# On Certain Subclass of Harmonic Starlike Functions

Authors: A. Y. Lashin
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.


---

## Body

## 1. Introduction

A continuous complex-valued functionf
=
u
+
i
v is said to be harmonic in a simply connected domain D
⊂
ℂ if both u and v are real harmonic in D. There is a close interrelation between analytic functions and harmonic functions. For example, for real harmonic functions u and v, there exist analytic functions U and V so that u
=
Re
(
U
) and v
=
Im
⁡
(
V
). Then f
(
z
)
=
h
(
z
)
+
g
(
z
)
¯, where h and g are, respectively, the analytic functions (
U
+
V
)
/
2 and (
U
-
V
)
/
2. In this case, the Jacobian of f
=
h
+
g
¯ is given by J
f
=
|
h
′
(
z
)
|
2
-
|
g
′
(
z
)
|
2. The mapping z
→
f
(
z
) is orientation preserving and locally one-to-one in D if and only if J
f
>
0 in D. The function f
=
h
+
g
¯ is said to be harmonic univalent in D if the mapping z
→
f
(
z
) is orientation preserving, harmonic, and one-to-one in D. We call h the analytic part and g the coanalytic part of f (see Clunie and Sheil-Small [1]).Denote byℋ the class of functions f
=
h
+
g
¯ that are harmonic univalent and orientation preserving in the open unit disk 𝕌
=
{
z
:
|
z
|
<
1
} for which f
(
0
)
=
f
z
(
0
)
-
1
=
0. Then for f
=
h
+
g
¯
∈
ℋ, we may express the analytic functions f and g as
(1)
h
(
z
)
=
z
+
∑
n
=
2
∞
‍
a
n
z
n
,
g
(
z
)
=
∑
n
=
1
∞
‍
b
n
z
n
,
|
b
1
|
<
1
.
Note that ℋ reduces to the class 𝒮 of normalized analytic univalent functions if the coanalytic part of its members is zero. For this class the function f
(
z
) may be expressed as
(2)
f
(
z
)
=
z
+
∑
n
=
2
∞
‍
a
n
z
n
.
A function f
=
h
+
g
¯ with h and g given by (1) is said to be harmonic starlike of order β
(
0
≤
β
<
1
) for |
z
|
=
r
<
1, if
(3)
∂
∂
θ
(
arg
f
(
r
e
i
θ
)
)
=
Re
(
z
h
′
(
z
)
-
z
g
′
(
z
)
¯
h
(
z
)
+
g
(
z
)
¯
)
≥
β
.
The class of all harmonic starlike functions of order β is denoted by 𝒮
H
*
(
β
) and extensively studied by Jahangiri [2]. The cases β
=
0 and b
1
=
0 were studied by Silverman and Silvia [3] and Silverman [4]. Other related works of the class ℋ also appeared in [5–16].Definition 1.
Letf
=
h
+
g
¯ where h and g are given by (1). Let 0
≤
β
<
1 and α
≥
0. Then f
∈
𝒮
H
*
(
α
,
β
) if and only if
(4)
Re
(
α
z
2
h
′′
(
z
)
+
z
h
′
(
z
)
+
α
z
2
g
′′
(
z
)
+
(
2
α
-
1
)
z
g
′
(
z
)
¯
h
(
z
)
+
g
(
z
)
¯
)
≥
β
.We note that forα
=
0, the class 𝒮
H
*
(
α
,
β
) reduces to the class 𝒮
H
*
(
β
). Further, if the coanalytic part g
(
z
) is zero, the class 𝒮
H
*
(
α
,
β
) reduces to the class 𝒫
(
α
,
β
) of functions f
∈
𝒮 which satisfy the condition
(5)
Re
(
α
z
2
f
′′
(
z
)
+
z
f
′
(
z
)
f
(
z
)
)
≥
β
for some β
(
0
≤
β
<
1
), α
≥
0, f
(
z
)
/
z
≠
0 and z
∈
𝕌. Observe that the classes 𝒫
(
α
,
β
) and 𝒫
(
α
,
0
) were introduced and studied by many authors and these include, for example, by Obradovic and Joshi [17], Padmanabhan [18], Li and Owa [19], Xu and Yang [20], Singh and Gupta [21], and Lashin [22]. We also note that for α
=
0, the class 𝒫
(
0
,
β
) was studied by Silverman [23].


## 2. Main Results

The first theorem of this section determines the sufficient coefficient condition for functionsf
=
h
+
g
¯ to belong to the class 𝒮
H
*
(
α
,
β
). The following lemma obtained by Jahangiri is needed.Lemma 2 (see [2, Theorem 1]).
Letf
=
h
+
g
¯ with h and g of the form (1) and let
(6)
∑
n
=
2
∞
‍
n
-
β
1
-
β
|
a
n
|
+
∑
n
=
1
∞
‍
n
+
β
1
-
β
|
b
n
|
≤
1
,
where 0
≤
β
<
1. Then f is harmonic, orientation preserving, and univalent in 𝕌, and f
∈
𝒮
H
*
(
β
).Theorem 3.
Letf
=
h
+
g
¯ where h and g are of the form (1). If
(7)
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
≤
1
for some β, (
0
≤
β
<
1
) and α
≥
0, then f is harmonic, orientation preserving, and univalent in 𝕌 and f
∈
𝒮
H
*
(
α
,
β
).Proof.
Since(
n
-
β
)
≤
α
n
(
n
-
1
)
+
n
-
β and n
+
β
≤
α
n
(
n
+
1
)
+
n
+
β, (
n
≥
1
), it follows from Lemma 2 that f
∈
𝒮
H
*
(
β
) and hence f is harmonic, orientation preserving, and univalent in 𝕌. Now, we only need to show that if (7) holds then
(8)
Re
(
α
z
2
h
′′
(
z
)
+
z
h
′
(
z
)
+
α
z
2
g
′′
(
z
)
+
(
2
α
-
1
)
z
g
′
(
z
)
¯
h
(
z
)
+
g
(
z
)
¯
)
=
Re
A
(
z
)
B
(
z
)
≥
β
.
Using the fact that Re
(
w
)
≥
β if and only if |
1
-
β
+
w
|
≥
|
1
+
β
-
w
|, it suffices to show that
(9)
|
A
(
z
)
+
(
1
-
β
)
B
(
z
)
|
-
|
A
(
z
)
-
(
1
+
β
)
B
(
z
)
|
≥
0
,
where
(10)
A
(
z
)
=
α
z
2
h
′′
(
z
)
+
z
h
′
(
z
)
+
α
z
2
g
′′
(
z
)
+
(
2
α
-
1
)
z
g
′
(
z
)
¯
,
B
(
z
)
=
h
(
z
)
+
g
(
z
)
¯
.
Substituting for A
(
z
) and B
(
z
) in (9), we obtain
(11)
|
A
(
z
)
+
(
1
-
β
)
B
(
z
)
|
-
|
A
(
z
)
-
(
1
+
β
)
B
(
z
)
|
=
|
(
2
-
β
)
z
+
∑
n
=
2
∞
‍
(
(
α
n
+
1
)
(
n
-
1
)
+
2
-
β
)
a
n
z
n
+
∑
n
=
1
∞
‍
(
α
n
(
n
+
1
)
-
(
n
-
1
+
β
)
)
b
n
¯
z
n
¯
|
-
|
-
β
z
+
∑
n
=
2
∞
‍
(
(
α
n
+
1
)
(
n
-
1
)
-
β
)
a
n
z
n
+
∑
n
=
1
∞
‍
(
α
n
(
n
+
1
)
-
(
n
+
1
+
β
)
)
b
¯
n
z
¯
n
|
≥
(
2
-
β
)
|
z
|
-
∑
n
=
2
∞
‍
(
(
α
n
+
1
)
(
n
-
1
)
+
2
-
β
)
|
a
n
|
|
z
|
n
-
∑
n
=
1
∞
‍
|
α
n
(
n
+
1
)
-
(
n
-
1
+
β
)
|
|
b
n
|
|
z
|
n
-
β
|
z
|
-
∑
n
=
2
∞
‍
(
(
α
n
+
1
)
(
n
-
1
)
-
β
)
|
a
n
|
|
z
|
n
-
∑
n
=
1
∞
‍
|
α
n
(
n
+
1
)
-
(
n
+
1
+
β
)
|
|
b
n
|
|
z
|
n
≥
2
(
1
-
β
)
|
z
|
-
2
∑
n
=
2
∞
‍
(
α
n
(
n
-
1
)
+
n
-
β
)
|
a
n
|
|
z
|
n
-
2
∑
n
=
1
∞
‍
(
α
n
(
n
+
1
)
+
(
n
+
β
)
)
|
b
n
|
|
z
|
n
≥
2
(
1
-
β
)
|
z
|
×
(
1
-
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
-
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
(
n
+
β
)
1
-
β
|
b
n
|
)
≥
0
,
by the given condition (7).
The harmonic function(12)
f
(
z
)
=
z
+
∑
n
=
2
∞
‍
1
-
β
α
n
(
n
-
1
)
+
n
-
β
x
n
z
n
+
∑
n
=
2
∞
‍
1
-
β
α
n
(
n
+
1
)
+
(
n
+
β
)
y
¯
n
z
¯
n
,
where ∑
n
=
2
∞
|
x
n
|
+
∑
n
=
1
∞
|
y
n
|
=
1 shows that the coefficient bound given in (7) is sharp. The functions of the form (12) are in 𝒮
H
*
(
α
,
β
) since
(13)
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
=
∑
n
=
2
∞
‍
|
x
n
|
+
∑
n
=
1
∞
‍
|
y
n
|
=
1
.Remark 4.
Settingg
(
z
)
=
0 in Theorem 3 yields the result obtained by Lashin [22, Theorem 2.1].We denote by𝒮
¯
H
*
(
α
,
β
) the class of functions f
∈
𝒮
H
*
(
α
,
β
) whose coefficients satisfy the condition (7).Theorem 5.
Let0
≤
α
1
<
α
2 and 0
≤
β
<
1. Then 𝒮
¯
H
*
(
α
2
,
β
)
⊂
𝒮
¯
H
*
(
α
1
,
β
).Proof.
Forf
∈
𝒮
¯
H
*
(
α
2
,
β
), it follows from (7) that
(14)
∑
n
=
2
∞
‍
α
1
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
+
∑
n
=
1
∞
‍
α
1
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
<
∑
n
=
2
∞
‍
α
2
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
+
∑
n
=
1
∞
‍
α
2
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
≤
1
.
Hence f
∈
𝒮
¯
H
*
(
α
1
,
β
).As a consequence of Theorem5, the functions in 𝒮
¯
H
*
(
α
,
β
) are starlike harmonic in 𝕌.Corollary 6.
Forα
≥
0 and 0
≤
β
<
1,𝒮
¯
H
*
(
α
,
β
)
⊂
𝒮
H
*
(
β
).


## 3. Distortion Bounds and Extreme Points

In this section, we obtain the distortion bounds and extreme points for functions in the class𝒮
¯
H
*
(
α
,
β
).Theorem 7.
Letf
=
h
+
g
¯ where h and g are of the form (1) and f
∈
𝒮
¯
H
*
(
α
,
β
). Then for |
z
|
=
r
<
1, we have
(15)
|
f
(
z
)
|
≤
(
1
+
|
b
1
|
)
r
+
(
1
-
β
2
α
+
2
-
β
-
2
α
+
1
+
β
2
α
+
2
-
β
|
b
1
|
)
r
2
,
(16)
|
f
(
z
)
|
≥
(
1
-
|
b
1
|
)
r
-
(
1
-
β
2
α
+
2
-
β
-
2
α
+
1
+
β
2
α
+
2
-
β
|
b
1
|
)
r
2
,
where
(17)
|
b
1
|
≤
1
-
β
2
α
+
1
+
β
.
The result is sharp.Proof.
We shall prove the first inequality. Letf
∈
𝒮
¯
H
*
(
α
,
β
). Then we have
(18)
|
f
(
z
)
|
≤
(
1
+
|
b
1
|
)
r
+
∑
n
=
2
∞
‍
(
|
a
n
|
+
|
b
n
|
)
r
n
≤
(
1
+
|
b
1
|
)
r
+
r
2
∑
n
=
2
∞
‍
(
|
a
n
|
+
|
b
n
|
)
=
(
1
+
|
b
1
|
)
r
+
1
-
β
2
α
+
2
-
β
×
∑
n
=
2
∞
‍
2
α
+
2
-
β
1
-
β
(
|
a
n
|
+
|
b
n
|
)
r
2
and so
(19)
|
f
(
z
)
|
≤
(
1
+
|
b
1
|
)
r
+
1
-
β
2
α
+
2
-
β
×
∑
n
=
2
∞
‍
(
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
+
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
)
r
2
≤
(
1
+
|
b
1
|
)
r
+
1
-
β
2
α
+
2
-
β
(
1
-
2
α
+
1
+
β
1
-
β
|
b
1
|
)
r
2
=
(
1
+
|
b
1
|
)
r
+
(
1
-
β
2
α
+
2
-
β
-
2
α
+
1
+
β
2
α
+
2
-
β
|
b
1
|
)
r
2
.
The proof of the inequality (16) is similar to the proof of the inequality (15), thus we omit it.
The upper bound given forf
∈
𝒮
¯
H
*
(
α
,
β
) is sharp and the equality occurs for the function
(20)
f
(
z
)
=
z
+
|
b
1
|
z
¯
+
(
1
-
β
2
α
+
2
-
β
-
2
α
+
1
+
β
2
α
+
2
-
β
|
b
1
|
)
z
¯
2
(
z
=
r
)
,
where |
b
1
|
≤
(
1
-
β
)
/
(
2
α
+
1
+
β
). This completes the proof of Theorem 7.Now, we determine the extreme points of the closed convex hull of the class𝒮
¯
H
*
(
α
,
β
) denoted by clco
⁡
𝒮
¯
H
*
(
α
,
β
).Theorem 8.
Letf
=
h
+
g
¯ where h and g are given by (1). Then f
∈
c
l
c
o
𝒮
¯
H
*
(
α
,
β
) if and only if
(21)
f
(
z
)
=
∑
n
=
1
∞
‍
(
X
n
h
n
+
Y
n
g
n
)
,
where
(22)
h
1
(
z
)
=
z
,
h
n
(
z
)
=
z
+
1
-
β
α
n
(
n
-
1
)
+
n
-
β
z
n
(
n
=
2,3
,
…
)
;
g
n
(
z
)
=
z
+
1
-
β
α
n
(
n
+
1
)
+
n
+
β
z
¯
n
(
n
=
1,2
,
3
,
…
)
,
∑
n
=
1
∞
‍
(
X
n
+
Y
n
)
=
1
,
X
n
≥
0
,
Y
n
≥
0
.
In particular, the extreme points of the class 𝒮
¯
H
*
(
α
,
β
) are {
h
n
} and {
g
n
}, respectively.Proof.
For a functionf of the form (21), we have
(23)
f
(
z
)
=
∑
n
=
1
∞
‍
(
X
n
h
n
+
Y
n
g
n
)
=
∑
n
=
1
∞
‍
(
X
n
+
Y
n
)
z
+
∑
n
=
2
∞
‍
1
-
β
α
n
(
n
-
1
)
+
n
-
β
X
n
z
n
+
∑
n
=
1
∞
‍
1
-
β
α
n
(
n
+
1
)
+
n
+
β
Y
n
z
¯
n
=
z
+
∑
n
=
2
∞
‍
1
-
β
α
n
(
n
-
1
)
+
n
-
β
X
n
z
n
+
∑
n
=
1
∞
‍
1
-
β
α
n
(
n
+
1
)
+
n
+
β
Y
n
z
¯
n
.
But
(24)
∑
n
=
2
∞
α
n
(
n
-
1
)
+
n
-
β
1
-
β
(
1
-
β
α
n
(
n
-
1
)
+
n
-
β
X
n
)
‍
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
(
1
-
β
α
n
(
n
+
1
)
+
n
+
β
Y
n
)
=
∑
n
=
2
∞
‍
X
n
+
∑
n
=
1
∞
‍
Y
n
=
1
-
X
1
≤
1
.
Thus f
∈
clco
⁡
𝒮
¯
H
*
(
α
,
β
).
Conversely, suppose thatf
∈
clco
⁡
𝒮
¯
H
*
(
α
,
β
). Set
(25)
X
n
=
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
(
n
=
2,3
,
…
)
,
Y
n
=
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
(
n
=
1,2
,
3
,
…
)
.
Then by the inequality (7), we have 0
≤
X
n
≤
1
(
n
=
2,3
,
…
) and 0
≤
Y
n
≤
1
(
n
=
1,2
,
…
). Define X
1
=
1
-
∑
n
=
2
∞
X
n
-
∑
n
=
1
∞
Y
n and note that X
1
≥
0. Thus we obtain f
(
z
)
=
∑
n
=
1
∞
(
X
n
h
n
+
Y
n
g
n
). This completes the proof of the theorem.


## 4. Convolution and Convex Combinations

For two harmonic functions(26)
f
(
z
)
=
z
+
∑
n
=
2
∞
‍
a
n
z
n
+
∑
n
=
1
∞
‍
b
n
¯
z
¯
n
,
F
(
z
)
=
z
+
∑
n
=
2
∞
‍
A
n
z
n
+
∑
n
=
1
∞
‍
B
n
¯
z
¯
n
we define their convolution
(27)
(
f
*
F
)
(
z
)
=
z
+
∑
n
=
2
∞
‍
a
n
A
n
z
n
+
∑
n
=
1
∞
‍
b
n
B
n
¯
z
¯
n
.
Using this definition, we show that the class 𝒮
¯
H
*
(
α
,
β
) is closed under convolution.Theorem 9.
For0
≤
β
<
1 and α
≥
0, let f
,
F
∈
𝒮
¯
H
*
(
α
,
β
). Then f
*
F
∈
𝒮
¯
H
*
(
α
,
β
).Proof.
We note that|
A
n
|
≤
1 and |
B
n
|
≤
1. For the convolution (
f
*
F
), we have
(28)
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
A
n
a
n
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
B
n
b
n
|
≤
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
≤
1
.
Therefore f
*
F
∈
𝒮
¯
H
*
(
α
,
β
).We show that the class𝒮
¯
H
*
(
α
,
β
) is closed under convex combination of its members.Theorem 10.
The class𝒮
¯
H
*
(
α
,
β
) is closed under convex combination.Proof.
Fori
=
1,2
,
3
,
…, let f
i
∈
𝒮
¯
H
*
(
α
,
β
) where f
i
(
z
) is given by
(29)
f
i
(
z
)
=
z
+
∑
n
=
2
∞
‍
a
n
i
z
n
+
∑
n
=
1
∞
‍
b
¯
n
i
z
¯
n
.
Then by (7), we have
(30)
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
i
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
i
|
≤
1
.
For ∑
i
=
1
∞
t
i
=
1, 0
≤
t
i
≤
1, the convex combination of f
i may be written as
(31)
∑
i
=
1
∞
‍
t
i
f
i
=
z
+
∑
n
=
2
∞
‍
(
∑
i
=
1
∞
‍
t
i
a
n
i
)
z
n
+
∑
n
=
1
∞
‍
(
∑
i
=
1
∞
‍
t
i
b
n
i
¯
)
z
¯
n
.
Then by (7), we have
(32)
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
∑
i
=
1
∞
‍
t
i
a
n
i
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
∑
i
=
1
∞
‍
t
i
b
¯
n
i
|
≤
∑
i
=
1
∞
‍
t
i
(
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
i
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
i
|
)
≤
∑
i
=
1
∞
‍
t
i
=
1
.
Therefore ∑
i
=
1
∞
t
i
f
i
∈
𝒮
¯
H
*
(
α
,
β
).


## 5. Neighborhood Results

Following the earlier investigations by Goodman [24], Ruscheweyh [25], Altintas et al. [26], and Porwal and Aouf [27], we define the δ-neighborhood of function f
(
z
)
∈
ℋ by
(33)
N
δ
(
f
)
=
{
F
∈
ℋ
:
F
(
z
)
=
z
+
∑
n
=
2
∞
‍
A
n
z
n
+
∑
n
=
1
∞
‍
B
n
¯
z
¯
n
,
∑
n
=
2
∞
‍
n
|
a
n
-
A
n
|
+
∑
n
=
1
∞
‍
n
|
b
n
-
B
n
|
≤
δ
}
.
In particular, for the identity function e
(
z
)
=
z, we immediately have
(34)
N
δ
(
e
)
=
{
F
∈
ℋ
:
F
(
z
)
=
z
+
∑
n
=
2
∞
‍
A
n
z
n
+
∑
n
=
1
∞
‍
B
¯
n
z
¯
n
,
∑
n
=
2
∞
‍
n
|
A
n
|
+
∑
n
=
1
∞
‍
n
|
B
n
|
≤
δ
}
.Theorem 11.
𝒮
¯
H
*
(
α
,
β
)
⊆
N
δ
(
e
), where
(35)
δ
=
(
1
-
β
)
(
2
α
+
1
)
(
α
+
1
)
(
2
α
+
1
-
β
)
.Proof.
Letf
∈
𝒮
¯
H
*
(
α
,
β
). Then, in view of (7), since α
n
(
n
-
1
)
+
n
-
β and α
n
(
n
+
1
)
+
n
+
β are increasing functions of n
(
n
≥
1
), we have
(36)
(
2
α
+
1
-
β
)
(
∑
n
=
2
∞
‍
|
a
n
|
+
∑
n
=
1
∞
‍
|
b
n
|
)
≤
(
2
α
+
2
-
β
)
∑
n
=
2
∞
‍
|
a
n
|
+
(
2
α
+
1
+
β
)
∑
n
=
1
∞
‍
|
b
n
|
≤
∑
n
=
2
∞
‍
(
α
n
(
n
-
1
)
+
n
-
β
)
|
a
n
|
+
∑
n
=
1
∞
‍
(
α
n
(
n
+
1
)
+
n
+
β
)
|
b
n
|
≤
(
1
-
β
)
,
which yields
(37)
∑
n
=
2
∞
‍
|
a
n
|
+
∑
n
=
1
∞
‍
|
b
n
|
≤
1
-
β
(
2
α
+
1
-
β
)
.
On the other hand, we also find from (7)
(38)
(
(
α
+
1
)
∑
n
=
2
∞
‍
n
|
a
n
|
-
β
∑
n
=
2
∞
‍
|
a
n
|
)
+
(
(
α
+
1
)
∑
n
=
1
∞
‍
n
|
b
n
|
-
β
∑
n
=
1
∞
‍
|
b
n
|
)
≤
∑
n
=
2
∞
‍
(
(
α
(
n
-
1
)
+
1
)
n
-
β
)
|
a
n
|
+
∑
n
=
1
∞
‍
(
(
α
(
n
+
1
)
+
1
)
n
+
β
)
|
b
n
|
≤
(
1
-
β
)
.
From (37) and (38), we obtain
(39)
(
α
+
1
)
(
∑
n
=
2
∞
‍
n
|
a
n
|
+
∑
n
=
1
∞
‍
n
|
b
n
|
)
≤
(
1
-
β
)
+
β
(
∑
n
=
2
∞
‍
|
a
n
|
+
∑
n
=
1
∞
‍
|
b
n
|
)
=
(
1
-
β
)
+
β
1
-
β
(
2
α
+
1
-
β
)
,
which is equivalent to
(40)
∑
n
=
2
∞
‍
n
|
a
n
|
+
∑
n
=
1
∞
‍
n
|
b
n
|
≤
(
1
-
β
)
(
2
α
+
1
)
(
α
+
1
)
(
2
α
+
1
-
β
)
=
δ
.


## 6. A Family of Class Preserving Integral Operator

In this section, we consider the closure property of the class𝒮
¯
H
*
(
α
,
β
) under the Bernardi integral operator F
(
z
), which is defined by
(41)
F
(
z
)
=
c
+
1
z
c
∫
0
z
t
c
-
1
h
(
t
)
d
t
+
c
+
1
z
c
∫
0
z
t
c
-
1
g
(
t
)
d
t
¯
F
(
z
)
=
c
+
1
z
c
∫
0
z
t
c
-
1
h
(
t
)
d
t
F
=
c
+
1
z
c
+
(
c
>
-
1
)
.Theorem 12.
Letf
=
h
+
g
¯ be in the class 𝒮
¯
H
*
(
α
,
β
), where h and g are given by (1). Then F
(
z
) defined by (41) also belongs to the class 𝒮
¯
H
*
(
α
,
β
).Proof.
From the representation ofF, it follows that
(42)
F
(
z
)
=
z
+
∑
n
=
2
∞
‍
c
+
1
c
+
n
a
n
z
n
+
∑
n
=
1
∞
‍
c
+
1
c
+
n
b
n
¯
z
¯
n
.
Now
(43)
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
(
c
+
1
c
+
n
|
a
n
|
)
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
(
c
+
1
c
+
n
|
b
n
|
)
≤
∑
n
=
2
∞
‍
α
n
(
n
-
1
)
+
n
-
β
1
-
β
|
a
n
|
+
∑
n
=
1
∞
‍
α
n
(
n
+
1
)
+
n
+
β
1
-
β
|
b
n
|
≤
1
by (7). Thus F
(
z
)
∈
𝒮
¯
H
*
(
α
,
β
).



---

*Source: 467929-2014-04-09.xml*