 𝑟 < 1, we must have
(1 − 𝛾) −
∞
∑
𝑛=2
1−
∞
∑
(𝑛 − 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑎𝑛 𝑟𝑛−1 −
(𝑛 + 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑏𝑛 𝑟𝑛−1
𝑛=1
∞
∑
𝑛=2
𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑎𝑛 𝑟𝑛−1 +
∞
∑
≥ 0. (2.6)
𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)𝑏𝑛 𝑟𝑛−1
𝑛=1
If the condition (2.5) does not hold, then the numerator in (2.6) is negative for 𝑟 sufficiently
close to 1. Hence, there exist 𝑧0 = 𝑟0 in (0,1) for which the quotient of (2.6) is negative.
This contradicts the required condition for 𝑓 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾). This completes the proof
of the theorem.
□
3. Distortion bounds and Extreme Points
By applying the condition (2.5) and employing similar steps of derivation as
given in [5, 6, 8, 7], we state the following results without proof.
74
GANGADHARAN. MURUGUSUNDARAMOORTHY AND KALIEPPAN.UMA
Theorem 3.1. (Distortion bounds) Let 𝑓 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾), then (for ∣𝑧∣ = 𝑟 < 1)
(
)
1
1−𝛾
1+𝛾
(1 − 𝑏1 )𝑟 − 𝜏
−
𝑏1 𝑟2 ≤ ∣𝑓 (𝑧)∣
𝜔2 (𝛼1 ; 𝜆; 𝑙; 𝑚) 2 − 𝛾
2−𝛾
(
)
1−𝛾
1+𝛾
1
−
𝑏1 𝑟 2 .
≤ (1 + 𝑏1 )𝑟 + 𝜏
𝜔2 (𝛼1 ; 𝜆; 𝑙; 𝑚) 2 − 𝛾
2−𝛾
Corollary 3.2. If 𝑓 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾), then
}
{
2𝜔2𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) − 1 − [𝜔2𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) + 1]𝛾
2𝜔2𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) − 1 − [𝜔2𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) − 1]𝛾
−
𝑏1 ⊂ 𝑓 (𝑈 ).
𝑤 : ∣𝑤∣ <
𝜏
𝜏
(2 − 𝛾)𝜔2 (𝛼1 ; 𝜆; 𝑙; 𝑚)
(2 − 𝛾)𝜔2 (𝛼1 ; 𝜆; 𝑙; 𝑚)
The extreme points of closed convex hulls of ℒ𝐻 (𝜏 ; 𝜆; 𝛾) denoted by 𝑐𝑙𝑐𝑜ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
Theorem 3.3. (Extreme Points) A function 𝑓 (𝑧) ∈ 𝑐𝑙𝑐𝑜ℒ𝐻 (𝜏 ; 𝜆; 𝛾) if and only
∞
∑
if 𝑓 (𝑧) =
[𝑋𝑛 ℎ𝑛 (𝑧) + 𝑌𝑛 𝑔𝑛 (𝑧)] , where
𝑛=1
ℎ1 (𝑧) = 𝑧, ℎ𝑛 (𝑧) = 𝑧 −
∞
∑
1−𝛾
1−𝛾
𝑧 𝑛 (𝑛 ≥ 2), 𝑔𝑛 (𝑧) = 𝑧 +
𝑧 𝑛 (𝑛 ≥ 2),
(𝑛 − 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
(𝑛 + 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
(𝑋𝑛 + 𝑌𝑛 ) = 1, 𝑋𝑛 ≥ 0 𝑎𝑛𝑑 𝑌𝑛 ≥ 0.
𝑛=1
In particular, the extreme points of ℒ𝐻 (𝜏 ; 𝜆; 𝛾) are {ℎ𝑛 } and {𝑔𝑛 }.
4. Inclusion Results
The following result gives the convex combinations of the class ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
Theorem 4.1. The family ℒ𝐻 (𝜏 ; 𝜆; 𝛾) is closed under convex combinations.
Proof. Let 𝑓𝑖 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾) (𝑖 = 1, 2, . . . ),where
𝑓𝑖 (𝑧) = 𝑧 −
∞
∑
𝑛=2
∣𝑎𝑖,𝑛 ∣𝑧 𝑛 +
∞
∑
∣𝑏𝑖,𝑛 ∣𝑧 𝑛 .
𝑛=2
The convex combination of 𝑓𝑖 may be written as
(∞
)
(∞
)
∞
∞
∞
∑
∑
∑
∑
∑
𝑛
𝑡𝑖 𝑓𝑖 (𝑧) = 𝑧 −
𝑡𝑖 ∣𝑎𝑖,𝑛 ∣ 𝑧 +
𝑡𝑖 ∣𝑏𝑖,𝑛 ∣ 𝑧 𝑛 ,
𝑛=2
𝑖=1
provided that
∞
∑
𝑖=1
𝑛=1
𝑖=1
𝑡𝑖 = 1 (0 ≤ 𝑡𝑖 ≤ 1).
𝑖=1
Applying the inequality (2.5) of Theorem 2.3, we obtain
(∞
)
(∞
)
∞
∞
∑
∑
(𝑛 + 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) ∑
(𝑛 − 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚) ∑
𝑡𝑖 ∣𝑎𝑖,𝑛 ∣ +
𝑡𝑖 ∣𝑏𝑖,𝑛 ∣
1−𝛾
1−𝛾
𝑛=1
𝑛=2
𝑖=1
𝑖=1
(∞
)
∞
∞
∑
∑ (𝑛 − 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
∑
(𝑛 + 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
=
𝑡𝑖
∣𝑎𝑖,𝑛 ∣ +
∣𝑏𝑖,𝑛 ∣
1−𝛾
1−𝛾
𝑛=2
𝑛=1
𝑖=1
≤
∞
∑
𝑡𝑖 = 1,
𝑖=1
and therefore,
∞
∑
𝑖=1
𝑡𝑖 𝑓𝑖 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
□
HARMONIC UNIVALENT FUNCTIONS . . .
75
Theorem 4.2. Let 𝑓 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾) and 𝐹 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛿), where 0 ≤ 𝛿 ≤ 𝛾 <
1, then
𝑓 (𝑧) ∗ 𝐹 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾) ⊂ ℒ𝐻 (𝜏 ; 𝜆; 𝛿).
∞
∑
Proof. Let 𝑓 (𝑧) = 𝑧 −
∞
∑
∣𝐴𝑛 ∣𝑧 𝑛 +
𝑛=2
∣𝑎𝑛 ∣𝑧 𝑛 +
𝑛=2
∞
∑
𝑛=1
∞
∑
𝑛=1
∣𝑏𝑛 ∣𝑧 𝑛 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾) and 𝐹 (𝑧) = 𝑧 −
∣𝐵𝑛 ∣𝑧 𝑛 ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛿), then
∞
∑
𝑓 (𝑧) ∗ 𝐹 (𝑧) = 𝑧 −
∣𝑎𝑛 ∣∣𝐴𝑛 ∣𝑧 𝑛 +
𝑛=2
∞
∑
∣𝑏𝑛 ∣∣𝐵𝑛 ∣𝑧 𝑛 .
𝑛=1
From the assertion that 𝑓 (𝑧) ∗ 𝐹 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛿), we note that ∣𝐴𝑛 ∣ ≤ 1 and
∣𝐵𝑛 ∣ ≤ 1. In view of Theorem 2.3 and the inequality 0 ≤ 𝛿 ≤ 𝛾 < 1, we have
∞
∞
∑
∑
(𝑛 − 𝛿)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
(𝑛 + 𝛿)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
∣𝑎𝑛 ∣ ∣𝐴𝑛 ∣ +
∣𝑏𝑛 ∣ ∣𝐵𝑛 ∣
1−𝛿
1−𝛿
𝑛=2
𝑛=1
≤
∞
∞
∑
∑
(𝑛 − 𝛿)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
(𝑛 + 𝛿)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣
1−𝛿
1−𝛿
𝑛=2
𝑛=1
≤
∞
∞
∑
∑
(𝑛 − 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
(𝑛 + 𝛾)𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ ≤ 1,
1−𝛾
1−𝛾
𝑛=2
𝑛=1
by Theorem 2.3, 𝑓 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾). Hence 𝑓 (𝑧) ∗ 𝐹 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾) ⊂ ℒ𝐻 (𝜏 ; 𝜆; 𝛿).
□
Lastly, we consider the closure property of the class ℒ𝐻 (𝜏 ; 𝜆; 𝛾) under the generalized Bernardi-Libera -Livingston integral operatorℒ𝑐 [𝑓 (𝑧)] which is defined by
𝑐+1
ℒ𝑐 [𝑓 (𝑧)] =
𝑧𝑐
∫𝑧
𝑡𝑐−1 𝑓 (𝑡)𝑑𝑡 (𝑐 > −1).
0
We prove the following result.
Theorem 4.3. Let 𝑓 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾), then ℒ𝑐 [𝑓 (𝑧)] ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾)
Proof. Using (1.1) and (1.3), we get
𝑐+1
ℒ𝑐 [𝑓 (𝑧)] =
𝑧𝑐
∫𝑧
𝑡
𝑐−1
𝑐+1
ℎ(𝑡)𝑑𝑡 + 𝑐
𝑧
0
=
𝑐+1
𝑧𝑐
∫𝑧
(
𝑐−1
𝑡
𝑡−
∞
∑
𝑛=2
𝐴𝑛 𝑧 𝑛 +
∞
∑
𝑛=1
𝑡𝑐−1 𝑔(𝑡)𝑑𝑡.
0
)
∣𝑎𝑛 ∣𝑡
𝑛
𝑛=2
0
=𝑧−
∞
∑
∫𝑧
𝑐+1
𝑑𝑡 + 𝑐
𝑧
∫𝑧
0
𝐵𝑛 𝑧 𝑛 ,
(
𝑡𝑐−1
∞
∑
𝑛=1
)
∣𝑏𝑛
∣𝑡𝑛
𝑑𝑡
76
GANGADHARAN. MURUGUSUNDARAMOORTHY AND KALIEPPAN.UMA
𝑐+1
𝑐+𝑛
∞ [
∑
𝑐+1
∣𝑎𝑛 ∣ and 𝐵𝑛 = 𝑐+𝑛
∣𝑏𝑛 ∣. Hence
(
)
(
)]
𝑛+𝛾 𝑐+1
𝑛−𝛾 𝑐+1
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ 𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
1−𝛾 𝑐+𝑛
1−𝛾 𝑐+𝑛
𝑛=1
]
[
∞
∑
𝑛+𝛾
𝑛−𝛾
≤
∣𝑎𝑛 ∣ +
∣𝑏𝑛 ∣ 𝜔𝑛𝜏 (𝛼1 ; 𝜆; 𝑙; 𝑚)
1
−
𝛾
1
−
𝛾
𝑛=1
where 𝐴𝑛 =
≤ 2,
since 𝑓 (𝑧) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾), therefore by Theorem 2.3, ℒ𝑐 (𝑓 (𝑧)) ∈ ℒ𝐻 (𝜏 ; 𝜆; 𝛾).
□
Concluding Remarks: By choosing 𝜏 = 1 and 𝜆 = 0 the various results
presented in this paper would provide interesting extensions and generalizations of
those considered earlier for simpler harmonic function classes (see [5, 6, 8, 12]). The
details involved in the derivations of such specializations of the results presented in
this paper are fairly straight- forward.
Acknowledgements : The authors would like to thank the referee and Prof.R.K.Raina
for their insightful suggestions.
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Gangadharan. Murugusundaramoorthy and Kalieppan.Uma
School of Advanced Sciences , VIT University,, Vellore - 632014, India.,
E-mail address: gmsmoorthy@yahoo.com