# A Subclass of Harmonic Functions Related to a Convolution Operator

Authors: Saqib Hussain; Akhter Rasheed; Maslina Darus
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We introduce a new subclass of harmonic functions by using a certain linear operator. For this class we derive coefficient bounds, extreme points, and inclusion results and also show that this class is closed under an integral operator.


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## Body

## 1. Introduction

Harmonic functions have long been used in the representation of minimal surface; for example, Heinz [1] in 1952 used such mappings in the study of the Gaussian curvature of nonparametric minimal surface over the unit disc. Such mappings have vast application in the field of engineering, physics, electronics, medicine, operations research, aerodynamics, and other branches of applied mathematics (see [2]).f
z
=
u
(
x
,
y
)
+
i
v
(
x
,
y
) is said to be complex valued harmonic function if both u and v are continuous and real harmonic; that is, u
x
x
+
u
y
y
=
0 and v
x
x
+
v
y
y
=
0. In simply connected domain [3], we can write f
z
=
h
z
+
g
z
¯, where h and g are analytic in E
=
z
:
z
<
1
. We call h the analytic part and g the coanalytic part of f. The necessary and sufficient condition for a function f
z
=
h
z
+
g
z
¯ to be univalent and sense preserving in E is h
′
z
>
g
′
z (see [3]). A function f
z
=
h
z
+
g
z
¯ is in class S
H if it is harmonic, univalent, and sense preserving in E, where h and g have the following series:(1)
h
z
=
z
+
∑
k
=
2
∞
a
k
z
k
,
g
z
=
∑
k
=
1
∞
b
k
z
k
,
b
1
<
1
.Hence(2)
f
z
=
z
+
∑
k
=
2
∞
a
k
z
k
+
∑
k
=
1
∞
b
k
z
k
¯
b
1
<
1
.Note that if the coanalytic partg is identically zero, then S
H reduces to well-known class of normalized univalent analytic functions S
. For this class, the function f
z may be expressed as(3)
f
z
=
z
+
∑
k
=
2
∞
a
k
z
k
.A functionf
z given by (2) is said to be harmonic starlike of order β
(
0
≤
β
<
1
) for |
z
|
=
r
<
1, if (4)
∂
∂
θ
arg
⁡
f
r
e
i
θ
>
β
,or equivalently (5)
R
e
z
h
′
z
-
z
g
′
z
¯
h
z
+
g
z
>
β
,
z
=
r
e
i
θ
.The class of all harmonic starlike functions of orderβ is denoted by S
H
∗
β
. This class was studied by Jahangiri [4]. The cases β
=
0 and b
1
=
0 were studied by Silverman and Silvia [5] and Silverman [6].ForF
z
=
z
+
∑
k
=
2
∞
‍
A
k
z
k
+
∑
k
=
1
∞
‍
B
k
z
k and f
z given by (2), the convolution is denoted by f
∗
F and defined as(6)
f
∗
F
=
z
+
∑
k
=
2
∞
a
k
A
k
z
k
+
∑
k
=
1
∞
b
k
B
k
z
k
.Letα
1
,
A
1
,
…
,
α
p
,
A
p and β
1
,
B
1
,
…
,
β
q
,
B
q
p
,
q
∈
N
=
{
1,2
,
3
,
…
} be positive real parameter such that(7)
1
+
∑
k
=
1
∞
B
k
-
∑
k
=
1
∞
A
k
≥
0
.The Wright generalized hypergeometric function [7] is defined as(8)
p
Ψ
q
α
1
,
A
1
,
…
,
α
p
,
A
p
;
β
1
,
B
1
,
…
,
β
q
,
B
q
;
z
=
p
Ψ
q
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
;
zwhich is defined by(9)
p
Ψ
q
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
;
z
=
∑
k
=
0
∞
Π
i
=
1
p
Γ
α
i
+
k
A
i
Π
i
=
1
q
Γ
β
i
+
k
B
i
z
k
k
!
z
∈
E
.IfA
i
=
1
i
=
1,2
,
…
,
p and B
i
=
1
i
=
1,2
,
…
,
q, we have the relationship(10)
Ω
p
Ψ
q
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
;
z
=
F
p
q
α
1
,
…
,
α
p
;
β
1
,
…
,
β
q
;
z
=
∑
k
=
0
∞
α
1
k
⋯
α
p
k
β
1
k
⋯
β
q
k
z
k
k
!
.p
≤
q
+
1
;
p
,
q
∈
N
0
=
N
∪
{
0
}
;
z
∈
E is the generalized hypergeometric function [8], where N denotes the set of all positive integers and (
α
)
k is the Pochhammer symbol and (11)
Ω
=
Π
i
=
1
p
Γ
α
i
Π
i
=
1
q
Γ
β
i
.Dziok and Srivastava [8] introduced a linear operator which is generalization of Dziok-Srivastava operator [8–10], Carlson-Shaffer operator [11], and the generalized Bernardi’s integral operator [12].Dziok and Raina [13] considered the linear operator W
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
:
S
→
S defined by (12)
W
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
f
z
=
p
Φ
q
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
;
z
∗
f
z
,where p
Φ
q
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
;
z is given by (13)
p
Φ
q
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
;
z
=
Ω
z
p
Ψ
q
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
;
z
.Forf
z
∈
S, we have(14)
W
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
f
z
=
z
+
∑
k
=
2
∞
Ω
σ
k
α
1
a
k
z
k
,where σ
k
α
1 is given by(15)
σ
k
α
1
=
Γ
α
1
+
A
1
k
-
1
⋯
Γ
α
p
+
A
p
k
-
1
Γ
β
1
+
β
1
k
-
1
⋯
Γ
β
q
+
β
q
k
-
1
k
-
1
!
.For our convenience we writeW
α
i
,
A
i
1
,
p
;
β
i
,
B
i
1
,
q
=
W
α
1
.Motivated by the work of [4, 14–21], we extend the work of Chandrashekar et al. [22] by introducing some new subclasses of S
H using the generalized hypergeometric function.Definition 1.
A functionf
∈
S
H is in class G
H
α
1
,
A
1
,
β
1
,
B
1
,
γ if(16)
R
e
1
+
1
+
e
i
ϕ
z
2
W
α
1
h
z
′
′
+
2
W
α
1
g
z
′
+
z
2
W
α
1
g
z
′
′
¯
z
W
α
1
h
z
′
-
z
W
g
z
′
¯
≥
γ
.Definition 2.
LetT
H denote the class of functions of the form(17)
f
z
=
z
-
∑
k
=
2
∞
a
k
z
k
-
∑
k
=
1
∞
b
k
z
k
¯
,
b
1
<
1
,and T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
=
G
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
∩
T
H.Throughout this paper, we shall assume0
≤
r
<
1, f
=
h
+
g
¯, Ω, and σ
k
α
1 as given in (11) and (15), respectively, unless otherwise mentioned.


## 2. Main Results

In Theorem3, we shall present a sufficient condition for f
∈
S
H to be in class G
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.Theorem 3.
Letf
=
h
+
g
¯ be given by (2). If(18)
∑
k
=
1
∞
k
2
k
-
1
-
γ
1
-
γ
a
k
+
2
k
+
1
+
γ
1
-
γ
b
k
σ
k
α
1
Ω
≤
2
,then f
∈
G
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.Proof.
When inequality (18) holds for the coefficients of f
=
h
+
g
¯ given in (2), we have to show that inequality (16) is satisfied. Arranging the left side inequality (16), we have (19)
R
e
z
W
α
1
h
z
′
+
1
+
e
i
ϕ
z
2
W
α
1
h
z
′
′
+
1
+
2
e
i
ϕ
z
W
α
1
g
z
′
¯
+
1
+
e
i
ϕ
z
2
W
α
1
g
z
′
′
¯
z
W
α
1
h
z
′
-
z
W
α
1
g
z
′
¯
≥
γ
=
R
e
A
z
B
z
.
As we knowR
e
w
≥
γ if and only if 1
-
γ
+
w
≥
1
+
γ
-
w, it is sufficient to show that(20)
A
z
+
1
-
γ
B
z
-
A
z
-
1
+
γ
B
z
≥
0
.
Substituting the valuesA
(
z
) and B
(
z
) in left side of (20), we obtain (21)
A
z
+
1
-
γ
B
z
-
A
z
-
1
+
γ
B
z
≥
2
-
γ
z
-
∑
k
=
2
∞
k
2
k
-
γ
a
k
σ
k
α
1
Ω
z
k
-
∑
k
=
1
∞
k
2
k
+
γ
b
k
σ
k
α
1
Ω
z
k
-
γ
z
-
∑
k
=
2
∞
k
2
k
-
2
-
γ
a
k
σ
k
α
1
Ω
z
k
-
∑
k
=
1
∞
k
2
k
+
2
+
γ
a
k
σ
k
α
1
Ω
z
k
≥
2
1
-
γ
1
-
∑
k
=
2
∞
k
2
k
-
1
-
γ
1
-
γ
a
k
σ
k
α
1
Ω
+
∑
k
=
1
∞
k
2
k
+
1
+
γ
1
-
γ
b
k
σ
k
α
1
Ω
=
2
1
-
γ
2
-
∑
k
=
1
∞
k
2
k
-
1
-
γ
1
-
γ
a
k
+
∑
k
=
1
∞
k
2
k
+
1
+
γ
1
-
γ
b
k
σ
k
α
1
Ω
≥
0
by hypothesis in (18), which implies that f
∈
G
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.Now we obtain the necessary and sufficient condition for the functionf
=
h
+
g
¯ given by (16) to be in T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.Theorem 4.
Letf
=
h
+
g
¯ be given by (16). Then f
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ if and only if(22)
∑
k
=
1
∞
k
2
k
-
1
-
γ
1
-
γ
a
k
+
2
k
+
1
+
γ
1
-
γ
b
k
σ
k
α
1
Ω
≤
2
.Proof.
SinceT
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
⊂
G
H
α
1
,
A
1
,
β
1
,
B
1
,
γ, we only have to prove the necessary part of theorem. Assume that f
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ, and then by virtue of (16), we obtain (23)
R
e
1
-
γ
+
1
+
e
i
ϕ
z
2
W
α
1
h
z
′
′
+
2
W
α
1
g
z
′
+
z
2
W
α
1
g
z
′
′
¯
z
W
α
1
h
z
′
-
z
W
α
1
g
z
′
¯
≥
0
.
This is equivalent to(24)
R
e
1
-
γ
z
-
∑
k
=
2
∞
k
k
1
+
e
i
ϕ
-
γ
-
e
i
ϕ
σ
k
α
1
Ω
a
k
z
k
+
∑
k
=
1
∞
k
k
1
+
e
i
ϕ
+
γ
+
e
i
ϕ
σ
k
α
1
Ω
b
k
z
¯
k
z
-
∑
k
=
2
∞
k
σ
k
α
1
Ω
a
k
z
k
-
∑
k
=
1
∞
k
σ
k
α
1
Ω
b
k
z
¯
k
=
R
e
1
-
γ
-
∑
k
=
2
∞
k
k
1
+
e
i
ϕ
-
γ
-
e
i
ϕ
σ
k
α
1
Ω
a
k
z
k
-
1
-
z
¯
/
z
∑
k
=
1
∞
k
k
1
+
e
i
ϕ
+
γ
+
e
i
ϕ
σ
k
α
1
Ω
b
k
z
¯
k
-
1
1
-
∑
k
=
2
∞
k
σ
k
α
1
Ω
a
k
z
k
-
1
+
z
¯
/
z
∑
k
=
1
∞
k
σ
k
α
1
Ω
b
k
z
¯
k
-
1
≥
0
.
This condition must hold for all values ofz
∈
E and for real ϕ, so that, by taking z
=
r
<
1 and ϕ
=
0, the above inequality reduces to(25)
1
-
γ
-
∑
k
=
2
∞
k
2
k
-
1
-
γ
σ
k
α
1
Ω
a
k
r
k
-
1
+
z
¯
/
z
∑
k
=
1
∞
k
2
k
+
1
+
γ
σ
k
α
1
Ω
b
k
r
k
-
1
1
-
∑
k
=
2
∞
k
σ
k
α
1
Ω
a
k
r
k
-
1
+
z
¯
/
z
∑
k
=
1
∞
k
σ
k
α
1
Ω
b
k
r
k
-
1
≥
0
.
Lettingr
→
1
- through real values, we obtain condition (22). This completes the proof.We determine the extreme points of closed convex hulls ofT
H
α
1
,
A
1
,
β
1
,
B
1
,
γ denoted by c
l
c
o
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.Theorem 5.
A functionf
(
z
)
∈
c
l
c
o
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ if and only if (26)
f
z
=
∑
k
=
1
∞
X
k
h
k
z
+
Y
k
g
k
z
,where (27)
h
1
z
=
z
,
h
k
z
=
z
-
1
-
γ
k
2
k
-
1
-
γ
σ
k
α
1
Ω
z
k
;
k
≥
2
,
g
k
z
=
z
-
1
-
γ
k
2
k
+
1
+
γ
σ
k
Ω
z
¯
k
;
k
≥
2
,
∑
k
=
1
∞
‍
X
k
+
Y
k
=
1, X
k
≥
0, and Y
k
≥
0
. In particular, the extreme points of T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ are h
k and g
k
.Proof.
First, we consider(28)
f
z
=
∑
k
=
1
∞
X
k
h
k
z
+
Y
k
g
k
z
=
∑
k
=
1
∞
X
k
+
Y
k
z
-
∑
k
=
2
∞
1
-
γ
k
2
k
-
1
-
γ
σ
k
α
1
Ω
X
k
z
k
-
∑
k
=
1
∞
1
-
γ
k
2
k
+
1
+
γ
σ
k
α
1
Ω
Y
k
z
¯
k
,
(29)
f
z
=
z
-
∑
k
=
1
∞
A
k
z
k
-
∑
k
=
1
∞
B
k
z
k
¯
,where A
k
=
1
-
γ
/
k
2
k
-
1
-
γ
σ
k
α
1
Ω
X
k, and B
k
=
1
-
γ
/
k
2
k
+
1
+
γ
σ
k
α
1
Ω
Y
k
.
Using (22) for the coefficients in (29), we have (30)
∑
k
=
2
∞
k
2
k
-
1
-
γ
σ
k
α
1
Ω
1
-
γ
A
k
+
∑
k
=
1
∞
k
2
k
+
1
+
γ
σ
k
α
1
Ω
1
-
γ
B
k
=
∑
k
=
2
∞
X
k
+
∑
k
=
1
∞
Y
k
=
1
-
x
1
≤
1
,and hence f
(
z
)
∈
c
l
c
o
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.
Conversely, suppose thatf
(
z
)
∈
c
l
c
o
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ, and set (31)
X
k
=
k
2
k
-
1
-
γ
σ
k
α
1
Ω
1
-
γ
A
k
,
k
≥
2
,
Y
k
=
k
2
k
+
1
+
γ
σ
k
α
1
Ω
1
-
γ
B
k
,
k
≥
1
,where ∑
k
=
1
∞
‍
X
k
+
Y
k
=
1
. Then (32)
f
z
=
z
-
∑
k
=
2
∞
A
k
z
k
-
∑
k
=
1
∞
B
k
z
¯
k
,
A
k
≥
0
,
B
k
≥
0
=
z
-
∑
k
=
2
∞
1
-
γ
k
2
k
-
1
-
γ
σ
k
α
1
Ω
X
k
z
k
-
∑
k
=
1
∞
1
-
γ
k
2
k
+
1
+
γ
σ
k
α
1
Ω
Y
k
z
¯
k
=
z
+
∑
k
=
2
∞
h
k
z
-
z
X
k
+
∑
k
=
1
∞
g
k
z
-
z
k
Y
k
=
∑
k
=
1
∞
X
k
h
k
z
+
Y
k
g
k
z
,which is the required result.Next we show thatT
H
α
1
,
A
1
,
β
1
,
B
1
,
γ is close under convex combinations of its members.Theorem 6.
The familyT
H
α
1
,
A
1
,
β
1
,
B
1
,
γ is closed under convex combination.Proof.
Fori
=
1,2
,
…, suppose that f
i
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ, where(33)
f
i
z
=
z
-
∑
k
=
2
∞
a
i
,
k
z
k
-
∑
k
=
2
∞
b
i
,
k
z
k
¯
.
For coefficients in (22) the relation given in (33) takes the form(34)
∑
k
=
2
∞
k
2
k
-
1
-
γ
σ
k
Ω
1
-
γ
a
i
,
k
+
∑
k
=
1
∞
k
2
k
+
1
+
γ
σ
k
Ω
1
-
γ
b
i
,
k
≤
1
.
For∑
i
=
1
∞
‍
t
i
=
1, 0
≤
t
i
≤
1, the convex combination of f
i may be written as(35)
∑
i
=
1
∞
t
i
f
i
z
=
z
-
∑
k
=
2
∞
∑
i
=
1
∞
t
i
a
i
,
k
z
k
-
∑
k
=
2
∞
∑
i
=
1
∞
t
i
b
i
,
k
¯
z
k
.
From inequality (22) for (35), (36)
∑
k
=
2
∞
k
2
k
-
1
-
γ
σ
k
α
1
Ω
1
-
γ
∑
i
=
1
∞
t
i
a
i
,
k
+
∑
k
=
1
∞
k
2
k
+
1
+
γ
σ
k
α
1
Ω
1
-
γ
∑
i
=
1
∞
t
i
a
i
,
k
∑
i
=
1
∞
t
i
∑
k
=
2
∞
k
2
k
-
1
-
γ
σ
k
α
1
Ω
1
-
γ
a
i
,
k
+
∑
k
=
1
∞
k
2
k
+
1
+
γ
σ
k
α
1
Ω
1
-
γ
b
i
,
k
≤
∑
t
=
1
∞
t
i
=
1
,and therefore ∑
t
=
1
∞
‍
t
i
f
i
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.Theorem 7.
For0
≤
δ
≤
γ
≤
1, let f
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ and F
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
δ
. Then (37)
f
z
∗
F
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
⊂
T
H
α
1
,
A
1
,
β
1
,
B
1
,
δ
.Proof.
Let(38)
f
z
=
z
-
∑
k
=
2
∞
a
k
z
k
-
∑
k
=
1
∞
b
k
z
k
¯
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
,
F
z
=
z
-
∑
k
=
2
∞
A
k
z
k
-
∑
k
=
1
∞
B
k
z
k
¯
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
δ
,and then(39)
f
z
∗
F
z
=
z
+
∑
k
=
2
∞
a
k
A
k
z
k
+
∑
k
=
1
∞
b
k
B
k
z
k
¯
.
We note thatA
k
≤
1 and B
k
≤
1. Therefore (40)
∑
k
=
2
∞
k
2
k
-
1
-
δ
σ
k
α
1
Ω
1
-
δ
a
k
A
k
+
∑
k
=
1
∞
k
2
k
+
1
+
δ
σ
k
α
1
Ω
1
-
δ
b
k
B
k
≤
∑
k
=
2
∞
k
2
k
-
1
-
δ
σ
k
α
1
Ω
1
-
δ
a
k
+
∑
k
=
1
∞
k
2
k
+
1
+
δ
σ
k
α
1
Ω
1
-
δ
b
k
≤
∑
k
=
2
∞
k
2
k
-
1
-
γ
σ
k
α
1
Ω
1
-
γ
a
k
+
∑
k
=
1
∞
k
2
k
+
1
+
γ
σ
k
α
1
Ω
1
-
γ
b
k
≤
1
,by using (22), since f
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ and 0
≤
δ
≤
γ
≤
1. This proves that f
z
∗
F
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
δ.Now for the classT
H
α
1
,
A
1
,
β
1
,
B
1
,
γ the closure property under the generalized Bernardi-Livingston integral operator L
c
f is examined which is defined by(41)
L
c
f
=
c
+
1
z
c
∫
0
z
t
c
-
1
f
t
d
t
,
c
>
-
1
.Theorem 8.
Letf
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ and then L
c
f
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.Proof.
Consider the generalized Bernardi-Livingston integral operatorL
c
f given in (41): (42)
L
c
f
=
c
+
1
z
c
∫
0
z
t
c
-
1
h
t
+
g
t
¯
d
t
=
c
+
1
z
c
∫
0
z
t
c
-
1
t
-
∑
k
=
2
∞
a
k
t
k
d
t
-
∫
0
z
t
c
-
1
∑
k
=
1
∞
b
k
t
k
¯
,
=
z
-
∑
k
=
2
∞
A
k
z
k
-
∑
k
=
1
∞
B
k
z
k
¯
,where (43)
A
k
=
c
+
1
c
+
n
a
k
;
B
k
=
c
+
1
c
+
n
b
k
.
Therefore(44)
∑
k
=
1
∞
k
2
k
-
1
-
γ
1
-
γ
c
+
1
c
+
n
a
k
+
2
k
+
1
+
γ
1
-
γ
c
+
1
c
+
n
b
k
σ
k
α
1
Ω
≤
∑
k
=
1
∞
k
2
k
-
1
-
γ
1
-
γ
a
k
+
2
k
+
1
+
γ
1
-
γ
b
k
σ
k
α
1
Ω
≤
2
1
-
γ
.
Sincef
z
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ, therefore, by Theorem 4, L
c
f
∈
T
H
α
1
,
A
1
,
β
1
,
B
1
,
γ
.



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*Source: 7123907-2016-09-29.xml*