# On Harmonic Functions Defined by Differential Operator with Respect tok-Symmetric Points

Authors: Afaf A. Ali Abubaker; Maslina Darus
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

We introduce new classesM
H
k
σ
,
s
(
λ
,
δ
,
α
) and M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
) of harmonic univalent functions with respect to k-symmetric points defined by differential operator. We determine a sufficient coefficient condition, representation theorem, and distortion theorem.


---

## Body

## 1. Introduction

A continuous functionf
=
u
+
i
v is a complex valued harmonic function in a complex domain C if both u and v are real harmonic in C. In any simply connected domain D
⊂
C we can write f
(
z
)
=
h
+
g
¯, where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |
h
′
(
z
)
|
>
|
g
′
(
z
)
| in D. See Clunie and Shell-Small (see [1]).Thus forf
=
h
+
g
¯
∈
S
*
H, we may write
(1)
h
(
z
)
=
z
+
∑
n
=
2
∞
‍
a
n
z
n
,
g
(
z
)
=
∑
n
=
1
∞
‍
b
n
z
n
,
|
b
1
|
<
1
.
Note that S
*
H reduces to S
*, the class of normalized analytic univalent functions if the coanalytic part of f
=
h
+
g
¯ is identically zero. Also, denote by S
H the subclasses of S
*
H consisting of functions f that map U onto starlike domain.A functionf is said to be starlike of order α in U denoted by S
H
(
α
) (see [2]) if
(2)
∂
∂
θ
(
arg
⁡
f
(
r
e
i
θ
)
)
=
Im
⁡
{
(
∂
/
∂
θ
)
f
(
r
e
i
θ
)
f
(
r
e
i
θ
)
}
=
R
{
z
h
′
(
z
)
-
z
g
′
(
z
)
¯
h
(
z
)
+
g
(
z
)
¯
}
≥
α
,
|
z
|
=
r
<
1
.
A function f of normalized univalent analytic functions is said to be starlike with respect to symmetrical points in U if it satisfies
(3)
R
{
z
f
′
(
z
)
f
(
z
)
-
f
(
-
z
)
}
>
0
,
z
∈
U
;
this class was introduced and studied by Sakaguchi in 1959 [3]. Some related classes are studied by Shanmugam et al. [4].In 1979, Chand and Singh [5] defined the class of starlike functions with respect to k-symmetric points of order α  (0
≤
α
<
1). Related classes are also studied by das and Singh [6]. Ahuja and Jahangiri [7] discussed the class S
H
(
α
) which denotes the class of complex-valued, sense-preserving, harmonic univalent functions f of the form (1) and satisfying the condition
(4)
Im
⁡
{
2
(
∂
/
∂
θ
)
f
(
r
e
i
θ
)
f
(
r
e
i
θ
)
-
f
(
-
r
e
i
θ
)
}
≥
α
.
In [8], the authors introduced and studied the class S
H
k
(
α
) which denotes the class of complex-valued, sense-preserving, harmonic univalent functions f of the form (1) and
(5)
h
k
(
z
)
=
z
+
∑
n
=
2
∞
‍
ϕ
n
a
n
z
n
,
g
k
(
z
)
=
∑
n
=
1
∞
‍
ϕ
n
b
n
z
n
,
|
b
1
|
<
1
,
where
(6)
ϕ
n
=
1
k
∑
ν
=
0
k
-
1
‍
ε
(
n
-
1
)
ν
,
(
k
≥
1
,
ε
k
=
1
)
.
From the definition of ϕ
n we know
(7)
ϕ
n
=
{
1
,
n
=
ι
k
+
1
,
0
,
n
≠
ι
k
+
1
,
(
n
≥
2
,
ι
,
k
≥
1
)
.
The differential operator D
λ
,
δ
σ
,
s was introduced by Ali Abubaker and Darus [9]. We define the differential operator of the harmonic function f
=
h
+
g
¯ given by (5) as
(8)
D
λ
,
δ
σ
,
s
f
(
z
)
=
D
λ
,
δ
σ
,
s
h
(
z
)
+
(
-
1
)
s
D
λ
,
δ
σ
,
s
g
(
z
)
¯
,
where
(9)
D
λ
,
δ
σ
,
s
h
(
z
)
=
z
+
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
a
n
z
n
,
D
λ
,
δ
σ
,
s
g
(
z
)
=
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
b
n
z
n
,
and also ψ
n
(
λ
,
δ
,
σ
,
s
)
=
n
s
(
C
(
δ
,
n
)
[
1
+
λ
(
n
-
1
)
]
)
σ, λ
≥
0, C
(
δ
,
n
)
=
(
δ
+
1
)
n
-
1
/
(
n
-
1
)
!, for δ
,
σ
,
s
∈
N
0
=
{
0,1
,
2
,
…
}, and (
x
)
n is the Pochhammer symbol defined by
(10)
(
x
)
n
=
Γ
(
x
+
n
)
Γ
(
x
)
=
{
1
,
n
=
0
x
(
x
+
1
)
⋯
(
x
+
n
-
1
)
,
n
=
{
1,2
,
3
,
…
}
.
We note that when s
=
0, σ
=
1, and λ
=
0 we obtain the Ruscheweyh derivative for harmonic functions (see [7]), when σ
=
0 we obtain the Salagean operator for harmonic functions (see [10]), and when σ
=
1, s
=
0 we obtain the operator for harmonic functions given by Al-Shaqsi and Darus [11].LetM
H
k
σ
,
s
(
λ
,
δ
,
α
) denote the class of complex-valued, sense-preserving, harmonic univalent functions f of the form (5) which satisfy the condition
(11)
Im
⁡
{
(
∂
/
∂
θ
)
D
λ
,
δ
σ
,
s
f
(
r
e
i
θ
)
D
λ
,
δ
σ
,
s
f
k
(
r
e
i
θ
)
}
=
R
{
z
(
D
λ
,
δ
σ
,
s
h
(
z
)
)
′
-
(
-
1
)
s
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
D
λ
,
δ
σ
,
s
h
k
(
z
)
+
(
-
1
)
s
D
λ
,
δ
σ
,
s
g
k
(
z
)
¯
}
≥
α
,
where z
=
r
e
i
θ, 0
≤
r
<
1, 0
≤
θ
<
π, 0
≤
α
<
1 and the functions D
λ
,
δ
σ
,
s
h
k and D
λ
,
δ
σ
,
s
g
k are of the form
(12)
D
λ
,
δ
σ
,
s
h
k
(
z
)
=
z
+
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
a
n
z
n
,
D
λ
,
δ
σ
,
s
g
k
(
z
)
=
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
b
n
z
n
.
Further, denote by M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
) the subclasses of M
H
k
σ
,
s
(
λ
,
δ
,
α
), such that the functions h and g in f
=
h
+
g
¯ are of the form
(13)
h
(
z
)
=
z
-
∑
n
=
2
∞
‍
|
a
n
|
z
n
,
g
(
z
)
=
∑
n
=
1
∞
‍
|
b
n
|
z
n
,
|
b
1
|
<
1
,
and the functions h
k and g
k in f
k
=
h
k
+
g
¯
k are of the form
(14)
h
k
(
z
)
=
z
-
∑
n
=
2
∞
‍
ϕ
n
|
a
n
|
z
n
,
g
k
(
z
)
=
∑
n
=
1
∞
‍
ϕ
n
|
b
n
|
z
n
,
|
b
1
|
<
1
.
In this paper, we obtain inclusion properties and coefficient conditions for the class M
H
k
σ
,
s
(
λ
,
δ
,
α
). A representation theorem and distortion bounds for the class M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
) are also established.Lemma 1 (see [12]).
Letf
=
h
+
g
¯
∈
S
H; if
(15)
∑
n
=
2
∞
‍
1
-
α
2
-
α
|
a
n
|
+
∑
n
=
1
∞
‍
1
+
α
2
-
α
|
b
n
|
≤
1
,
where 0
≤
α
<
1, then f is harmonic, sense-preserving, univalent in U and f is starlike harmonic of order α.


## 2. Main Results

First, we give a meaningful conclusion about the classM
H
k
σ
,
s
(
λ
,
δ
,
α
).Theorem 2.
Letf
∈
M
H
k
σ
,
s
(
λ
,
δ
,
α
), where f is given by (1); then f
k defined by (5) is in M
H
1
σ
,
s
(
λ
,
δ
,
α
)
=
M
H
σ
,
s
(
λ
,
δ
,
α
).Proof.
Letf
∈
M
H
k
σ
,
s
(
λ
,
δ
,
α
). Then substituting r
e
i
θ by ε
ν
r
e
i
θ, where ε
k
=
1 (ν
=
0,1
,
…
,
k
-
1) in (11), respectively, we have
(16)
Im
⁡
{
(
∂
/
∂
θ
)
D
λ
,
δ
σ
,
s
f
(
ε
ν
r
e
i
θ
)
D
λ
,
δ
σ
,
s
f
k
(
ε
ν
r
e
i
θ
)
}
≥
α
.
According to the definition of f
k and ε
k
=
1, we know f
k
(
ε
ν
r
e
i
θ
)
=
ε
ν
f
k
(
r
e
i
θ
) for any ν
=
0,1
,
…
,
k
-
1 and summing up we can get
(17)
Im
⁡
{
1
k
∑
ν
=
0
k
-
1
‍
(
∂
/
∂
θ
)
D
λ
,
δ
σ
,
s
f
(
ε
ν
r
e
i
θ
)
ε
ν
D
λ
,
δ
σ
,
s
f
k
(
r
e
i
θ
)
}
=
Im
⁡
{
(
∂
/
∂
θ
)
D
λ
,
δ
σ
,
s
f
k
(
r
e
i
θ
)
D
λ
,
δ
σ
,
s
f
k
(
r
e
i
θ
)
}
≥
α
;
that is, f
k
∈
M
H
σ
,
s
(
λ
,
δ
,
α
).Next, a sufficient coefficient condition for harmonic functions inM
H
σ
,
s
(
λ
,
δ
,
α
) is given.Theorem 3.
Letf
=
h
+
g
¯ with h and g given by (1) and f
k
=
h
k
+
g
¯
k with h
k and g
k given by (5). Let
(18)
∑
n
=
1
∞
‍
ψ
(
n
-
1
)
k
+
1
(
λ
,
δ
,
σ
,
s
)
[
(
n
-
1
)
k
-
α
ϕ
n
+
1
1
-
α
|
a
(
n
-
1
)
k
+
1
|
+
(
n
-
1
)
k
+
α
ϕ
n
+
1
1
-
α
|
b
(
n
-
1
)
k
+
1
|
]
+
∑
n
=
2
n
≠
ι
k
+
1
∞
n
ψ
n
(
λ
,
δ
,
σ
,
s
)
1
-
α
[
|
a
n
|
+
|
b
n
|
]
≤
2
,
where a
1
=
1, ι
≥
1, and λ
≥
0, for δ
,
σ
,
s
∈
N
0
=
{
0,1
,
2
,
…
} and k
≥
1. Then f is sense-preserving harmonic univalent in U and f
∈
M
H
k
σ
,
s
(
λ
,
δ
,
α
).Proof.
Since(19)
∑
n
=
1
∞
‍
[
n
-
α
1
-
α
|
a
n
|
+
n
+
α
1
-
α
|
b
n
|
]
≤
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
-
α
ϕ
n
1
-
α
|
a
n
|
+
n
+
α
ϕ
n
1
-
α
|
b
n
|
]
,
ϕ
n
=
1
k
∑
ν
=
0
k
-
1
‍
ε
(
n
-
1
)
ν
,
ε
k
=
1
,
=
∑
n
=
1
∞
‍
ψ
(
n
-
1
)
k
+
1
(
λ
,
δ
,
σ
,
s
)
×
[
(
n
-
1
)
k
-
α
ϕ
n
+
1
1
-
α
|
a
(
n
-
1
)
k
+
1
|
+
(
n
-
1
)
k
-
α
ϕ
n
+
1
1
-
α
|
b
(
n
-
1
)
k
+
1
|
]
+
∑
n
=
2
n
≠
ι
k
+
1
∞
n
ψ
n
(
λ
,
δ
,
σ
,
s
)
1
-
α
[
|
a
n
|
+
|
b
n
|
]
≤
2
,
by Lemma 1, we conclude that f is sense-preserving, harmonic univalent, and starlike in U. To prove f
∈
M
H
k
σ
,
s
(
λ
,
δ
,
α
), according to the condition (11), we need to show that
(20)
Im
⁡
{
(
∂
/
∂
θ
)
D
λ
,
δ
σ
,
s
f
(
r
e
i
θ
)
D
λ
,
δ
σ
,
s
f
k
(
r
e
i
θ
)
}
=
R
{
z
(
D
λ
,
δ
σ
,
s
h
(
z
)
)
′
-
(
-
1
)
s
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
D
λ
,
δ
σ
,
s
h
k
(
z
)
+
(
-
1
)
s
D
λ
,
δ
σ
,
s
g
k
(
z
)
¯
}
=
R
A
(
z
)
B
(
z
)
≥
α
,
A
(
z
)
=
z
(
D
λ
,
δ
σ
,
s
h
(
z
)
)
′
-
(
-
1
)
s
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
=
z
+
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
a
n
z
n
-
(
-
1
)
s
×
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
b
n
z
n
,
(21)
B
(
z
)
=
D
λ
,
δ
σ
,
s
h
k
(
z
)
+
(
-
1
)
s
D
λ
,
δ
σ
,
s
g
k
(
z
)
¯
=
z
+
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
a
n
z
n
+
(
-
1
)
s
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
b
n
z
n
,
where
(22)
ϕ
n
=
1
k
∑
ν
=
0
k
-
1
‍
ε
(
n
-
1
)
ν
,
ε
k
=
1
.
Using the fact thatR
(
ω
)
≥
α if and only if |
1
-
α
+
ω
|
≥
|
1
+
α
-
ω
|, it suffices to show that
(23)
|
A
(
z
)
+
(
1
-
α
)
B
(
z
)
|
-
|
A
(
z
)
-
(
1
+
α
)
B
(
z
)
|
≥
0
.
On the other hand, forA
(
z
) and B
(
z
) as given in (20) and (21), respectively, we have
(24)
|
A
(
z
)
+
(
1
-
α
)
B
(
z
)
|
-
|
A
(
z
)
-
(
1
+
α
)
B
(
z
)
|
=
|
[
(
1
-
α
)
D
λ
,
δ
σ
,
s
g
k
(
z
)
-
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
]
(
1
-
α
)
D
λ
,
δ
σ
,
s
h
k
(
z
)
+
z
(
D
λ
,
δ
σ
,
s
h
(
z
)
)
′
+
(
-
1
)
s
[
(
1
-
α
)
D
λ
,
δ
σ
,
s
g
k
(
z
)
-
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
]
|
-
|
[
(
1
-
α
)
D
λ
,
δ
σ
,
s
g
k
(
z
)
-
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
]
(
1
+
α
)
D
λ
,
δ
σ
,
s
h
k
(
z
)
-
z
(
D
λ
,
δ
σ
,
s
h
(
z
)
)
′
+
(
-
1
)
s
[
(
1
+
α
)
D
λ
,
δ
σ
,
s
g
k
(
z
)
+
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
]
|
=
|
[
(
1
-
α
)
D
λ
,
δ
σ
,
s
g
k
(
z
)
-
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
]
(
2
-
α
)
z
+
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
+
(
1
-
α
)
ϕ
n
]
a
n
z
n
-
(
-
1
)
s
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
-
(
1
-
α
)
ϕ
n
]
b
n
z
n
¯
|
-
|
[
(
1
-
α
)
D
λ
,
δ
σ
,
s
g
k
(
z
)
-
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
]
-
α
z
+
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
-
(
1
+
α
)
ϕ
n
]
a
n
z
n
-
(
-
1
)
s
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
+
(
1
+
α
)
ϕ
n
]
b
n
z
n
¯
|
≥
(
2
-
α
)
|
z
|
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
+
(
1
-
α
)
ϕ
n
]
|
a
n
|
|
z
|
n
-
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
+
(
1
-
α
)
ϕ
n
]
∥
b
n
∥
|
z
|
n
-
α
|
z
|
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
-
(
1
+
α
)
ϕ
n
]
|
a
n
|
|
z
|
n
-
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
+
(
1
+
α
)
ϕ
n
]
|
b
n
|
|
z
|
n
=
(
2
-
α
)
|
z
|
{
1
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
-
α
ϕ
n
1
-
α
]
|
a
n
|
|
z
|
n
-
1
-
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
+
α
ϕ
n
1
-
α
]
|
b
n
|
|
z
|
n
-
1
}
≥
2
(
1
-
α
)
{
1
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
-
α
ϕ
n
1
-
α
]
|
a
n
|
-
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
[
n
+
α
ϕ
n
1
-
α
]
|
b
n
|
}
≥
0
.
Note that, by substituting the value of ϕ
n given by (7) in the previous inequality above, then
(25)
|
A
(
z
)
+
(
1
-
α
)
B
(
z
)
|
-
|
A
(
z
)
-
(
1
+
α
)
B
(
z
)
|
≥
2
(
1
-
α
)
{
+
∑
n
=
2
n
≠
ι
k
+
1
∞
n
ψ
n
(
λ
,
δ
,
σ
,
s
)
1
-
α
[
|
a
n
|
+
|
b
n
|
]
1
-
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
×
[
(
n
-
1
)
k
-
α
+
1
1
-
α
|
a
(
n
-
1
)
k
+
1
|
-
(
n
-
1
)
k
+
α
+
1
1
-
α
|
b
(
n
-
1
)
k
+
1
|
]
+
∑
n
=
2
n
≠
ι
k
+
1
∞
n
ψ
n
(
λ
,
δ
,
σ
,
s
)
1
-
α
[
|
a
n
|
+
|
b
n
|
]
1
-
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
}
≥
0
,
by (18). Thus concludes the proof of the theorem.Next, the condition (18) is also necessary for functions in M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
), which is clarified.Theorem 4.
Letf
=
h
+
g
¯ with h and g given by (13) and f
k
=
h
k
+
g
¯
k with h
k and g
k given by (14). Then f
∈
M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
) if and only if
(26)
∑
n
=
1
∞
‍
ψ
(
n
-
1
)
k
+
1
(
λ
,
δ
,
ρ
,
s
)
×
[
(
n
-
1
)
k
-
α
ϕ
n
+
1
1
-
α
|
a
(
n
-
1
)
k
+
1
|
+
(
n
-
1
)
k
+
α
ϕ
n
+
1
1
-
α
|
b
(
n
-
1
)
k
+
1
|
]
+
∑
n
=
2
n
≠
ι
k
+
1
∞
n
ψ
n
(
λ
,
δ
,
σ
,
s
)
1
-
α
[
|
a
n
|
+
|
b
n
|
]
≤
2
,
where a
1
=
1, ι
≥
1, and λ
≥
0, for δ
,
σ
,
s
∈
N
0
=
{
0,1
,
2
,
…
} and k
≥
1 given by (6).Proof.
The if part follows from Theorem3 upon noting that if the analytic and coanalytic parts of f
=
h
+
g
¯
∈
M
H
k
σ
,
s
(
λ
,
δ
,
α
) are of the form (13), then f
∈
M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
). For the only if part, we show that f
∉
M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
), if the condition (26) does not hold. Thus we can write
(27)
R
{
z
(
D
λ
,
δ
σ
,
s
h
(
z
)
)
′
-
(
-
1
)
s
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
D
λ
,
δ
σ
,
s
h
k
(
z
)
+
(
-
1
)
s
D
λ
,
δ
σ
,
s
g
k
(
z
)
¯
}
≥
α
;
this is equivalent to
(28)
0
≤
R
{
z
(
D
λ
,
δ
σ
,
s
h
(
z
)
)
′
-
(
-
1
)
s
z
(
D
λ
,
δ
σ
,
s
g
(
z
)
)
′
¯
D
λ
,
δ
σ
,
s
h
k
(
z
)
+
(
-
1
)
s
D
λ
,
δ
σ
,
s
g
k
(
z
)
¯
}
-
α
.
That is,
(29)
R
{
(
+
(
-
1
)
s
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
|
b
n
|
z
n
¯
)
-
1
(
1
-
α
)
z
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
(
n
-
α
ϕ
n
)
|
a
n
|
z
n
-
(
-
1
)
s
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
(
n
+
α
ϕ
n
)
|
b
n
|
z
n
¯
)
×
(
z
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
|
a
n
|
z
n
+
(
-
1
)
s
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
|
b
n
|
z
n
¯
)
-
1
}
≥
0
.
The above-required condition must hold for all values of z, |
z
|
=
r
<
1. Upon choosing the values of z on the positive real axis where 0
≤
z
=
r
<
1, we must have
(30)
(
(
1
-
α
)
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
(
n
-
α
ϕ
n
)
|
a
n
|
r
n
-
1
-
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
(
n
+
α
ϕ
n
)
|
b
n
|
r
n
-
1
)
×
(
1
-
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
|
a
n
|
r
n
-
1
+
∑
n
=
1
∞
‍
ψ
n
(
λ
,
δ
,
σ
,
s
)
ϕ
n
|
b
n
|
r
n
-
1
)
-
1
≥
0
.
If the condition (26) does not hold, then the numerator in (30) is negative for r sufficiently close to 1. Hence there exists a z
0
=
r
0 in (
0,1
) for which the quotient in (30) is negative. This contradicts the required condition for f
∈
M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
) and the proof is complete.Now, the distortion result is given.Theorem 5.
Iff
∈
M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
), then
(31)
|
f
(
z
)
|
≥
(
1
-
|
b
1
|
)
-
1
ψ
2
(
λ
,
δ
,
σ
,
s
)
[
1
-
α
2
-
α
-
1
+
α
2
-
α
|
b
1
|
]
r
2
,
|
f
(
z
)
|
≤
(
1
+
|
b
1
|
)
+
1
ψ
2
(
λ
,
δ
,
σ
,
s
)
[
1
-
α
2
-
α
-
1
+
α
2
-
α
|
b
1
|
]
r
2
.Proof.
We will only prove the left-hand inequality of the above theorem. The arguments for the right-hand inequality are similar and so we omit it. Letf
∈
M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
). Taking the absolute value of f
(
z
) we obtain
(32)
|
f
(
z
)
|
≥
(
1
-
|
b
1
|
)
r
-
∑
n
=
2
∞
‍
[
|
a
n
|
+
|
b
n
|
]
r
n
≥
(
1
-
|
b
1
|
)
r
-
∑
n
=
2
∞
‍
[
|
a
n
|
+
|
b
n
|
]
r
2
≥
(
1
-
|
b
1
|
)
r
-
1
-
α
ψ
2
(
λ
,
δ
,
ρ
,
s
)
(
2
-
α
ϕ
2
)
×
∑
n
=
2
∞
‍
ψ
2
(
λ
,
δ
,
ρ
,
s
)
(
2
-
α
ϕ
2
)
1
-
α
[
|
a
n
|
+
|
b
n
|
]
r
2
≥
(
1
-
|
b
1
|
)
r
-
1
-
α
2
-
α
∑
n
=
2
∞
‍
ψ
n
(
λ
,
δ
,
ρ
,
s
)
×
[
(
n
-
α
ϕ
n
)
1
-
α
|
a
n
|
+
(
n
+
α
ϕ
n
)
1
-
α
|
b
n
|
]
r
2
by (26):
(33)
=
(
1
-
|
b
1
|
)
-
1
ψ
2
(
λ
,
δ
,
σ
,
s
)
[
1
-
α
2
-
α
-
1
+
α
2
-
α
|
b
1
|
]
r
2
.
The following covering result follows from the left-hand inequality in Theorem 5.Corollary 6.
Iff
∈
M
¯
H
k
σ
,
s
(
λ
,
δ
,
α
) then
(34)
{
ω
:
|
ω
|
<
2
ψ
2
-
1
-
(
ψ
2
(
λ
,
δ
,
ρ
,
s
)
-
1
)
α
ψ
2
(
λ
,
δ
,
ρ
,
s
)
(
2
-
α
)
-
2
ψ
2
(
λ
,
δ
,
ρ
,
s
)
-
1
-
(
ψ
2
(
λ
,
δ
,
ρ
,
s
)
+
1
)
α
ψ
2
(
λ
,
δ
,
ρ
,
s
)
(
2
-
α
)
|
b
1
|
}
⊂
f
(
U
)
.Note that other work related to Sakaguchi and classes of functions with respect to symmetric points can be found in [13–16].



---

*Source: 628972-2014-07-23.xml*