# On Certain Subclasses of Analytic Functions Involving Carlson-Shaffer Operator and Related to Lemniscate of Bernoulli

Authors: Jagannath Patel; Ashok Kumar Sahoo
Journal: Journal of Complex Analysis
 (2014)

---

## Abstract

The object of the present investigation is to solve the Fekete-Szegö problem and determine the sharp upper bound to the second Hankel determinant for a new classR
(
a
,
c
) of analytic functions involving the Carlson-Shaffer operator in the unit disk. We also obtain a sufficient condition for normalized analytic functions in the unit disk to be in this class.


---

## Body

## 1. Introduction and Preliminaries

LetA be the class of functions f of the form
(1)
f
(
z
)
=
z
+
∑
k
=
2
∞
‍
a
k
z
k
which are analytic in the open unit disk U
=
{
z
∈
C
:
|
z
|
<
1
}.A functionf
∈
A is said to be starlike of order ρ, if
(2)
Re
z
f
′
(
z
)
f
(
z
)
>
ρ
(
0
≤
ρ
<
1
;
z
∈
U
)
.Similarly, a functionf
∈
A is said to be convex of order ρ, if
(3)
Re
1
+
z
f
′
′
(
z
)
f
′
(
z
)
>
ρ
(
0
≤
ρ
<
1
;
z
∈
U
)
.
By usual notations, we write these classes of functions by S
⋆
(
ρ
) and K
(
ρ
), respectively. We denote S
⋆
(
0
)
=
S
⋆ and K
(
0
)
=
K, the familiar subclasses of starlike, convex functions in U.Furthermore, letP denote the class of analytic functions ϕ normalized by
(4)
ϕ
z
=
1
+
p
1
z
+
p
2
z
2
+
⋯
(
z
∈
U
)
such that Re
{
ϕ
(
z
)
}
>
0 in U.For functionsf and g, analytic in U, we say that f is subordinate to g, written as f
≺
g or f
(
z
)
≺
g
(
z
)
(
z
∈
U
), if there exists a Schwarz function ω, which (by definition) is analytic in U with ω
(
0
)
=
0, |
ω
(
z
)
|
<
1, and f
(
z
)
=
g
(
ω
(
z
)
), z
∈
U. Furthermore, if the function g is univalent in U, then we have the following equivalence relation (cf., e.g., [1]; see also [2]):
(5)
f
z
≺
g
z
⟺
f
0
=
g
0
,
f
(
U
)
⊂
g
(
U
)
.For functionsf
j
(
z
)
=
∑
k
=
0
∞
‍
a
k
,
j
z
k
(
j
=
1,2
) analytic in U, we define the Hadamard product (or convolution) of f
1 and f
2 by
(6)
(
f
1
⋆
f
2
)
(
z
)
=
∑
k
=
0
∞
‍
a
k
,
1
a
k
,
2
z
k
=
(
f
2
⋆
f
1
)
(
z
)
(
z
∈
U
)
.
Note that (
f
1
⋆
f
2
) is also analytic in U.Carlson and Shaffer [3] defined the linear operator L
(
a
,
c
)
:
A
→
A in terms of the incomplete beta function φ by
(7)
L
(
a
,
c
)
f
(
z
)
=
φ
(
a
,
c
;
z
)
⋆
f
(
z
)
(
f
∈
A
;
z
∈
U
)
,
where
(8)
φ
a
,
c
;
z
=
∑
k
=
0
∞
‍
a
k
c
k
z
k
+
1
a
∈
C
,
c
∈
C
∖
Z
0
-
,
Z
0
-
=
…
,
-
2
,
-
1,0
;
z
∈
U
and ϰ
n denotes the Pochhammer symbol (or shifted factorial) given, in terms of the Gamma function Γ, by
(9)
ϰ
n
=
Γ
(
ϰ
+
n
)
Γ
(
ϰ
)
=
ϰ
ϰ
+
1
ϰ
+
2
⋯
(
ϰ
+
n
-
1
)
,
n
∈
N
1
,
n
=
0
.Iff
∈
A is given by (1), then it follows from (7) that
(10)
L
a
,
c
f
z
=
z
+
∑
k
=
1
∞
‍
a
k
c
n
a
k
+
1
z
k
+
1
z
∈
U
,
(11)
z
L
(
a
,
c
)
f
′
(
z
)
=
a
L
(
a
+
1
,
c
)
f
(
z
)
-
(
a
-
1
)
L
(
a
,
c
)
f
(
z
)
hhhhhhhhhhhhhhhhhhhhhhhhhh
(
z
∈
U
)
.
We note that for f
∈
A
(i)
L
(
a
,
a
)
f
(
z
)
=
f
(
z
);
(ii)
L
(
2,1
)
f
(
z
)
=
z
f
′
(
z
);
(iii)
L
(
3,1
)
f
(
z
)
=
z
f
′
(
z
)
+
(
1
/
2
)
z
2
f
′
′
(
z
);
(iv)
L
(
m
+
1,1
)
f
(
z
)
=
D
m
f
(
z
)
=
(
z
/
1
-
z
m
+
1
)
⋆
f
(
z
)
(
m
∈
Z
;
m
>
-
1
), the well-known Ruscheweyh derivative [4] of f;
(v)
L
(
2,2
-
μ
)
f
(
z
)
=
Ω
z
μ
f
(
z
)
(
0
≤
μ
<
1
;
z
∈
U
), the well-known Owa-Srivastava fractional differential operator [5]. We also observe that Ω
z
0
f
(
z
)
=
f
(
z
) and Ω
z
1
f
(
z
)
=
lim
⁡
z
→
1
Ω
z
μ
f
(
z
)
=
z
f
′
(
z
).With the aid of the linear operatorL
(
a
,
c
), we introduce a subclass of A as follows.Definition 1.
A functionf
∈
A is said to be in the class R
(
a
,
c
), if it satisfies the condition
(12)
L
(
a
,
c
)
f
(
z
)
z
2
-
1
<
1
(
z
∈
U
)
.It follows from (12) and the definition of subordination that a function f
∈
R
(
a
,
c
) satisfies the following subordination relation:
(13)
L
(
a
,
c
)
f
(
z
)
z
≺
1
+
z
(
z
∈
U
)
.
We further note that if f
∈
R
(
a
,
c
), then the function L
(
a
,
c
)
f
(
z
)
/
z lies in the region bounded by the right half of the lemniscate of Bernoulli given by
(14)
ω
∈
C
:
ω
2
-
1
<
1
=
u
2
+
v
2
2
u
+
i
v
:
u
2
+
v
2
2
=
2
(
u
2
-
v
2
)
.Noonan and Thomas [6] defined the qth Hankel determinant of a sequence a
k
,
a
k
+
1
,
a
k
+
2
,
… of real or complex numbers by
(15)
H
q
(
k
)
=
a
k
a
k
+
1
⋯
a
k
+
q
-
1
a
k
+
1
a
k
+
2
⋯
a
k
+
q
⋮
⋮
⋮
⋮
a
k
+
q
-
1
a
k
+
q
⋯
a
k
+
2
q
-
2
(
k
,
q
∈
N
)
.
This determinant has been studied by several authors including Noor [7] with the subject of inquiry ranging from the rate of growth of H
q
(
k
) (as k
→
∞) to the determination of precise bounds with specific values of k and q for certain subclasses of analytic functions in the unit disc U.Fork
=
1, q
=
2, a
1
=
1, and k
=
q
=
2, the Hankel determinant simplifies to
(16)
H
2
(
1
)
=
a
3
-
a
2
2
,
H
2
(
2
)
=
a
2
a
4
-
a
3
2
.The Hankel determinantH
2
(
1
) was considered by Fekete and Szegö [8] and we refer to H
2
(
2
) as the second Hankel determinant. It is known [9] that if f given by (1) is analytic and univalent in U, then the sharp inequality H
2
(
1
)
=
|
a
3
-
a
2
2
|
≤
1 holds. For a family F of functions in A of the form (1), the more general problem of finding the sharp upper bounds for the functionals |
a
3
-
λ
a
2
2
|
(
λ
∈
C
or
λ
∈
R
) is popularly known as Fekete-Szegö problem for the class F. The Fekete-Szegö problem for various known subclasses of univalent functions (i.e., starlike, convex, close-to-convex, etc.) has been completely settled [8, 10–12]. Recently, Janteng et al. [13, 14] have obtained the sharp upper bounds to the second Hankel determinant H
2
(
2
) for the family of functions in A whose derivatives have positive real part in U. For initial work on this class of functions, one may refer to the work of MacGregor [15].In our present investigation, we follow the techniques adopted by Libera and Złotkiewicz [16, 17] to solve the Fekete-Szegö problem and also determine the sharp upper bound to the second Hankel determinant for the class R
(
a
,
c
).To establish our main results, we will need the following lemma for functions belonging to the classP.Lemma 2.
Let the functionϕ, given by (4), be a member of the class P. Then
(17)
|
p
k
|
≤
2
(
k
≥
1
)
,
(18)
p
2
-
ν
p
1
2
≤
2
max
⁡
{
1
,
2
ν
-
1
}
(
ν
∈
C
)
,
(19)
p
2
=
1
2
p
1
2
+
(
4
-
p
1
2
)
x
,
p
3
=
1
4
p
1
3
+
2
(
4
-
p
1
2
)
p
1
x
h
h
i
-
4
-
p
1
2
p
1
x
2
+
2
4
-
p
1
2
1
-
x
2
z
,
for some complex numbers x
,
z satisfying |
x
|
≤
1 and |
z
|
≤
1. The estimates in (17) and (18) are sharp for the functions defined in U by
(20)
f
z
=
1
+
z
1
-
z
,
g
(
z
)
=
1
+
z
2
1
-
z
2
.We note that the estimate (17) is contained in [9]; the estimate (18) is due to Ma and Minda [18], whereas the results in (19) are obtained by Libera and Złotkiewicz [17] (see also [16]).


## 2. Main Results

Unless otherwise mentioned, we assume throughout the sequel thata
≥
c
>
0.Now, we solve the Fekete-Szegö problem for the classR
(
a
,
c
).Theorem 3.
If the functionf, given by (1), belongs to the class R
(
a
,
c
), then for any λ
∈
C
(21)
a
3
-
λ
a
2
2
≤
1
2
c
2
a
2
max
⁡
1
,
|
a
(
c
+
1
)
+
2
λ
c
(
a
+
1
)
|
4
a
(
c
+
1
)
.
The estimate is sharp.Proof.
From (13), it follows that
(22)
L
(
a
,
c
)
f
(
z
)
z
=
1
+
ω
(
z
)
(
z
∈
U
)
,
where ω is analytic and satisfies the conditions ω
(
0
)
=
0 and |
ω
(
z
)
|
<
1 in U. Setting
(23)
ϕ
(
z
)
=
1
+
ω
(
z
)
1
-
ω
(
z
)
=
1
+
p
1
z
+
p
2
z
2
+
⋯
(
z
∈
U
)
,
we see that ϕ
∈
P. From the above expression, we get
(24)
ω
(
z
)
=
ϕ
(
z
)
-
1
ϕ
(
z
)
+
1
(
z
∈
U
)
so that, by (22), we get
(25)
L
(
a
,
c
)
f
(
z
)
z
=
2
ϕ
(
z
)
1
+
ϕ
(
z
)
1
/
2
(
z
∈
U
)
.
Now, it is easily seen that
(26)
2
ϕ
(
z
)
1
+
ϕ
(
z
)
1
/
2
=
1
+
1
4
p
1
z
+
1
4
p
2
-
5
32
p
1
2
z
2
+
1
4
p
3
-
5
16
p
1
p
2
+
13
128
p
1
3
z
3
+
⋯
.
Differentiating the series expansion of f given by (1) with respect to z and comparing the coefficients of z, z
2, and z
3 in (26), we deduce that
(27)
a
2
=
c
4
a
p
1
(28)
a
3
=
c
2
a
2
1
4
p
2
-
5
32
p
1
2
.
(29)
a
4
=
c
3
a
3
1
4
p
3
-
5
16
p
1
p
2
+
13
128
p
1
3
.
Thus, by using (27) and (28), we get
(30)
a
3
-
λ
a
2
2
=
1
4
c
2
a
2
p
2
-
5
a
(
c
+
1
)
+
2
λ
c
(
a
+
1
)
8
a
(
c
+
1
)
p
1
2
which with the aid of (18) yields the required estimate (21). The estimate (21) is sharp for the function f
∈
A defined in U by
(31)
f
(
z
)
=
φ
(
c
,
a
;
z
)
⋆
z
1
+
z
2
,
a
(
c
+
1
)
+
2
λ
c
(
a
+
1
)
4
a
(
c
+
1
)
≤
1
φ
(
c
,
a
;
z
)
⋆
z
1
+
z
,
a
(
c
+
1
)
+
2
λ
c
(
a
+
1
)
4
a
(
c
+
1
)
>
1
.Remark 4.
If the functionf, given by (1), belongs to the class R
(
a
,
c
), then it follows at once from (27) that |
a
2
|
≤
c
/
2
a and Theorem 3 gives |
a
3
|
≤
(
c
)
2
/
2
(
a
)
2. The inequality for |
a
2
| is sharp when f is defined by
(32)
f
(
z
)
=
φ
(
c
,
a
;
z
)
⋆
z
1
+
z
(
z
∈
U
)
,
and the estimate for |
a
3
| is sharp for the function g defined by
(33)
g
(
z
)
=
φ
c
,
a
;
z
⋆
z
1
+
z
2
(
z
∈
U
)
.For the caseλ
∈
R, Theorem 3 reduces to the following result.Corollary 5.
Letλ
∈
R. If the function f, given by (1), belongs to the class R
(
a
,
c
), then
(34)
a
3
-
λ
a
2
2
≤
-
c
{
a
(
c
+
1
)
+
2
λ
(
a
+
1
)
c
}
8
a
a
2
,
λ
<
-
5
a
(
c
+
1
)
2
(
a
+
1
)
c
c
2
2
a
2
,
-
5
a
(
c
+
1
)
2
(
a
+
1
)
c
≤
λ
≤
3
a
(
c
+
1
)
2
(
a
+
1
)
c
c
{
a
(
c
+
1
)
+
2
λ
c
(
a
+
1
)
}
8
a
a
2
,
λ
>
3
a
(
c
+
1
)
2
(
a
+
1
)
c
.
The estimate is sharp for the function f defined in U by
(35)
f
(
z
)
=
φ
c
,
a
;
z
⋆
z
1
+
z
,
λ
<
-
5
a
c
+
1
2
a
+
1
c
o
r
λ
>
3
a
(
c
+
1
)
2
(
a
+
1
)
c
φ
(
c
,
a
;
z
)
⋆
z
1
+
z
2
,
-
5
a
(
c
+
1
)
2
(
a
+
1
)
c
≤
λ
≤
3
a
(
c
+
1
)
2
(
a
+
1
)
c
.Puttinga
=
2 and c
=
1 in Corollary 5, we get the following.Corollary 6.
If the functionf, given by (1), satisfies the subordination relation
(36)
f
′
(
z
)
≺
1
+
z
(
z
∈
U
)
,
then
(37)
a
3
-
λ
a
2
2
≤
-
2
+
3
λ
48
,
λ
<
-
10
3
1
6
,
-
10
3
≤
λ
≤
2
2
+
3
λ
48
,
λ
>
2
.
The estimate is sharp for the function f defined in U by
(38)
f
(
z
)
=
∫
0
z
‍
d
t
1
-
t
⋆
z
1
+
z
,
λ
<
-
10
3
o
r
λ
>
2
∫
0
z
‍
d
t
1
-
t
⋆
z
1
+
z
2
,
-
10
3
≤
λ
≤
2
.In the following theorem, we find the sharp upper bound to the second Hankel determinant for the classR
(
a
,
c
).Theorem 7.
Ifa
≥
c
≥
1
/
2 and the function f, given by (1), belongs to the class R
(
a
,
c
), then
(39)
a
2
a
4
-
a
3
2
≤
c
2
2
a
2
2
.
The estimate in (39) is sharp for the function g, given by (33).Proof.
From (27), (28), and (29), we deduce that
(40)
a
2
a
4
-
a
3
2
=
1
16
c
a
c
3
a
3
p
1
p
3
-
5
4
p
1
2
p
2
+
13
32
p
1
4
h
h
h
-
c
2
a
2
2
p
2
2
-
5
4
p
1
2
p
2
+
25
64
p
1
4
.
Since the functions ϕ
(
z
) and ϕ
(
e
i
θ
z
)
(
θ
∈
R
) are in the class P simultaneously, we assume without loss of generality that p
1
>
0. For convenience of notation, we write p
1
=
p
(
0
≤
p
≤
2
). Now, by using (19) in (40), we get
(41)
a
2
a
4
-
a
3
2
=
1
16
c
4
a
c
3
a
3
p
4
+
2
(
4
-
p
2
)
p
2
x
h
h
h
h
h
h
h
h
h
h
-
(
4
-
p
2
)
p
2
x
2
h
h
h
h
h
h
h
h
h
h
+
2
(
4
-
p
2
)
(
1
-
x
2
)
p
z
h
h
h
-
5
(
a
-
c
)
c
c
2
8
a
2
a
3
p
4
+
(
4
-
p
2
)
p
2
x
h
h
h
-
1
4
c
2
a
2
2
h
h
h
×
p
4
+
4
-
p
2
2
x
2
+
2
p
2
(
4
-
p
2
)
x
h
h
h
+
(
a
-
c
)
c
c
2
64
a
2
a
3
p
4
for some x
(
|
x
|
≤
1
) and for some z
(
|
z
|
≤
1
). Applying the triangle inequality in (41) and replacing |
x
| by y in the resulting equation, we get
(42)
a
2
a
4
-
a
3
2
≤
c
c
2
32
a
a
2
×
a
c
+
3
a
+
2
32
a
+
1
2
p
4
h
+
a
-
c
4
a
+
1
2
(
4
-
p
2
)
p
2
y
h
+
(
2
-
p
)
(
4
-
p
2
)
{
2
(
a
+
2
)
(
c
+
1
)
-
(
a
-
c
)
p
}
2
a
+
1
2
y
2
h
+
c
+
2
a
+
2
4
-
p
2
p
=
G
p
,
y
0
≤
p
≤
2
,
0
≤
y
≤
1
say
.
We next maximize the function G
(
p
,
y
) on the closed rectangle [
0,2
]
×
[
0,1
]. Since
(43)
∂
G
∂
y
=
c
c
2
32
a
a
2
×
2
-
p
4
-
p
2
a
+
2
c
+
1
-
a
-
c
p
2
a
+
1
2
y
a
-
c
4
a
+
1
2
(
4
-
p
2
)
p
2
h
+
2
-
p
4
-
p
2
a
+
2
c
+
1
-
a
-
c
p
2
a
+
1
2
y
>
0
for 0
<
p
<
2 and 0
<
y
<
1, the function G
(
p
,
y
) cannot have a maximum value in the interior on the closed rectangle [
0,2
]
×
[
0,1
]. Therefore, for fixed p
∈
[
0,2
],
(44)
max
⁡
0
≤
y
≤
1
G
(
p
,
y
)
=
G
(
p
,
1
)
=
F
(
p
)
(
say
)
,
where
(45)
F
p
=
c
c
2
32
a
a
2
×
a
c
-
21
a
+
24
c
+
2
32
a
+
1
2
p
4
h
+
a
-
7
c
-
2
a
c
-
4
a
+
1
2
p
2
h
+
8
(
c
+
1
)
a
+
1
(
0
≤
p
≤
2
)
.
Setting
(46)
F
′
(
p
)
=
c
c
2
256
a
a
2
a
+
1
2
×
(
a
c
-
21
a
+
24
c
+
2
)
p
2
hhi
+
16
(
a
-
7
c
-
2
a
c
-
4
)
(
a
c
-
21
a
+
24
c
+
2
)
p
2
p
=
0
,
we note that either p
=
0 or
(47)
p
2
=
16
(
2
a
c
+
7
c
-
a
+
4
)
a
c
-
21
a
+
24
c
+
2
>
4
.
Since a
<
2
a
c
+
7
c
+
4, we further observe that F
′
′
(
0
)
<
0. Thus, the maximum value of F is attained at p
=
0 so that the upper bound in (42) corresponds to p
=
0 and y
=
1 from which we get the assertion of the theorem.Lettinga
=
2 and c
=
1 in Theorem 7, we get the following.Corollary 8.
If the functionf, given by (1), satisfies the condition (36), then
(48)
a
2
a
4
-
a
3
2
≤
1
36
.
and the estimate is sharp for the function f defined by
(49)
f
(
z
)
=
∫
0
z
‍
d
t
1
-
t
⋆
z
1
+
z
2
(
z
∈
U
)
.Next, we find the sharp upper bound for the fourth coefficient of functions belonging to the classR
(
a
,
c
).Theorem 9.
If the functionf, given by (1), belongs to the class R
(
a
,
c
), then
(50)
a
4
≤
1
2
c
3
a
3
.
The estimate is sharp.Proof.
Using (19) in (29) and following the lines of proof of Theorem 7, we deduce that
(51)
a
4
=
c
3
a
3
×
p
3
128
-
(
4
-
p
2
)
p
32
x
h
-
(
4
-
p
2
)
p
16
x
2
+
(
4
-
p
2
)
8
(
1
-
x
2
)
z
for some (
|
x
|
≤
1
) and z
(
|
z
|
≤
1
). By an application of the triangle inequality in the above expression followed by replacement of |
x
| by y in the resulting equation, we obtain
(52)
a
4
≤
c
3
a
3
×
p
3
128
+
(
4
-
p
2
)
p
32
y
h
-
4
-
p
2
2
-
p
16
y
2
+
4
-
p
2
8
=
G
(
p
,
y
)
(
0
≤
p
≤
2
,
0
≤
y
≤
1
)
(
say
)
.
We next maximize the function G
(
p
,
y
) on the closed rectangle [
0,2
]
×
[
0,1
]. Since
(53)
∂
G
∂
y
=
c
3
32
a
3
(
4
-
p
2
)
p
-
4
(
2
-
p
)
y
<
0
for 0
<
p
<
2 and 0
<
y
<
1, it follows that G
(
p
,
y
) cannot have a maximum value in the interior of the closed rectangle 0,2
×
0,1. Thus, for fixed p
∈
0,2,
(54)
max
⁡
0
≤
y
≤
1
G
(
p
,
y
)
=
G
(
p
,
0
)
=
F
(
p
)
(
say
)
,
where
(55)
F
(
p
)
=
c
3
128
a
3
(
p
3
-
16
p
2
+
64
)
.
We further note that
(56)
F
′
(
p
)
=
c
3
128
a
3
(
3
p
-
32
)
p
=
0
for p
=
0 or p
=
32
/
3. Since F
′
′
(
0
)
=
-
(
c
)
3
/
4
(
a
)
3
<
0, the function F attains its maximum value at p
=
0. Thus, the upper bound of the function G corresponds to p
=
y
=
0. Putting p
=
y
=
0 in (52), we get our desired estimate (50).
The estimate in (50) is sharp for the function f defined by
(57)
f
(
z
)
=
φ
c
,
a
;
z
⋆
z
1
+
z
3
(
z
∈
U
)
.Lettinga
=
2 and c
=
1 in Theorem 9, we obtain the following.Corollary 10.
If the functionf, given by (1), satisfies the condition (36), then
(58)
|
a
4
|
≤
1
8
and the estimate is sharp for the function f defined by
(59)
f
(
z
)
=
∫
0
z
‍
d
t
1
-
t
⋆
z
1
+
z
3
(
z
∈
U
)
.Finally, we obtain a sufficient condition for a function inA to be in the class R
(
a
,
c
).Theorem 11.
Letγ
>
0. If f
∈
A satisfies
(60)
Re
L
(
a
+
1
,
c
)
f
(
z
)
L
(
a
,
c
)
f
(
z
)
<
1
+
1
2
a
γ
(
z
∈
U
)
,
then
(61)
L
(
a
,
c
)
f
(
z
)
z
≺
1
+
z
1
/
γ
(
z
∈
U
)
and the result is the best possible.Proof.
Setting(62)
L
(
a
,
c
)
f
(
z
)
z
=
1
+
w
z
1
/
γ
(
z
∈
U
)
and choosing the principal branch in (62), we see that w is analytic in U with w
(
0
)
=
0. Taking the logarithmic differentiation in (62) and using the identity (11) in the resulting equation, we deduce that
(63)
L
(
a
+
1
,
c
)
f
(
z
)
L
(
a
,
c
)
f
(
z
)
=
1
+
1
a
γ
z
w
′
(
z
)
1
+
w
(
z
)
(
z
∈
U
)
.
We claim that |
w
(
z
)
|
<
1 for all z
∈
U. Otherwise, there exists a point z
0
∈
U such that
(64)
max
⁡
|
z
|
≤
|
z
0
|
w
z
=
w
z
0
=
1
(
w
(
z
0
)
≠
1
)
.
Letting w
(
z
0
)
=
e
i
θ
(
-
π
<
θ
≤
π
) and applying Jack’s lemma [19], we have
(65)
z
0
w
′
(
z
0
)
=
k
w
(
z
0
)
(
k
≥
1
)
.
Using (65) in (63), we get
(66)
Re
L
(
a
+
1
,
c
)
f
(
z
)
L
(
a
,
c
)
f
(
z
)
=
1
+
1
a
γ
Re
z
0
w
′
(
z
0
)
1
+
w
(
z
0
)
=
1
+
k
a
γ
Re
e
i
θ
1
+
e
i
θ
≥
1
+
1
2
a
γ
,
which contradicts the hypothesis (60). Thus, we conclude that |
w
(
z
)
|
<
1 for all z
∈
U and the assertion of the theorem follows from (62).
To see that the result is the best possible, we consider the principal branch of the functionf
0 defined by
(67)
f
0
(
z
)
=
φ
z
⋆
z
1
+
z
1
/
γ
(
z
∈
U
)
.
Then, it follows from (67) that
(68)
L
(
a
,
c
)
f
0
(
z
)
z
=
1
+
z
1
/
γ
(
z
∈
U
)
.
On differentiating the above expression logarithmically followed by the use of the identity (11), we obtain
(69)
L
(
a
+
1
,
c
)
f
0
(
z
)
L
(
a
,
c
)
f
0
(
z
)
=
1
+
1
a
γ
z
1
+
z
⟶
1
+
1
2
a
γ
as
z
⟶
1
-
.
The proof of the theorem is thus completed.Forγ
=
2 in Theorem 11, we have the following.Corollary 12.
Iff
∈
A satisfies
(70)
Re
L
(
a
+
1
,
c
)
f
(
z
)
L
(
a
,
c
)
f
(
z
)
<
1
+
1
4
a
(
z
∈
U
)
,
then f
∈
R
(
a
,
c
). The result is the best possible.Remark 13.
Further, by specializing the parametersa and c, one can obtain interesting subclasses of A involving the various operators discussed in the introduction and the corresponding results obtained here can be extended to these classes.



---

*Source: 295703-2014-12-16.xml*