# Fekete-Szegö Inequalities for Starlike Functions with respect tok-Symmetric Points of Complex Order

Authors: M. K. Aouf; R. M. El-Ashwah; S. M. El-Deeb
Journal: Journal of Complex Analysis
 (2014)

---

## Abstract

Sharp upper bounds ofa
3
-
μ
a
2
2 for the function f
z
=
z
+
∑
m
=
2
∞
a
m
z
m belonging to certain subclass of starlike functions with respect to k-symmetric points of complex order are obtained. Also, applications of our results to certain functions defined through convolution  with a normalized analytic function are given. In particular, Fekete-Szegö inequalities for certain classes of functions defined through fractional derivatives are obtained.


---

## Body

## 1. Introduction

Let𝒜 denote the class of analytic functions of the following form:
(1)
f
(
z
)
=
z
+
∑
m
=
2
∞
‍
a
m
z
m
(
z
∈
U
=
{
z
:
z
∈
ℂ
,
|
z
|
<
1
}
)
.
And let 𝒮 be the subclass of 𝒜, which are univalent functions.Letg
(
z
)
∈
𝒮 be given by the following:
(2)
g
(
z
)
=
z
+
∑
m
=
2
∞
‍
g
m
z
m
.
The Hadamard product (or convolution) of f
(
z
) and g
(
z
) is given by
(3)
(
f
*
g
)
(
z
)
=
z
+
∑
m
=
2
∞
‍
a
m
b
m
z
m
=
(
g
*
f
)
(
z
)
.Iff and g are analytic functions in U, we say that f is subordinate to g, written f
≺
g if there exists a Schwarz function w, which is analytic in U with w
(
0
)
=
0 and |
w
(
z
)
|
<
1 for all z
∈
U, such that f
(
z
)
=
g
(
w
(
z
)
). Furthermore, if the function g is univalent in U, then we have the following equivalence (see [1, 2]):
(4)
f
(
z
)
≺
g
(
z
)
⟺
f
(
0
)
=
g
(
0
)
,
f
(
U
)
⊂
g
(
U
)
.Sakaguchi [3] introduced a class S
s
* of functions starlike with respect to symmetric points, which consists of functions f
(
z
)
∈
𝒮 satisfying the inequality
(5)
Re
{
2
z
f
′
(
z
)
f
(
z
)
-
f
(
-
z
)
}
>
0
(
z
∈
U
)
.Chand and Singh [4] introduced a class S
s
(
k
) of functions starlike with respect to k-symmetric points, which consists of functions f
(
z
)
∈
𝒮 satisfying the inequality
(6)
Re
{
z
f
′
(
z
)
f
k
(
z
)
}
>
0
(
z
∈
U
)
,
where
(7)
f
k
(
z
)
=
1
k
∑
ν
=
0
k
-
1
ϵ
-
ν
f
(
ϵ
ν
z
)
(
k
∈
ℕ
;
ϵ
k
=
1
)
.Al-Shaqsi and Darus [5] defined the linear operator D
λ
n
f
(
z
)
:
𝒜
→
𝒜 as follows:
(8)
D
λ
0
f
(
z
)
=
(
1
-
λ
)
f
(
z
)
+
λ
z
f
′
(
z
)
=
D
λ
f
(
z
)
,
D
λ
1
f
(
z
)
=
(
1
-
λ
)
z
f
′
(
z
)
+
λ
z
(
z
f
′
(
z
)
)
′
(
λ
≥
0
)
,
D
λ
2
f
(
z
)
=
z
2
!
[
(
1
-
λ
)
(
z
f
′
(
z
)
)
′′
+
λ
(
z
(
z
f
′
(
z
)
)
)
′′
]
=
z
2
!
[
z
(
D
λ
f
(
z
)
)
]
′′
(
λ
≥
0
)
,
⋮
D
λ
n
f
(
z
)
=
z
n
!
[
z
n
-
1
(
D
λ
f
(
z
)
)
]
(
n
)
(
λ
≥
0
;
n
∈
ℕ
)
,
and in general
(9)
D
λ
n
f
(
z
)
=
z
+
∑
m
=
2
∞
‍
[
1
+
λ
(
m
-
1
)
]
C
(
n
,
m
)
a
m
z
m
11111
(
λ
≥
0
;
n
∈
ℕ
0
=
ℕ
∪
{
0
}
;
ℕ
=
{
1,2
,
…
}
)
,
where
(10)
C
(
n
,
m
)
=
(
m
+
n
-
1
n
)
,
m
=
2,3
,
4
,
…
.In this paper, we define the following classS
s
(
k
)
(
b
,
λ
,
n
;
Φ
) (b
∈
ℂ
*
=
ℂ
∖
{
0
}) as follows.Definition 1.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯ be univalent starlike function with respect to 1 which maps the unit disk U onto a region in the right half plane which is symmetric with respect to the real axis. Let b
≠
0 be a complex number and let B
1
>
0. Then functions f
(
z
)
∈
𝒜 are in the class S
s
(
k
)
(
b
,
λ
,
n
;
Φ
) if
(11)
1
+
1
b
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
k
(
z
)
-
1
)
≺
Φ
(
z
)
(
b
∈
ℂ
*
;
n
∈
ℕ
0
;
λ
≥
0
;
z
∈
U
)
,
where D
λ
n
f
(
z
) is defined by (9) and f
k
(
z
) is defined by (7).We note that for suitable choices ofb, k, n, λ, and Φ
(
z
) we obtain the following subclasses:(i)
S
s
(
k
)
(
1,0
,
0
;
Φ
)
=
S
s
(
k
)
(
Φ
) (see Al-Shaqsi and Darus [6]),
(ii)
S
s
(
k
)
(
1,0
,
0
;
(
1
+
A
z
)
/
(
1
+
B
z
)
)
=
S
s
(
k
)
[
A
,
B
]
(
-
1
≤
B
<
A
≤
1
) (see Al-Shaqsi and Darus [6]),
(iii)
S
s
(
k
)
(
1,0
,
0
;
(
1
+
(
1
-
2
α
)
z
)
/
(
1
-
z
)
)
=
S
s
(
k
)
(
α
)
(
0
≤
α
<
1
) (see Al-Shaqsi and Darus [6]),
(iv)
S
s
(
2
)
(
1,0
,
0
;
Φ
)
=
S
s
*
(
Φ
) (see Shanmugam et al. [7]),
(v)
S
s
(
2
)
(
1,0
,
0
;
(
1
+
z
)
/
(
1
-
z
)
)
=
S
s
* (see Sakaguchi [3]),
(vi)
S
s
(
1
)
(
1
,
λ
,
n
;
Φ
)
=
M
λ
n
(
Φ
) (see Shanthi et al. [8] and Al-Shaqsi and Darus [9]),
(vii)
S
s
(
1
)
(
1,0
,
0
;
Φ
)
=
S
*
(
Φ
) (see Ma and Minda [10]),
(viii)
S
s
(
1
)
(
1,0
,
0
;
(
1
+
A
z
)
/
(
1
+
B
z
)
)
=
S
*
[
A
,
B
]
(
-
1
≤
B
<
A
≤
1
) (see Janowski [11]),
(ix)
S
s
(
1
)
(
b
,
0,0
;
Φ
)
=
S
b
*
(
Φ
) and S
s
(
1
)
(
b
,
0,1
;
Φ
)
=
C
b
(
Φ
) (see Ravichandran et al. [12]),
(x)
S
s
(
1
)
(
b
,
0,0
;
(
1
+
z
)
/
(
1
-
z
)
)
=
S
(
b
)
(
b
∈
ℂ
*
) (see Nasr and Aouf [13]),
(xi)
S
s
(
1
)
(
b
,
0,1
;
(
1
+
z
)
/
(
1
-
z
)
)
=
C
(
b
)
(
b
∈
ℂ
*
) (see Nasr and Aouf [14] and Aouf et al. [15]),
(xii)
S
s
(
1
)
(
(
1
-
ρ
)
e
-
i
λ
cos
⁡
λ
,
0,0
;
(
1
+
z
)
/
(
1
-
z
)
)
=
S
λ
(
ρ
)
(
|
λ
|
<
π
/
2
;
0
≤
ρ
<
1
) (see Libera [16]),
(xiii)
S
s
(
1
)
(
(
1
-
ρ
)
e
-
i
λ
cos
⁡
λ
,
0,1
;
(
1
+
z
)
/
(
1
-
z
)
)
=
C
λ
(
ρ
)
(
|
λ
|
<
π
/
2
;
0
≤
ρ
<
1
) (see Chichra [17]),
(xiv)
S
s
(
1
)
(
(
1
-
ρ
)
e
-
i
λ
cos
⁡
λ
,
0,0
;
Φ
)
=
S
ρ
λ
(
Φ
) and S
s
(
1
)
(
(
1
-
ρ
)
e
-
i
λ
cos
⁡
λ
,
0,1
;
Φ
)
=
C
ρ
λ
(
Φ
)
(
|
λ
|
<
π
/
2
;
0
≤
ρ
<
1
) (see Aouf and Silverman [18]),
(xv)
S
s
(
1
)
(
e
-
i
γ
cos
⁡
γ
,
0,0
;
(
1
+
(
1
-
2
δ
)
z
)
/
(
1
-
z
)
)
=
S
γ
(
δ
)
(
|
γ
|
<
π
/
2
,
0
≤
δ
<
1
) (see Keogh and Markes [19]).Also, we note the following:
(12)
(
i
)
S
s
(
1
)
(
b
,
λ
,
n
;
Φ
)
=
M
λ
n
(
b
;
Φ
)
=
{
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
1
)
f
(
z
)
∈
𝒜
:
1
+
1
b
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
1
)
≺
Φ
(
z
)
(
b
∈
ℂ
*
;
n
∈
ℕ
0
;
λ
≥
0
;
z
∈
U
)
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
1
)
}
;
(
ii
)
S
s
(
1
)
(
(
1
-
ρ
)
e
-
i
α
cos
⁡
α
,
λ
,
n
;
Φ
)
=
M
λ
n
(
ρ
,
α
;
Φ
)
=
{
e
i
α
(
z
(
D
λ
n
f
(
z
)
)
′
/
D
λ
n
f
(
z
)
)
-
ρ
cos
⁡
α
-
i
sin
α
(
1
-
ρ
)
cos
⁡
α
f
(
z
)
∈
𝒜
:
(
e
i
α
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
)
111111111111111111111
-
ρ
cos
⁡
α
-
i
sin
α
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
)
)
×
(
(
1
-
ρ
)
cos
⁡
α
)
-
1
≺
Φ
(
z
)
(
|
α
|
<
π
2
;
0
≤
ρ
<
1
;
n
∈
ℕ
0
;
λ
≥
0
;
z
∈
U
π
2
)
e
i
α
(
z
(
D
λ
n
f
(
z
)
)
′
/
D
λ
n
f
(
z
)
)
-
ρ
cos
⁡
α
-
i
sin
α
(
1
-
ρ
)
cos
⁡
α
}
;
(
iii
)
S
s
(
2
)
(
b
,
λ
,
n
;
Φ
)
=
S
s
*
(
b
,
λ
,
n
;
Φ
)
=
{
(
2
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
-
1
)
f
(
z
)
∈
𝒜
:
1
+
1
b
(
2
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
11111111111111111111111111
-
1
2
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
)
≺
Φ
(
z
)
(
b
∈
ℂ
*
;
n
∈
ℕ
0
;
λ
≥
0
;
z
∈
U
)
(
2
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
-
1
)
}
;
(
iv
)
S
s
(
2
)
(
(
1
-
ρ
)
e
-
i
α
cos
⁡
α
,
λ
,
n
;
Φ
)
=
S
s
*
(
ρ
,
α
,
λ
,
n
;
Φ
)
=
{
(
2
e
i
α
z
(
D
λ
n
f
(
z
)
)
′
/
(
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
)
)
-
ρ
cos
⁡
α
-
i
sin
α
(
1
-
ρ
)
cos
⁡
α
f
(
z
)
∈
𝒜
:
(
(
2
e
i
α
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
)
111111111111111111111
-
ρ
cos
⁡
α
-
i
sin
α
(
2
e
i
α
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
)
)
×
(
(
1
-
ρ
)
cos
⁡
α
)
-
1
≺
Φ
(
z
)
(
|
α
|
<
π
2
;
0
≤
ρ
<
1
;
n
∈
ℕ
0
;
λ
≥
0
;
z
∈
U
π
2
)
(
2
e
i
α
z
(
D
λ
n
f
(
z
)
)
′
/
(
D
λ
n
f
(
z
)
-
D
λ
n
f
(
-
z
)
)
)
-
ρ
cos
⁡
α
-
i
sin
α
(
1
-
ρ
)
cos
⁡
α
}
;
(
v
)
S
s
(
k
)
(
b
,
0,0
;
Φ
)
=
S
s
(
k
)
(
b
;
Φ
)
=
{
f
(
z
)
∈
𝒜
:
1
+
1
b
(
z
f
′
(
z
)
f
k
(
z
)
-
1
)
≺
Φ
(
z
)
(
b
∈
ℂ
;
z
∈
U
)
(
z
f
′
(
z
)
f
k
(
z
)
-
1
)
}
;
(
vi
)
S
s
(
k
)
(
(
1
-
ρ
)
e
-
i
α
cos
⁡
α
,
0,0
;
Φ
)
=
S
s
(
k
)
(
ρ
,
α
;
Φ
)
=
{
f
(
z
)
∈
𝒜
:
(
e
i
α
(
z
f
′
(
z
)
f
k
(
z
)
)
-
ρ
cos
⁡
α
-
i
sin
α
)
1111111111111111111111
×
(
(
1
-
ρ
)
cos
⁡
α
)
-
1
≺
Φ
(
z
)
(
|
α
|
<
π
2
;
0
≤
ρ
<
1
;
z
∈
U
)
(
e
i
α
(
z
f
′
(
z
)
f
k
(
z
)
)
-
ρ
cos
⁡
α
-
i
sin
α
)
}
.
In this paper, we obtain the Fekete-Szegö inequalities for the functions in the class S
s
(
k
)
(
b
,
λ
,
n
;
Φ
). We also give application of our results to certain functions defined through convolution and, in particular, we consider the class S
s
(
k
)
,
γ
(
b
,
λ
,
n
;
Φ
) defined by fractional derivatives.


## 2. Fekete-Szegö Problem

To prove our results, we need the following lemmas.Lemma 2 (see [10]).
Ifp
(
z
)
=
1
+
c
1
z
+
c
2
z
2
+
⋯ is an analytic function with positive real part in U and ν is a complex number, then
(13)
|
c
2
-
ν
c
1
2
|
≤
2
max
⁡
{
1
;
|
2
ν
-
1
|
}
.
The result is sharp for the functions given by
(14)
p
(
z
)
=
1
+
z
2
1
-
z
2
,
p
(
z
)
=
1
+
z
1
-
z
.Lemma 3 (see [10]).
Ifp
(
z
)
=
1
+
c
1
z
+
c
2
z
2
+
⋯ is an analytic function with positive real part in U
, then
(15)
|
c
2
-
ν
c
1
2
|
≤
{
-
4
ν
+
2
,
i
f
ν
≤
0
,
2
,
i
f
0
≤
ν
≤
1
,
4
ν
-
2
,
i
f
ν
≥
1
.
When ν
<
0 or ν
>
1, the equality holds if and only if p
(
z
)
=
(
1
+
z
)
/
(
1
-
z
) or one of its rotations. If  0
<
ν
<
1, then the equality holds if and only if p
(
z
)
=
(
1
+
z
2
)
/
(
1
-
z
2
) or one of its rotations. If ν
=
0
, the equality holds if and only if
(16)
p
(
z
)
=
(
1
2
+
1
2
ϑ
)
1
+
z
1
-
z
+
(
1
2
-
1
2
ϑ
)
1
-
z
1
+
z
(
0
≤
ϑ
≤
1
)
or one of its rotations. If ν
=
1, the equality holds if and only if
(17)
1
p
(
z
)
=
(
1
2
+
1
2
ϑ
)
1
+
z
1
-
z
+
(
1
2
-
1
2
ϑ
)
1
-
z
1
+
z
(
0
≤
ϑ
≤
1
)
.
Also the above upper bound is sharp and it can be improved as follows when 0
<
ν
<
1:
(18)
|
c
2
-
ν
c
1
2
|
+
ν
|
c
1
|
2
≤
2
(
0
<
ν
<
1
2
)
,
(19)
|
c
2
-
ν
c
1
2
|
+
(
1
-
ν
)
|
c
1
|
2
≤
2
(
1
2
<
ν
<
1
)
.Theorem 4.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
). If f
(
z
) given by (1) belongs to the class S
s
(
k
)
(
b
,
λ
,
n
;
Φ
) and b
∈
ℂ
*, then
(20)
|
a
3
-
μ
a
2
2
|
≤
2
|
b
|
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
·
max
⁡
{
1
,
|
B
2
B
1
+
b
B
1
ψ
2
(
2
-
ψ
2
)
-
(
(
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
)
-
1
(
μ
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
×
(
3
-
ψ
3
)
)
×
(
2
(
1
+
λ
)
2
×
(
n
+
1
)
(
2
-
ψ
2
)
2
)
-
1
)
B
2
B
1
+
b
B
1
ψ
2
(
2
-
ψ
2
)
|
}
,
where
(21)
ψ
m
=
1
k
∑
ν
=
0
k
-
1
ϵ
(
m
-
1
)
ν
,
(
k
∈
ℕ
;
m
≥
2
;
ϵ
k
=
1
)
.
This result is sharp.Proof.
Letf
(
z
)
∈
S
s
(
k
)
(
b
,
λ
,
n
;
Φ
); then there is a Schwarz function w
(
z
) in U with w
(
0
)
=
0 and |
w
(
z
)
|
<
1 in U such that
(22)
1
+
1
b
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
k
(
z
)
-
1
)
=
Φ
(
w
(
z
)
)
.
If the function p
1
(
z
) is analytic and has positive real part in U and p
1
(
0
)
=
1, then
(23)
p
1
(
z
)
=
1
+
w
(
z
)
1
-
w
(
z
)
=
1
+
c
1
z
+
c
2
z
2
+
⋯
.
Since w
(
z
) is a Schwarz function, define
(24)
p
(
z
)
=
1
+
1
b
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
k
(
z
)
-
1
)
=
1
+
d
1
z
+
d
2
z
2
+
⋯
111111111111
(
z
∈
U
)
.
From (7), we obtain
(25)
f
k
(
z
)
=
1
k
∑
ν
=
0
k
-
1
ϵ
-
ν
f
(
ϵ
ν
z
)
=
1
k
∑
ν
=
0
k
-
1
ϵ
-
ν
[
ϵ
ν
z
+
∑
m
=
2
∞
‍
a
m
(
ϵ
ν
z
)
m
]
=
z
+
∑
m
=
2
∞
‍
a
m
ψ
m
z
m
,
where ψ
m is given by (21). In view of (22) and (23), we have
(26)
p
(
z
)
=
Φ
(
p
1
(
z
)
-
1
p
1
(
z
)
+
1
)
.
Since
(27)
p
1
(
z
)
-
1
p
1
(
z
)
+
1
=
1
2
[
c
1
z
+
(
c
2
-
c
1
2
2
)
z
2
+
(
c
3
+
c
1
3
4
-
c
1
c
2
)
z
3
+
⋯
]
,
therefore, we have
(28)
Φ
(
p
1
(
z
)
-
1
p
1
(
z
)
+
1
)
=
1
+
1
2
B
1
c
1
z
+
[
1
2
B
1
(
c
2
-
c
1
2
2
)
+
1
4
B
2
c
1
2
]
z
2
+
⋯
,
and, from this equation and (28), we obtain
(29)
d
1
=
1
2
B
1
c
1
,
d
2
=
1
2
B
1
(
c
2
-
c
1
2
2
)
+
1
4
B
2
c
1
2
.
Then, from (24), we see that
(30)
d
1
=
(
1
+
λ
)
(
n
+
1
)
(
2
-
ψ
2
)
a
2
b
,
d
2
=
(
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
a
3
-
2
(
1
+
λ
)
2
(
n
+
1
)
2
(
2
-
ψ
2
)
ψ
2
a
2
2
)
×
(
2
b
)
-
1
.
Now from (24), (28), and (30), we have
(31)
a
2
=
b
B
1
c
1
2
(
1
+
λ
)
(
n
+
1
)
(
2
-
ψ
2
)
,
a
3
=
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
×
{
c
2
-
c
1
2
2
+
1
2
B
2
B
1
c
1
2
+
b
B
1
ψ
2
c
1
2
2
(
2
-
ψ
2
)
}
.
Therefore, we have
(32)
a
3
-
μ
a
2
2
=
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
{
c
2
-
ν
c
1
2
}
,
where
(33)
ν
=
1
2
[
1
-
B
2
B
1
-
b
B
1
ψ
2
(
2

-

ψ
2
)

+

μ
b
B
1
(
1

+

2
λ
)
(
n

+

2
)
(
3

-

ψ
3
)
2
(
1

+

λ
)
2
(
n

+

1
)
(
2

-

ψ
2
)
2
]
.
Our result now follows by an application of Lemma 2. The result is sharp for the functions
(34)
1
+
1
b
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
k
(
z
)
-
1
)
=
Φ
(
z
2
)
,
1
+
1
b
(
z
(
D
λ
n
f
(
z
)
)
′
D
λ
n
f
k
(
z
)
-
1
)
=
Φ
(
z
)
.
This completes the proof of Theorem 4.Remark 5.
(i) Puttingb
=
cos
⁡
γ
e
-
i
γ
(
|
γ
|
<
π
/
2
), n
=
λ
=
0, k
=
2, and Φ
(
z
)
=
(
1
+
(
1
-
2
δ
)
z
)
/
(
1
-
z
)
(
0
≤
δ
<
1
) in Theorem 4, we obtain the result obtained by Keogh and Markes [19, Theorem 1].
(ii) Puttingb
=
1 and k
=
2 in Theorem 4, we obtain the result obtained by Shanthi et al. [8, Theorem 2.6].
(iii) Puttingb
=
1, n
=
λ
=
0, and k
=
1 in Theorem 4, we obtain the result obtained by Ma and Minda [10].Puttingb
=
1 and k
=
1 in Theorem 4, we obtain the following corollary.Corollary 6.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯. If f
(
z
) given by (1) belongs to the class M
λ
n
(
Φ
), then
(35)
|
a
3
-
μ
a
2
2
|
≤
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
·
max
⁡
{
1
,
|
B
2
B
1
-
(
(
(
1
+
λ
)
2
(
n
+
1
)
)
-
1
(
(
1
+
λ
)
2
μ
(
1
+
2
λ
)
(
n
+
2
)
1111111111111111
-
(
1
+
λ
)
2
(
n
+
1
)
)
1111111111
×
(
(
1
+
λ
)
2
(
n
+
1
)
)
-
1
)
B
1
B
2
B
1
|
}
.
This result is sharp.Puttingb
=
1, n
=
λ
=
0, and k
=
2 in Theorem 4, we obtain the following corollary.Corollary 7.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯  (
B
1
>
0
). If f
(
z
) given by (1) belongs to the class S
s
*
(
Φ
), then
(36)
|
a
3
-
μ
a
2
2
|
≤
B
1
2
max
⁡
{
1
,
|
B
2
B
1
-
μ
B
1
2
|
}
.
This result is sharp.Puttingk
=
1 in Theorem 4, we obtain the following corollary.Corollary 8.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
)
. If f
(
z
) given by (1) belongs to the class M
λ
n
(
b
;
Φ
) and b
∈
ℂ
*, then
(37)
|
a
3
-
μ
a
2
2
|
≤
2
|
b
|
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
·
max
⁡
{
1
,
|
B
2
B
1
-
(
(
(
1
+
λ
)
2
(
n
+
1
)
)
-
1
(
μ
b
(
1
+
2
λ
)
(
n
+
2
)
1111111111111111
-
b
(
1
+
λ
)
2
(
n
+
1
)
)
1111111111
×
(
(
1
+
λ
)
2
(
n
+
1
)
)
-
1
)
B
1
B
2
B
1
|
}
.
This result is sharp.Puttingb
=
(
1
-
ρ
)
e
-
i
α
cos
⁡
α
(
|
α
|
<
π
/
2
;
0
≤
ρ
<
1
) in Corollary 8, we obtain the following corollary.Corollary 9.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
). If f
(
z
) given by (1) belongs to the class M
λ
n
(
ρ
,
α
;
Φ
), then
(38)
|
a
3
-
μ
a
2
2
|
≤
2
B
1
(
1
-
ρ
)
cos
⁡
α
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
·
max
⁡
{
1
,
|
B
2
B
1
-
(
(
(
1
+
λ
)
2
(
n
+
1
)
)
-
1
(
(
1
+
λ
)
2
μ
(
1
+
2
λ
)
(
n
+
2
)
1111111111111111
-
(
1
+
λ
)
2
(
n
+
1
)
)
1111111
111111
111
×
(
(
1
+
λ
)
2
(
n
+
1
)
)
-
1
)
B
1
1111111111
×
(
1
-
ρ
)
e
-
i
α
cos
⁡
α
B
2
B
1
|
}
.
This result is sharp.Puttingn
=
λ
=
0 in Theorem 4, we obtain the following corollary.Corollary 10.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
). If f
(
z
) given by (1) belongs to the class S
s
(
k
)
(
b
;
Φ
) and b
∈
ℂ
*, then
(39)
|
a
3
-
μ
a
2
2
|
≤
|
b
|
B
1
(
3
-
ψ
3
)
·
max
⁡
{
1
,
|
B
2
B
1
-
μ
(
3
-
ψ
3
)
-
(
2
-
ψ
2
)
ψ
2
(
2
-
ψ
2
)
2
b
B
1
|
}
.
This result is sharp.Puttingb
=
(
1
-
ρ
)
e
-
i
α
cos
⁡
α
(
|
α
|
<
π
/
2
;
0
≤
ρ
<
1
) in Corollary 10, we obtain the following corollary.Corollary 11.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
). If f
(
z
) given by (1) belongs to the class S
s
(
k
)
(
ρ
,
α
;
Φ
), then
(40)
|
a
3
-
μ
a
2
2
|
≤
B
1
(
1
-
ρ
)
cos
⁡
α
(
3
-
ψ
3
)
·
max
⁡
{
1
,
|
B
2
B
1
-
μ
(
3
-
ψ
3
)
-
(
2
-
ψ
2
)
ψ
2
(
2
-
ψ
2
)
2
B
1
11111111111
×
(
1
-
ρ
)
e
-
i
α
cos
⁡
α
μ
(
3
-
ψ
3
)
-
(
2
-
ψ
2
)
ψ
2
(
2
-
ψ
2
)
2
|
}
.
This result is sharp.By using Lemma3, we can obtain the following theorem.Theorem 12.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
)
. If f
(
z
) given by (1) belongs to the class S
s
(
k
)
(
b
,
λ
,
n
;
Φ
) and b
>
0, then
(41)
|
a
3
-
μ
a
2
2
|
≤
{
2
b
B
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
+
2
b
2
B
1
2
ψ
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
(
2
-
ψ
2
)
-
μ
b
2
B
1
2
(
1
+
λ
)
2
(
n
+
1
)
2
(
2
-
ψ
2
)
2
1111111111111
i
f
μ
≤
σ
1
,
2
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
1111111111111
i
f
σ
1
≤
μ
≤
σ
2
,
-
2
b
B
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
-
2
b
2
B
1
2
ψ
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
(
2
-
ψ
2
)
+
μ
b
2
B
1
2
(
1
+
λ
)
2
(
n
+
1
)
2
(
2
-
ψ
2
)
2
1111111111111
i
f
μ
≥
σ
2
,
where
(42)
σ
1
=
2
(
2

-

ψ
2
)
(
1

+

λ
)
2
(
n

+

1
)
[
(
B
2

-

B
1
)
(
2

-

ψ
2
)

+

b
B
1
2
ψ
2
]
b
B
1
2
(
1

+

2
λ
)
(
n

+

2
)
(
3

-

ψ
3
)
,
σ
2
=
2
(
2

-

ψ
2
)
(
1

+

λ
)
2
(
n

+

1
)
[
(
B
2

+

B
1
)
(
2

-

ψ
2
)

+

b
B
1
2
ψ
2
]
b
B
1
2
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
,
and ψ
m is given by (21). The result is sharp.Proof.
To show that the bounds are sharp, we define the functionsK
φ
δ
(
δ
≥
2
) by
(43)
1
+
1
b
(
z
(
D
λ
n
K
φ
δ
(
z
)
)
′
D
λ
n
K
φ
δ
(
z
)
-
1
)
=
Φ
(
z
δ
-
1
)
,
1111111111111
K
φ
δ
(
0
)
=
0
=
K
φ
δ
′
(
0
)
-
1
,
and the functions F
ρ and G
ρ
(
0
≤
ρ
≤
1
) by
(44)
1
+
1
b
(
z
(
D
λ
n
F
ρ
(
z
)
)
′
D
λ
n
F
ρ
(
z
)
-
1
)
=
ϕ
(
z
(
z
+
ρ
)
1
+
ρ
z
)
,
F
ρ
(
0
)
=
0
=
F
ρ
′
(
0
)
-
1
,
1
+
1
b
(
z
(
D
λ
n
G
ρ
(
z
)
)
′
D
λ
n
G
ρ
(
z
)
-
1
)
=
Φ
(
-
z
(
z
+
ρ
)
1
+
ρ
z
)
,
G
ρ
(
0
)
=
0
=
G
ρ
′
(
0
)
-
1
.
Cleary the functions K
φ
δ, F
ρ, and G
ρ
∈
S
s
(
k
)
(
b
,
λ
,
n
;
Φ
)
. Also we write K
φ
=
K
φ
2. If μ
<
σ
1 or μ
>
σ
2, then the equality holds if and only if f is K
φ or one of its rotations. When σ
1
<
μ
<
σ
2
, then the equality holds iff is K
φ
3 or one of its rotations. If μ
=
σ
1
, then the equality holds if and only if f is F
ρ or one of its rotations. If μ
=
σ
2
, then the equality holds if and only if f is G
ρ or one of its rotations. If σ
1
≤
μ
≤
σ
2, in view of Lemma 3. This completes the proof of Theorem 12.Remark 13.
(i) Putting=
1,  n
=
0, and λ
=
0 in Theorem 12, we obtain the result obtained by Al-Shaqsi and Darus [6, Theorem 2.1].
(ii) Puttingb
=
1,  n
=
0,  λ
=
0, and k
=
2 in Theorem 12, we obtain the result obtained by Shanmugam et al. [7, Theorem 2.1].
(iii) Puttingb
=
1 and k
=
2 in Theorem 12, we obtain the result obtained by Al-Shaqsi and Darus [9, Theorem 2.1].
(iv) Puttingb
=
1, n
=
λ
=
0, and k
=
1 in Theorem 12, we obtain the result obtained by Ma and Minda [10].Puttingk
=
1 in Theorem 12, we obtain the following corollary.Corollary 14.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯  (
B
1
>
0
). If f
(
z
) given by (1) belongs to the class M
λ
n
(
b
;
Φ
) and if b
>
0, then
(45)
|
a
3
-
μ
a
2
2
|
≤
{
b
B
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
+
b
2
B
1
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
-
μ
b
2
B
1
2
(
1
+
λ
)
2
(
n
+
1
)
2
11111111111
i
f
μ
≤
σ
1
*
,
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
11111111111
i
f
σ
1
*
≤
μ
≤
σ
2
*
,
-
b
B
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
-
b
2
B
1
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
+
μ
b
2
B
1
2
(
1
+
λ
)
2
(
n
+
1
)
2
11111111111
i
f
μ
≥
σ
2
*
,
where
(46)
σ
1
*
=
(
1
+
λ
)
2
(
n
+
1
)
[
B
2
-
B
1
+
b
B
1
2
]
b
B
1
2
(
1
+
2
λ
)
(
n
+
2
)
,
σ
2
*
=
(
1
+
λ
)
2
(
n
+
1
)
[
B
2
+
B
1
+
b
B
1
2
]
b
B
1
2
(
1
+
2
λ
)
(
n
+
2
)
.
The result is sharp.The proof of Theorem15 is similar to the proof of Theorem 12, so the details are omitted.Theorem 15.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
)
. Let f
(
z
) given by (1) belong to the class S
s
(
k
)
(
b
,
λ
,
n
;
Φ
) and b
>
0. Let σ
3 be given by
(47)
σ
3
=
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
[
B
2
B
1
+
b
B
1
ψ
2
(
2
-
ψ
2
)
]
,
where ψ
m is given by (21). If σ
1
<
μ
≤
σ
3, then
(48)
|
a
3
-
μ
a
2
2
|
+
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
×
[
1
-
B
2
B
1
-
b
B
1
ψ
2
(
2
-
ψ
2
)
+
μ
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
]
|
a
2
|
2
≤
2
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
.
If σ
3
<
μ
≤
σ
2, then
(49)
|
a
3
-
μ
a
2
2
|
+
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
×
[
1
+
B
2
B
1
+
b
B
1
ψ
2
(
2
-
ψ
2
)
-
μ
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
]
|
a
2
|
2
≤
2
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
,
where σ
1 and σ
2 are given in Theorem 12.Remark 16.
(i) Puttingb
=
1,  n
=
0, and λ
=
0 in Theorem 15, we obtain the result obtained by Al-Shaqsi and Darus [6, Remark 2.1].
(ii) Puttingb
=
1,  n
=
λ
=
0, and k
=
2 in Theorem 15, we obtain the result obtained by Shanmugam et al. [7, Theorem 2.2].
(iii) Puttingb
=
1 and k
=
2 in Theorem 15, we obtain the result obtained by Al-Shaqsi and Darus [9, Remark 2.2].Puttingk
=
1 in Theorem 15, we obtain the following corollary.Corollary 17.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯  (
B
1
>
0
)
. Let f
(
z
) given by (1) belong to the class M
λ
n
(
b
;
Φ
) and b
>
0. Let σ
3
* be given by
(50)
σ
3
*
=
(
1
+
λ
)
2
(
n
+
1
)
b
B
1
2
(
1
+
2
λ
)
(
n
+
2
)
[
B
2
+
b
B
1
2
]
.
If σ
1
*
<
μ
≤
σ
3
*, then
(51)
|
a
3
-
μ
a
2
2
|
+
(
1
+
λ
)
2
(
n
+
1
)
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
×
[
1
-
B
2
B
1
+
μ
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
-
b
B
1
(
1
+
λ
)
2
(
n
+
1
)
(
1
+
λ
)
2
(
n
+
1
)
]
×
|
a
2
|
2
≤
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
.
If σ
3
*
<
μ
≤
σ
2
*, then
(52)
|
a
3
-
μ
a
2
2
|
+
(
1
+
λ
)
2
(
n
+
1
)
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
×
[
1
+
B
2
B
1
-
μ
b
B
1
(
1
+
2
λ
)
(
n
+
2
)
-
b
B
1
(
1
+
λ
)
2
(
n
+
1
)
(
1
+
λ
)
2
(
n
+
1
)
]
×
|
a
2
|
2
≤
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
,
where σ
1
* and σ
2
* are given in Corollary 14.


## 3. Applications to Functions Defined by Fractional Derivatives

Definition 18 (see [20]).
Letf
(
z
) be analytic in a simply connected region of the z-plane containing origin. The fractional derivative of f of order γ is defined by
(53)
D
z
γ
f
(
z
)
=
1
Γ
(
1
-
γ
)
d
d
z
∫
0
z
(
z
-
ζ
)
-
γ
f
(
ζ
)
d
ζ
(
0
≤
γ
<
1
)
,
where the multiplicity of (
z
-
ζ
)
-
γ is removed by requiring that log
⁡
(
z
-
ζ
) be real for (
z
-
ζ
)
>
0.Using Definition18, Owa and Srivastava (see [21–23]) introduced a fractional derivative operator Ω
γ
:
𝒜
→
𝒜 defined by
(54)
Ω
γ
f
(
z
)
=
Γ
(
2
-
γ
)
z
γ
D
z
γ
f
(
z
)
,
(
γ
≠
2,3
,
4
,
…
)
.Definition 19.
A fixedg
(
z
) given by (2) and f
(
z
) given by (1) belong to the class S
s
(
k
)
,
g
(
b
,
λ
,
n
;
Φ
) if
(55)
1
+
1
b
(
z
(
D
λ
n
(
f
*
g
)
(
z
)
)
′
D
λ
n
(
f
k
*
g
)
(
z
)
-
1
)
≺
Φ
(
z
)
111
.
(
b
∈
ℂ
*
;
n
∈
ℕ
0
;
λ
≥
0
;
z
∈
U
)
,
where Φ
(
z
) is defined in Definition 1 and D
λ
n is defined in (9).The classS
s
(
k
)
,
γ
(
b
,
λ
,
n
;
Φ
) consists of the functions f
∈
𝒜 for which Ω
γ
f
∈
S
s
(
k
)
(
b
,
λ
,
n
;
Φ
). The class S
s
(
k
)
,
γ
(
b
,
λ
,
n
;
Φ
) is a special case of the class S
s
(
k
)
,
g
(
b
,
λ
,
n
;
Φ
) when
(56)
g
(
z
)
=
z
+
∑
m
=
2
∞
‍
Γ
(
m
+
1
)
Γ
(
2
-
γ
)
Γ
(
m
+
1
-
γ
)
z
m
(
z
∈
U
)
.
Now applying Theorem 4 for the function (
f
*
g
)
(
z
)
=
z
+
g
2
a
2
z
2
+
g
3
a
3
z
3
+
⋯, we get the following theorem.Theorem 20.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯  (
B
1
>
0
). If g
(
z
) given by (2) and f
(
z
) given by (1) belong to the class S
s
(
k
)
,
g
(
b
,
λ
,
n
;
Φ
) and if b
∈
ℂ
*, then
(57)
|
a
3
-
μ
a
2
2
|
≤
2
|
b
|
B
1
g
3
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
·
max
⁡
{
1
,
|
μ
b
B
1
g
3
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
2
g
2
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
B
2
B
1
+
b
B
1
ψ
2
(
2
-
ψ
2
)
-
μ
b
B
1
g
3
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
2
g
2
2
(
1
+
λ
)
2
(
n
+
1
)
(
2
-
ψ
2
)
2
|
}
,
where ψ
m is given by (21). This result is sharp.Proof.
The proof of Theorem20 is similar to the proof of Theorem 4, so the details are omitted.Puttingb
=
1 and n
=
λ
=
0 in Theorem 20, we obtain the following corollary.Corollary 21.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
). If g
(
z
) given by (2) and f
(
z
) given by (1) belong to the class S
s
(
k
)
,
g
(
Φ
), then
(58)
|
a
3
-
μ
a
2
2
|
≤
B
1
g
3
(
3
-
ψ
3
)
×
max
⁡
{
1
,
|
B
2
B
1
+
B
1
ψ
2
(
2
-
ψ
2
)
-
μ
B
1
g
3
(
3
-
ψ
3
)
g
2
2
(
2
-
ψ
2
)
2
|
}
,
where ψ
m is given by (21). This result is sharp.Puttingb
=
1, n
=
λ
=
0, and k
=
2 in Theorem 20, we obtain the following corollary.Corollary 22.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
). If g
(
z
) given by (2) and f
(
z
) given by (1) belong to the class S
s
*
,
g
(
Φ
), then
(59)
|
a
3
-
μ
a
2
2
|
≤
B
1
2
g
3
max
⁡
{
1
,
|
B
2
B
1
-
μ
g
3
B
1
2
g
2
2
|
}
.
This result is sharp.Now applying Theorem12 for the function (
f
*
g
)
(
z
)
=
z
+
g
2
a
2
z
2
+
g
3
a
3
z
3
+
⋯, we get the following theorem.Theorem 23.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
). If f
(
z
) given by (1) and g
(
z
) given by (2), then f
(
z
) belongs to the class S
s
(
k
)
,
g
(
b
,
λ
,
n
;
Φ
) and b
>
0, then(60)
|
a
3
-
μ
a
2
2
|
≤
{
2
b
B
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
g
3
+
2
b
2
B
1
2
ψ
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
(
2
-
ψ
2
)
g
3
-
μ
b
2
B
1
2
(
1
+
λ
)
2
(
n
+
1
)
2
(
2
-
ψ
2
)
2
g
2
2
111111111111111111111111
i
f
μ
≤
η
1
,
2
b
B
1
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
g
3
111111111111111111111111
i
f
η
1
≤
μ
≤
η
2
,
-
2
b
B
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
g
3
-
2
b
2
B
1
2
ψ
2
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
(
2
-
ψ
2
)
g
3
+
μ
b
2
B
1
2
(
1
+
λ
)
2
(
n
+
1
)
2
(
2
-
ψ
2
)
2
g
2
2
111111111111111111111111
i
f
μ
≥
η
2
,
where
(61)
η
1
=
(
2
g
2
2
(
2
-
ψ
2
)
(
1
+
λ
)
2
(
n
+
1
)
×
[
(
B
2
-
B
1
)
(
2
-
ψ
2
)
+
b
B
1
2
ψ
2
]
)
×
(
b
B
1
2
g
3
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
)
-
1
,
η
2
=
(
2
g
2
2
(
2
-
ψ
2
)
(
1
+
λ
)
2
(
n
+
1
)
×
[
(
B
2
+
B
1
)
(
2
-
ψ
2
)
+
b
B
1
2
ψ
2
]
)
×
(
b
B
1
2
g
3
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
)
-
1
,
and ψ
m is given by (21). The result is sharp.Remark 24.
(i) Puttingb
=
1,  n
=
0, and λ
=
0 in Theorem 23, we obtain the result obtained by Al-Shaqsi and Darus [6, Theorem 3.1].
(ii) Puttingb
=
1,  n
=
λ
=
0, and k
=
2 in Theorem 23, we obtain the result obtained by Shanmugam et al. [7, Theorem 3.2].
(iii) Puttingb
=
1 in Theorem 23, we obtain the result obtained by Al-Shaqsi and Darus [9, Theorem 3.2].
Since(62)
Ω
γ
f
(
z
)
=
z
+
∑
m
=
2
∞
‍
Γ
(
m
+
1
)
Γ
(
2
-
γ
)
Γ
(
m
+
1
-
γ
)
a
m
z
m
,
we have
(63)
g
2
=
Γ
(
3
)
Γ
(
2
-
γ
)
Γ
(
3
-
γ
)
=
2
2
-
γ
,
(64)
g
3
=
Γ
(
4
)
Γ
(
2
-
γ
)
Γ
(
4
-
γ
)
=
6
(
2
-
γ
)
(
3
-
γ
)
.
For g
2, g
3 given by (63) and (64), respectively, Theorem 23 reduces to the following theorem.Theorem 25.
LetΦ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯
(
B
1
>
0
) and γ
<
2
. If f
(
z
) given by (1) belongs to the class S
s
(
k
)
,
γ
(
b
,
λ
,
n
;
Φ
)and b
>
0, then
(65)
|
a
3
-
μ
a
2
2
|
≤
{
b
B
2
(
2
-
γ
)
(
3
-
γ
)
3
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
+
b
2
B
1
2
ψ
2
(
2
-
γ
)
(
3
-
γ
)
3
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
(
2
-
ψ
2
)
-
μ
b
2
B
1
2
(
2
-
γ
)
2
4
(
1
+
λ
)
2
(
n
+
1
)
2
(
2
-
ψ
2
)
2
1111111111111111111111
i
f
μ
≤
η
1
*
,
b
B
1
(
2
-
γ
)
(
3
-
γ
)
3
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
1111111111111111111111
i
f
η
1
*
≤
μ
≤
η
2
*
,
-
b
B
2
(
2
-
γ
)
(
3
-
γ
)
3
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
-
b
2
B
1
2
ψ
2
(
2
-
γ
)
(
3
-
γ
)
3
(
1
+
2
λ
)
(
n
+
1
)
(
n
+
2
)
(
3
-
ψ
3
)
(
2
-
ψ
2
)
+
μ
b
2
B
1
2
(
2
-
γ
)
2
4
(
1
+
λ
)
2
(
n
+
1
)
2
(
2
-
ψ
2
)
2
1111111111111111111111
i
f
μ
≥
η
2
*
,
where
(66)
η
1
*
=
(
4
(
3
-
γ
)
(
2
-
ψ
2
)
(
1
+
λ
)
2
(
n
+
1
)
×
[
(
B
2
-
B
1
)
(
2
-
ψ
2
)
+
b
B
1
2
ψ
2
]
)
×
(
3
(
2
-
γ
)
b
B
1
2
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
)
-
1
,
η
2
*
=
(
4
(
3
-
γ
)
(
2
-
ψ
2
)
(
1
+
λ
)
2
(
n
+
1
)
×
[
(
B
2
+
B
1
)
(
2
-
ψ
2
)
+
b
B
1
2
ψ
2
]
)
×
(
3
(
2
-
γ
)
b
B
1
2
(
1
+
2
λ
)
(
n
+
2
)
(
3
-
ψ
3
)
)
-
1
,
and ψ
m is given by (21). The result is sharp.Remark 26.
(i) Puttingb
=
1,  n
=
0, and λ
=
0 in Theorem 25, we obtain the result obtained by Al-Shaqsi and Darus [6, Theorem 3.3].
(ii) Puttingb
=
1,  n
=
λ
=
0, and k
=
2 in Theorem 25, we obtain the result obtained by Shanmugam et al. [7, Theorem 3.4].
(iii) Puttingb
=
1 in Theorem 25, we obtain the result obtained by Al-Shaqsi and Darus [9, Theorem 3.3].



---

*Source: 131475-2014-03-19.xml*