# Fekete-Szegö Inequalities for Certain Classes of Biunivalent Functions

Authors: Şahsene Altınkaya; Sibel Yalçın
Journal: International Scholarly Research Notices
 (2014)

---

## Abstract

We obtain the Fekete-Szegö inequalities for the classesS
S
,
Σ
*
(
α
,
ϕ
) and £
S
,
Σ
(
α
,
ϕ
) of biunivalent functions denoted by subordination. The results presented in this paper improve the recent work of Crisan (2013).


---

## Body

## 1. Introduction and Definitions

LetA denote the class of analytic functions in the unit disk
(1)
U
=
z
∈
C
:
z
<
1
that have the form
(2)
f
(
z
)
=
z
+
∑
‍
n
=
2
∞
a
n
z
n
.
Further, by S we will denote the class of all functions in A which are univalent in U.The Koebe one-quarter theorem [1] states that the image of U under every function f from S contains a disk of radius 1
/
4. Thus every such univalent function has an inverse f
-
1 which satisfies
(3)
f
-
1
f
z
=
z
,
z
∈
U
,
f
f
-
1
w
=
w
,
w
<
r
0
f
,
r
0
f
≥
1
4
,
where
(4)
f
-
1
w
=
w
-
a
2
w
2
+
2
a
2
2
-
a
3
w
3
-
5
a
2
3
-
5
a
2
a
3
+
a
4
w
4
+
⋯
.A functionf
z
∈
A is said to be biunivalent in U if both f
z and f
-
1
z are univalent in U. Let Σ denote the class of biunivalent functions defined in the unit disk U.If the functionsf and g are analytic in U, then f is said to be subordinate to g, written as
(5)
f
z
≺
g
z
,
z
∈
U
if there exists a Schwarz function w
z, analytic in U, with
(6)
w
0
=
0
,
w
z
<
1
,
z
∈
U
such that
(7)
f
z
=
g
w
z
,
z
∈
U
.Lewin [2] studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient a
2. Subsequently, Brannan and Clunie [3] conjectured that a
2
≤
2 for f
∈
Σ. Netanyahu [4] showed that max
⁡
a
2
=
4
/
3 if f
z
∈
Σ. Brannan and Taha [5] introduced certain subclasses of the biunivalent function class Σ similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions. They introduced bistarlike functions and obtained estimates on the initial coefficients. Bounds for the initial coefficients of several classes of functions were also investigated in [6–12]. The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients a
n for n
∈
N
∖
1,2
;
N
=
1,2
,
3
,
… is presumably still an open problem.Letϕ be an analytic and univalent function with positive real part in U with ϕ
0
=
1, ϕ
′
0
>
0, and ϕ maps the unit disk U onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor’s series expansion of such function is of the form
(8)
ϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯
,
where all coefficients are real and B
1
>
0.ByS
*
(
ϕ
) and C
(
ϕ
) we denote the following classes of functions:
(9)
S
*
ϕ
=
f
:
f
∈
A
,
z
f
′
z
f
z
≺
ϕ
z
;
z
∈
U
,
C
ϕ
=
f
:
f
∈
A
,
1
+
z
f
′′
z
f
′
z
≺
ϕ
z
;
z
∈
U
.The classesS
*
(
ϕ
) and C
(
ϕ
) are the extensions of classical sets of starlike and convex functions and in such a form were defined and studied by Ma and Minda [13]. They investigated growth and distortion properties of functions in S
*
(
ϕ
) and C
(
ϕ
) as well as Fekete-Szegö inequalities for S
*
(
ϕ
) and C
(
ϕ
). Their proof of Fekete-Szegö inequalities requires the univalence of ϕ. Ali et al. [14] have investigated Fekete-Szegö problems for various other classes and their proof does not require the univalence or starlikeness of ϕ. So in this paper, we assume that ϕ has series expansion ϕ
(
z
)
=
1
+
B
1
z
+
B
2
z
2
+
⋯, B
1
,
B
2 are real, and B
1
>
0. A function f is bistarlike of Ma-Minda type or biconvex of Ma-Minda type if both f and f
-
1 are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by S
Σ
*
(
ϕ
) and C
Σ
(
ϕ
) (see [15]).In [16], Sakaguchi introduced the class S
S
* of starlike functions with respect to symmetric points in U, consisting of functions f
∈
A that satisfy the condition Re
(
z
f
′
(
z
)
/
(
f
(
z
)
-
f
(
-
z
)
)
)
>
0, z
∈
U. Similarly, in [17], Wang et al. introduced the class C
S of convex functions with respect to symmetric points in U, consisting of functions f
∈
A that satisfy the condition Re
(
(
z
f
′
(
z
)
)
′
/
(
f
′
(
z
)
+
f
′
(
-
z
)
)
)
>
0, z
∈
U. In the style of Ma and Minda, Ravichandran (see [18]) defined the classes S
S
*
(
ϕ
) and C
S
(
ϕ
).A functionf
∈
A is in the class S
S
*
(
ϕ
) if
(10)
2
z
f
′
z
f
z
-
f
-
z
≺
ϕ
z
,
z
∈
U
,
and in the class C
S
(
ϕ
) if
(11)
2
z
f
′
z
′
f
′
z
+
f
′
-
z
≺
ϕ
z
,
z
∈
U
.In this paper, motivated by the earlier work of Zaprawa [19], we obtain the Fekete-Szegö inequalities for the classes S
S
,
Σ
*
(
α
,
ϕ
) and £
S
,
Σ
(
α
,
ϕ
). These inequalities will result in bounds of the third coefficient which are, in some cases, better than these obtained in [7].In order to derive our main results, we require the following lemma.Lemma 1 (see [20]).
Ifp
z
=
1
+
p
1
z
+
p
2
z
2
+
p
3
z
3
+
⋯ is an analytic function in U with positive real part, then
(12)
p
n
≤
2
,
n
∈
N
=
1,2
,
…
,
p
2
-
p
1
2
2
≤
2
-
p
2
2
2
.


## 2. Fekete-Szegö Inequalities for the Function ClassS
S
,
Σ
*
α
,
ϕ

Definition 2 (see [7]).
A functionf
∈
Σ is said to be in the class S
S
,
Σ
*
α
,
ϕ if the following subordination holds:
(13)
2
1
-
α
z
f
′
z
+
α
z
z
f
′
z
′
1
-
α
f
z
-
f
-
z
+
α
z
f
′
z
+
f
′
-
z
≺
ϕ
z
,
2
1
-
α
w
g
′
w
+
α
w
w
g
′
w
′
1
-
α
g
w
-
g
-
w
+
α
w
g
′
w
+
g
′
-
w
≺
ϕ
w
,
where g
w
=
f
-
1
w.We note that, forα
=
0, the class S
S
,
Σ
*
α
,
ϕ reduces to the class S
S
*
ϕ introduced by Ravichandran [18].Theorem 3.
Letf given by (2) be in the class S
S
,
Σ
*
α
,
ϕ and μ
∈
R. Then
(14)
a
3
-
μ
a
2
2
≤
B
1
2
1
+
2
α
;
μ
-
1
≤
1
1
+
2
α
1
+
2
α
+
2
1
+
α
2
B
1
-
B
2
B
1
2
1
-
μ
B
1
3
2
1
+
2
α
B
1
2
+
4
1
+
α
2
B
1
-
B
2
;
μ
-
1
≥
1
1
+
2
α
1
+
2
α
+
2
1
+
α
2
B
1
-
B
2
B
1
2
.Letf
∈
S
S
,
Σ
*
α
,
ϕ and g be the analytic extension of f
-
1 to U. Then there exist two functions u and v, analytic in U with u
0
=
v
0
=
0, u
z
<
1, v
w
<
1, z
,
w
∈
U, such that
(15)
2
z
f
′
z
+
α
z
2
f
′′
z
1
-
α
f
z
-
f
-
z
+
α
z
f
′
z
+
f
′
-
z
=
ϕ
u
z
,
2
z
f
z
+
α
z
2
f
′′
z
1
-
α
f
z
-
f
-
z
+
α
f
′
z
+
f
′
-
z
z
∈
U
,
2
w
g
′
w
+
α
w
2
g
′′
w
1
-
α
g
w
-
g
-
w
+
α
w
g
′
w
+
g
′
-
w
=
ϕ
v
w
,
w
∈
U
.
Next, define the functions p and q by
(16)
p
z
=
1
+
u
z
1
-
u
z
=
1
+
p
1
z
+
p
2
z
2
+
⋯
,
q
w
=
1
+
v
w
1
-
v
w
=
1
+
q
1
w
+
q
2
w
2
+
⋯
.
Clearly, Re
p
z
>
0 and Re
q
w
>
0. From (16) one can derive
(17)
u
z
=
p
z
-
1
p
z
+
1
=
1
2
p
1
z
+
1
2
p
2
-
1
2
p
1
2
z
2
+
⋯
,
v
w
=
q
w
-
1
q
w
+
1
=
1
2
q
1
w
+
1
2
q
2
-
1
2
q
1
2
w
2
+
⋯
.
Combining (8), (15), and (17),
(18)
2
z
f
′
z
+
α
z
2
f
′′
z
1
-
α
f
z
-
f
-
z
+
α
z
f
′
z
+
f
′
-
z
=
1
+
1
2
B
1
p
1
z
+
1
4
B
2
p
1
2
+
1
2
B
1
p
2
-
1
2
p
1
2
z
2
+
⋯
,
2
w
g
′
w
+
α
w
2
g
′′
w
1
-
α
g
w
-
g
-
w
+
α
w
g
′
w
+
g
′
-
w
=
1
+
1
2
B
1
q
1
w
+
1
4
B
2
q
1
2
+
1
2
B
1
q
2
-
1
2
q
1
2
w
2
+
⋯
.
From (18), we deduce
(19)
2
1
+
α
a
2
=
1
2
B
1
p
1
,
(20)
2
1
+
2
α
a
3
=
1
4
B
2
p
1
2
+
1
2
B
1
p
2
-
1
2
p
1
2
, and(21)
-
2
1
+
α
a
2
=
1
2
B
1
q
1
,
(22)
2
1
+
2
α
2
a
2
2
-
a
3
=
1
4
B
2
q
1
2
+
1
2
B
1
q
2
-
1
2
q
1
2
.
From (19) and (21) we obtain
(23)
p
1
=
-
q
1
.
Subtracting (20) from (22) and applying (23) we have
(24)
a
3
=
a
2
2
+
1
8
1
+
2
α
B
1
p
2
-
q
2
.
By adding (20) to (22), we get
(25)
4
1
+
2
α
a
2
2
=
1
2
B
1
p
2
+
q
2
-
1
4
B
1
-
B
2
p
1
2
+
q
1
2
.
Combining this with (19) and (21) leads to
(26)
a
2
2
=
B
1
3
p
2
+
q
2
8
1
+
2
α
B
1
2
+
2
1
+
α
2
B
1
-
B
2
.
From (24) and (26) it follows that
(27)
a
3
-
μ
a
2
2
=
B
1
h
μ
+
1
8
1
+
2
α
p
2
=
=
=
=
=
=
=
=
=
=
+
h
μ
-
1
8
1
+
2
α
q
2
,
where
(28)
h
μ
=
B
1
2
1
-
μ
8
1
+
2
α
B
1
2
+
2
1
+
α
2
B
1
-
B
2
.
Then, in view of (8) and (12), we conclude that
(29)
a
3
-
μ
a
2
2
≤
B
1
2
1
+
2
α
,
0
≤
h
μ
≤
1
8
1
+
2
α
,
4
B
1
h
μ
,
h
μ
≥
1
8
1
+
2
α
.
Taking μ
=
1 or μ
=
0 we get the following.Corollary 4.
Iff
∈
S
S
,
Σ
*
α
,
ϕ then
(30)
a
3
-
a
2
2
≤
B
1
2
1
+
2
α
.Corollary 5.
Iff
∈
S
S
,
Σ
*
α
,
ϕ then
(31)
a
3
≤
B
1
2
1
+
2
α
;
B
1
-
B
2
B
1
2
∈
-
∞
,
-
1
+
2
α
1
+
α
2
∪
0
,
∞
B
1
3
2
1
+
2
α
B
1
2
+
4
1
+
α
2
B
1
-
B
2
;
B
1
-
B
2
B
1
2
∈
-
1
+
2
α
1
+
α
2
,
-
1
+
2
α
2
1
+
α
2
+
+
+
+
+
+
+
∪
-
1
+
2
α
2
1
+
α
2
,
0
.Corollary 6.
If(32)
ϕ
z
=
1
+
z
1
-
z
β
=
1
+
2
β
z
+
2
β
2
z
2
+
⋯
,
0
<
β
≤
1
,
then inequalities (30) and (31) become
(33)
a
3
-
a
2
2
≤
β
1
+
2
α
,
a
3
≤
β
1
+
2
α
.Corollary 7.
If(34)
ϕ
z
=
1
+
1
-
2
β
z
1
-
z
=
1
+
2
1
-
β
z
+
2
1
-
β
z
2
+
⋯
,
0
≤
β
<
1
,
then inequalities (30) and (31) become
(35)
a
3
-
a
2
2
≤
1
-
β
1
+
2
α
,
a
3
≤
1
-
β
1
+
2
α
.Remark 8.
Corollaries6 and 7 provide an improvement of the estimate a
3 obtained by Crisan [7].


## 3. Fekete-Szegö Inequalities for the Function Class£
S
,
Σ
α
,
ϕ

Definition 9 (see [7]).
A functionf
∈
Σ is said to be £
S
,
Σ
α
,
ϕ if the following subordination holds:
(36)
2
z
f
′
z
f
z
-
f
-
z
α
2
z
f
′
z
′
f
′
z
+
f
′
-
z
1
-
α
≺
ϕ
z
,
2
w
g
′
w
g
w
-
g
-
w
α
2
w
g
′
w
′
g
′
w
+
g
′
-
w
1
-
α
≺
ϕ
w
,
where g
w
=
f
-
1
w.We note that, forα
=
0, the class £
S
,
Σ
α
,
ϕ reduces to the class C
S
ϕ introduced by Ravichandran [18].Theorem 10.
Letf given by (2) be in the class £
S
,
Σ
α
,
ϕ and μ
∈
R. Then
(37)
a
3
-
μ
a
2
2
≤
B
1
2
3
-
2
α
;
μ
-
1
≤
1
3
-
2
α
+
+
+
+
+
×
3
-
3
α
+
α
2
+
2
2
-
α
2
B
1
-
B
2
B
1
2
1
-
μ
B
1
3
2
3
-
3
α
+
α
2
B
1
2
+
4
2
-
α
2
B
1
-
B
2
;
μ
-
1
≥
1
3
-
2
α
+
+
+
+
+
×
3
-
3
α
+
α
2
+
2
2
-
α
2
B
1
-
B
2
B
1
2
.Letf
∈
£
S
,
Σ
α
,
ϕ and g be the analytic extension of f
-
1 to U. Then there exist two functions u and v, analytic in U with u
0
=
v
0
=
0, u
z
<
1, v
w
<
1, z
,
w
∈
U, such that
(38)
2
z
f
′
z
′
f
′
z
+
f
′
-
z
=
ϕ
u
z
,
2
w
g
′
w
′
g
′
w
+
g
′
-
w
=
ϕ
v
w
.
From (38), we deduce
(39)
2
2
-
α
a
2
=
1
2
B
1
p
1
,
(40)
2
3
-
2
α
a
3
-
2
α
1
-
α
a
2
2
=
1
4
B
2
p
1
2
+
1
2
B
1
p
2
-
1
2
p
1
2
, and(41)
-
2
2
-
α
a
2
=
1
2
B
1
q
1
,
(42)
2
3
-
2
α
2
a
2
2
-
a
3
-
2
α
1
-
α
a
2
2
=
1
4
B
2
q
1
2
+
1
2
B
1
q
2
-
1
2
q
1
2
.
From (39) and (41) we obtain
(43)
p
1
=
-
q
1
.
Subtracting (40) from (42) and applying (43) we have
(44)
a
3
=
a
2
2
+
1
8
3
-
2
α
B
1
p
2
-
q
2
.
By adding (40) to (42), we get
(45)
4
3
-
3
α
+
α
2
a
2
2
=
1
2
B
1
p
2
+
q
2
-
1
4
B
1
-
B
2
p
1
2
+
q
1
2
.
Combining this with (39) and (41) leads to
(46)
a
2
2
=
B
1
3
p
2
+
q
2
8
3
-
3
α
+
α
2
B
1
2
+
2
2
-
α
2
B
1
-
B
2
.
From (44) and (46) it follows that
(47)
a
3
-
μ
a
2
2
=
B
1
h
μ
+
1
8
3
-
2
α
p
2
+
+
+
+
+
+
+
+
+
h
μ
-
1
8
3
-
2
α
q
2
,
where
(48)
h
μ
=
B
1
2
1
-
μ
8
3
-
3
α
+
α
2
B
1
2
+
2
2
-
α
2
B
1
-
B
2
.
Then, in view of (8) and (12), we conclude that
(49)
a
3
-
μ
a
2
2
≤
B
1
2
3
-
2
α
,
0
≤
h
μ
≤
1
8
3
-
2
α
,
4
B
1
h
μ
,
h
μ
≥
1
8
3
-
2
α
.
Taking μ
=
1 or μ
=
0 we get the following.Corollary 11.
Iff
∈
£
S
,
Σ
α
,
ϕ then
(50)
a
3
-
a
2
2
≤
B
1
2
3
-
2
α
.Corollary 12.
Iff
∈
£
S
,
Σ
α
,
ϕ then
(51)
a
3
≤
B
1
2
3
-
2
α
;
B
1
-
B
2
B
1
2
∈
-
∞
,
α
-
3
2
2
-
α
+
+
+
+
+
+
+
+
∪
α
1
-
α
2
2
-
α
2
,
∞
B
1
3
2
3
-
3
α
+
α
2
B
1
2
+
4
2
-
α
2
B
1
-
B
2
;
B
1
-
B
2
B
1
2
∈
α
-
3
2
2
-
α
,
3
α
-
α
2
-
3
2
2
-
α
2
+
+
+
+
+
+
+
∪
3
α
-
α
2
-
3
2
2
-
α
2
,
α
1
-
α
2
2
-
α
2
.Corollary 13.
If(52)
ϕ
z
=
1
+
z
1
-
z
β
=
1
+
2
β
z
+
2
β
2
z
2
+
⋯
,
0
<
β
≤
1
,
then inequalities (50) and (51) become
(53)
a
3
-
a
2
2
≤
β
3
-
2
α
,
a
3
≤
β
3
-
2
α
.Corollary 14.
If(54)
ϕ
z
=
1
+
1
-
2
β
z
1
-
z
=
1
+
2
1
-
β
z
+
2
1
-
β
z
2
+
⋯
,
0
≤
β
<
1
,
then inequalities (50) and (51) become
(55)
a
3
-
a
2
2
≤
1
-
β
3
-
2
α
,
a
3
≤
1
-
β
3
-
3
α
+
α
2
.Remark 15.
Corollaries13 and 14 provide an improvement of the estimate a
3 obtained by Crisan [7].



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*Source: 327962-2014-11-09.xml*