1. Introduction
A continuous function
f
=
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 for
f
=
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 function
f
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].
Let
M
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]).
Let
f
=
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 class
M
H
k
σ
,
s
(
λ
,
δ
,
α
)
.
Theorem 2.
Let
f
∈
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.
Let
f
∈
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 in
M
H
σ
,
s
(
λ
,
δ
,
α
)
is given.
Theorem 3.
Let
f
=
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 that
R
(
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, for
A
(
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.
Let
f
=
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 Theorem 3 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.
If
f
∈
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. Let
f
∈
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.
If
f
∈
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].