1. No category

COEFFICIENT ESTIMATES FOR A NEW SUBCLASS
OF ANALYTIC AND BI-UNIVALENT FUNCTIONS
DEFINED BY HADAMARD PRODUCT
SERAP BULUT
Abstract. In this paper, we introduce and investigate a new general subclass H ; ('; ) of analytic and bi-univalent functions in
the open unit disk U: For functions belonging to this class, we obtain estimates on the …rst two Taylor-Maclaurin coe¢ cients ja2 j
and ja3 j :
1. Introduction
Let A denote the class of all functions of the form
f (z) = z +
1
X
an z n
(1.1)
n=2
which are analytic in the open unit disk
U = fz : z 2 C and
jzj < 1g :
We also denote by S the class of all functions in the normalized analytic
function class A which are univalent in U:
Since univalent functions are one-to-one, they are invertible and the
inverse functions need not be de…ned on the entire unit disk U: In fact,
the Koebe one-quarter theorem [7] ensures that the image of U under
every univalent function f 2 S contains a disk of radius 1=4: Thus
every function f 2 A has an inverse f 1 ; which is de…ned by
f
1
(f (z)) = z
(z 2 U)
and
f f
1
(w) = w
jwj < r0 (f ) ; r0 (f )
1
4
:
2000 Mathematics Subject Classi…cation. Primary 30C45.
Key words and phrases. Analytic functions; Univalent functions; Bi-univalent
functions; Taylor-Maclaurin series expansion; Coe¢ cient bounds and coe¢ cient
estimates; Taylor-Maclaurin coe¢ cients; Hadamard product (convolution).
1
2
SERAP BULUT
1
In fact, the inverse function f
f
1
a2 w2 + 2a22
(w) = w
is given by
a3 w 3
5a32
5a2 a3 + a4 w4 +
:
Denote by f
the Hadamard product (or convolution) of the functions f and ; that is, if f is given by (1:1) and is given by
(z) = z +
1
X
bn z n
(1.2)
(bn > 0) ;
n=2
then
(f
) (z) = z +
1
X
an b n z n :
(1.3)
n=2
For two functions f and , analytic in U; we say that the function
f is subordinate to in U; and write
f (z)
(z 2 U) ;
(z)
if there exists a Schwarz function !; which is analytic in U with
! (0) = 0
and
j! (z)j < 1
(z 2 U)
such that
f (z) =
(! (z))
(z 2 U) :
Indeed, it is known that
f (z)
(z)
(z 2 U) ) f (0) =
Furthermore, if the function
following equivalence
f (z)
(z)
(0)
and f (U)
(U) :
is univalent in U; then we have the
(z 2 U) , f (0) =
(0)
and f (U)
(U) :
A function f 2 A is said to be bi-univalent in U if both f and f 1
are univalent in U: Let
denote the class of bi-univalent functions
in U given by (1:1) : For a brief history and interesting examples of
functions in the class ; see [15] (see also [3]). In fact, the aforecited
work of Srivastava et al. [15] essentially revived the investigation of
various subclasses of the bi-univalent function class in recent years;
it was followed by such works as those by Tang et al. [16], El-Ashwah
[8], Frasin and Aouf [9], Aouf et al. [1], and others (see, for example,
[4, 5, 6, 10, 13, 14, 15, 17, 18]).
Throughout this paper, we assume that ' is an analytic function with
positive real part in the unit disk U; satisfying ' (0) = 1; '0 (0) > 0; and
ON THE NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS
3
' (U) is symmetric with respect to the real axis. Such a function has
a series expansion of the form
' (z) = 1 + B1 z + B2 z 2 +
;
(1.4)
(B1 > 0) :
With this assumption on '; we now introduce the following subclass
of bi-univalent functions.
De…nition 1. Let the function f; de…ned by (1:1) ; be in the analytic
function class A and let 2 : We say that
f 2H
;
('; )
(
1;
0)
if the following conditions are satis…ed:
f2 ;
(1
(f
)
) (z)
z
(f
)0 (z)
+ (f
1
) (z)
' (z)
z
(z 2 U)
(1.5)
and
(1
(f
)
)
w
1
(w)
!
+
(f
1 0
)
(w)
(f
)
w
1
(w)
!
1
' (w)
(w 2 U) ;
(1.6)
where the function (f
(f
)
1
)
1
is given by
(w) = w a2 b2 w2 + 2a22 b22
a3 b 3 w 3
Remark 1: If we let
5a32 b32
5a2 b2 a3 b3 + a4 b4 w4 +
(1.7)
z
;
1 z
then the class H ; ('; ) reduces to the class denoted by H ( ; ')
which is the subclass of the functions f 2 satisfying
(z) =
f (z)
z
1
+ f 0 (z)
g (w)
(1
)
+ g 0 (w)
w
where the function g is de…ned by
g (w)
w
1
(1
)
f (z)
z
and
g (w) = w
a2 w2 + 2a22
a3 w 3
5a32
' (z)
' (w) ;
5a2 a3 + a4 w4 +
which was introduced and studied recently by Tang et al. [16].
; (1.8)
:
4
SERAP BULUT
Remark 2: If we let
1+z
1 z
' (z) =
(0 <
1) ;
then the class H ; ('; ) reduces to the new class denoted by H ; ( ; )
which is the subclass of the functions f 2 satisfying
)
(
1
(f
) (z)
(f
) (z)
0
+ (f
) (z)
arg (1
)
<
z
z
2
and
8
<
arg (1
:
)
(f
)
w
where the function (f
classes
(i)
H
1
(w)
)
;1
!
1
+
(f
)
1 0
(w)
(f
)
w
1
(w)
!
is de…ned by (1:7) : Also we have the
( ; )=B ( ; ; )
introduced and studied recently by El-Ashwah [8];
(ii)
z
=N ( ; )
H ;
;
1 z
introduced and studied recently by Ça¼
glar et al. [6];
(iii)
z
H ;1
;
=B ( ; )
1 z
introduced and studied by Frasin and Aouf [9];
(iv)
z
H1;0
;
=S [ ]
1 z
of strongly bi-starlike functions of order (0 <
1) introduced and
studied by Brannan and Taha [3];
(v)
!
1
X
(a
)
(a
)
1
1 n
q n
z n = Tq;s [a1 ; b1 ; ; ]
H ;1
;z +
(b1 )n
(bs )n n!
n=2
introduced and studied recently by Aouf et al. [1].
Remark 3: If we let
' (z) =
1 + (1
1
2 )z
z
(0
< 1) ;
1
9
=
;
<
2
;
ON THE NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS
5
then the class H ; ('; ) reduces to the new class denoted by H ; ( ; )
which is the subclass of the functions f 2 satisfying
(
)
1
(f
) (z)
(f
)
(z)
< (1
)
+ (f
)0 (z)
>
z
z
and
8
<
< (1
:
)
(f
)
w
1
(w)
where the function (f
classes
(i)
H
!
1
)
;1
+
(f
)
1 0
(w)
(f
)
w
1
(w)
!
is de…ned by (1:7) : Also we have the
( ; )=B ( ; ; )
introduced and studied recently by El-Ashwah [8];
(ii)
z
H ;
;
=N ( ; )
1 z
introduced and studied recently by Ça¼
glar et al. [6];
(iii)
z
=B ( ; )
H ;1
;
1 z
introduced and studied by Frasin and Aouf [9];
(iv)
z
H1;0
;
=S ( )
1 z
of bi-starlike functions of order
by Brannan and Taha [3];
(v)
H
;1
;z +
1
X
(a1 )
n
n=2
(b1 )n
(0
< 1) introduced and studied
(aq )n 1 n
z
(bs )n n!
!
= Tq;s [a1 ; b1 ; ; ]
introduced and studied recently by Aouf et al. [1].
For more results see also [2, 4, 11, 14, 15, 18].
Firstly, in order to derive our main results, we need to following
lemma.
1
9
=
;
> ;
6
SERAP BULUT
Lemma 1. [12] If p 2 P; then jcn j
2 for each n; where P is the
family of all functions p analytic in U for which
< (p (z)) > 0; p (z) = 1 + c1 z + c2 z 2 +
for z 2 U:
2. A set of general coefficient estimates
In this section, we state and prove our general results involving the
bi-univalent function class H ; ('; ) given by De…nition 1:
Theorem 1. Let the function f (z) given by the Taylor-Maclaurin series expansion (1:1) be in the function class
H
;
('; )
(
1;
with
2
and
1
min
b2
q
and
(
B1
+
;
s
(bn > 0) :
2 (B1 + jB2 B1 j)
;
( + 1) (2 + )
p
B1 2B1
B12
( + 1) (2 + )
B1
B12
2 +
2 +
( + )
1
min
b3
ja3 j
bn z n
n=2
Then
ja2 j
(z) = z +
1
X
0)
;
2B13
B12 ( + 1) (2 + ) 2 (B2
2
2 (B2
B1 ) ( + )
9
=
;
(2.1)
B1 [( + 3) + j1
j]
2 jB2 B1 j
+
;
2 ( + 1) (2 + )
( + 1) (2 + )
)
B1
+
:
(2.2)
2 +
B1 ) ( + )2
Proof. Let f 2 H ; ('; ) : Then there are analytic functions u; v :
U ! U; with u (0) = v (0) = 0; satisfying
(1
)
(f
) (z)
z
and
(1
)
(f
)
w
1
(w)
+ (f
!
+
(f
)0 (z)
)
1 0
(f
1
) (z)
= ' (u (z))
z
(w)
(f
)
w
1
(w)
!
1
= ' (v (w)) :
ON THE NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS
7
De…ne the functions p and q by
p(z) =
1 + u (z)
= 1 + p1 z + p2 z 2 +
1 u (z)
q(z) =
1 + v (z)
= 1 + q1 z + q 2 z 2 +
1 v (z)
(2.3)
and
;
(2.4)
z2 +
(2.5)
or, equivalently,
u (z) =
p21
2
p (z) 1
1
=
p1 z + p2
p (z) + 1
2
and
q (z) 1
q12
1
v (z) =
=
q1 z + q2
z2 +
:
(2.6)
q (z) + 1
2
2
Then p and q are analytic in U with p (0) = 1 = q (0) : Since u; v :
U ! U; the functions p and q have a positive real part in U; and by
Lemma 1 jpn j 2 and jqn j 2 for each n: Also it is clear that
(1
)
(f
) (z)
z
and
(1
)
(f
)
w
1
(w)
)0 (z)
+ (f
!
+
(f
)
(f
) (z)
1
(f
)
w
='
z
1 0
(w)
1
(w)
p (z) 1
p (z) + 1
(2.7)
!
1
='
(2.8)
Using (2:5) and (2:6) together with (1:4) ; we get
'
p (z) 1
p (z) + 1
1
= 1 + B1 p1 z +
2
1
B1 p2
2
p21
2
q (w) 1
q (w) + 1
1
= 1 + B1 q1 w +
2
1
B1 q2
2
q12
2
1
+ B2 p21 z 2 +
4
(2.9)
and
'
Now, upon equating the coe¢ cients in (2:7)
1
+ B2 q12 w2 +
:
4
(2.10)
(2:10) ; we get
1
( + ) b2 a2 = B1 p1 ;
2
(2 + ) b3 a3 + (
1)
+
1
b22 a22 = B1 p2
2
2
1
( + ) b2 a2 = B1 q1
2
(2.11)
p21
2
1
+ B2 p21 ;
4
(2.12)
(2.13)
q (w) 1
q (w) + 1
:
8
SERAP BULUT
and
(2 + ) b3 a3 + ( + 3)
+
1
b22 a22 = B1 q2
2
2
q12
2
1
+ B2 q12 :
4
(2.14)
From (2:11) and (2:13) ; it follows that
p1 =
(2.15)
q1
and
1
2 ( + )2 b22 a22 = B12 p21 + q12 :
4
Now (2:12) ; (2:14) and (2:15) yield
a22 =
(2.16)
B1 (p2 + q2 ) + (B2 B1 ) p21
:
2 ( + 1) (2 + ) b22
(2.17)
Also from (2:12) ; (2:14) and (2:16) ; we get
a22 =
B13 (p2 + q2 )
2 B12 ( + 1) (2 + ) 2 (B2
B1 ) ( + )2 b22
:
(2.18)
Applying Lemma 1 for (2:11) ; (2:17) and (2:18) ; we get the desired
estimate on the coe¢ cient ja2 j as asserted in (2:1) :
Next, in order to …nd the bound on the coe¢ cient ja3 j ; by subtracting
(2:14) from (2:12) ; we obtain
2 (2 + ) b3 a3
1
2 (2 + ) b22 a22 = B1 (p2
2
q2 ) :
(2.19)
Upon substituting the value of a22 from (2:11) ; (2:17) and (2:18) into
(2:19) ; it follows that
a3 =
B1 (p2 q2 )
B12 p21
+
;
2
4 ( + ) b3 4 (2 + ) b3
(2.20)
B1 (p2 + q2 ) + (B2 B1 ) p21
B1 (p2 q2 )
+
2 ( + 1) (2 + ) b3
4 (2 + ) b3
B1 [( + 3) p2 + (1
) q2 ] + 2 (B2 B1 ) p21
=
4 ( + 1) (2 + ) b3
a3 =
(2.21)
and
B13 (p2 + q2 )
2 B12 ( + 1) (2 + ) 2 (B2
B1 (p2 q2 )
;
B1 ) ( + ) b3 4 (2 + ) b3
(2.22)
respectively. Applying Lemma 1 for the (2:20) ; (2:21) and (2:22) ; we
get the desired estimate on the coe¢ cient ja3 j as asserted in (2:2) :
a3 =
2
+
ON THE NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS
If we take
9
z
(z) =
1 z
in Theorem 1; then we have the following corollary.
Corollary 1. Let the function f (z) given by the Taylor-Maclaurin
series expansion (1:1) be in the function class
H ( ; ')
Then
ja2 j
min
q
and
ja3 j
(
B1
+
;
(
s
1;
0) :
2 (B1 + jB2 B1 j)
;
( + 1) (2 + )
p
B1 2B1
B12 ( + 1) (2 + )
B12
B1
2 +
2 +
( + )
2 (B2
B1 ) ( + )2
9
=
;
B1 [( + 3) + j1
j]
2 jB2 B1 j
+
;
2 ( + 1) (2 + )
( + 1) (2 + )
)
B1
2B13
+
:
2 +
B12 ( + 1) (2 + ) 2 (B2 B1 ) ( + )2
min
;
Remark 4: Corollary 1 is an improvement of the estimates which were
given by Tang et al. [16, Theorem 2:1].
Remark 5: If we set = 1 and = 1 in Corollary 1; then we have
an improvement of the estimates which were given by Ali et al. [2,
Theorem 2:1].
If we take
1+z
(0 <
1)
1 z
in Theorem 1; then we have the following corollary.
' (z) =
Corollary 2. Let the function f (z) given by the Taylor-Maclaurin
series expansion (1:1) be in the function class
H
;
( ; )
(
1;
with
2
and
(z) = z +
0; 0 <
1
X
n=2
bn z n
1)
(bn > 0) :
10
SERAP BULUT
Then
1
min
b2
ja2 j
q
and
ja3 j
1
min
b3
(
2
+
;
2
( + )2 +
;
2 [( + 3) + j1
j]
4 (1
)
+
;
2 ( + 1) (2 + )
( + 1) (2 + )
)
2
:
+
2 +
2
+2
2
;
2
( + )2 +
4 (2
)
;
( + 1) (2 + )
9
=
+2
4 2
2
2 +
2 +
( + )
4
s
Remark 6: In Corollary 2;
(i) if we set = 1; then we have an improvement of the estimates
which were given by El-Ashwah [8, Theorem 1],
(ii) if we set (z) = 1 z z ; then we have an improvement of the estimates
which were given by Ça¼
glar et al. [6, Theorem 2:1],
z
(iii) if we set (z) = 1 z and = 1; then we have an improvement of
the estimates which were given by Frasin and Aouf [9, Theorem 2:2],
P
(a1 )n (aq )n 1 n
(iv) if we set (z) = z + 1
= 1; then we have
n=2 (b1 )n (bs )n n! z and
an improvement of the estimates which were given by Aouf et al. [1,
Theorem 4].
If we take
1 + (1 2 ) z
(0
< 1)
1 z
in Theorem 1; then we have the following corollary.
' (z) =
Corollary 3. Let the function f (z) given by the Taylor-Maclaurin
series expansion (1:1) be in the function class
B ( ; ; )
(
1; 0
with
2
and
bn z n
(bn > 0) :
n=2
Then
ja2 j
(z) = z +
1
X
< 1)
1
min
b2
(
2 (1
+
)
;
s
4 (1
)
( + 1) (2 + )
)
ON THE NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS
and
ja3 j
8
>
>
<
>
>
:
1
b3
min
n
4(1 )2
( + )2
+
2(1 )
2 +
;
4(1 )
( +1)(2 + )
2(1 )
(2 + )b3
o
; 0
11
<1
:
;
1
Remark 7: In Corollary 3;
(i) if we set = 1; then we have an improvement of the estimates
which were given by El-Ashwah [8, Theorem 2],
(ii) if we set (z) = 1 z z ; then we have the estimates which were given
by Ça¼
glar et al. [6, Theorem 3:1],
(iii) if we set (z) = 1 z z and = 1; then we have an improvement of
the estimates which were given by Frasin and Aouf [9, Theorem 3:2],
P
(a1 )n (aq )n 1 n
(iv) if we set (z) = z + 1
= 1; then we have
n=2 (b1 )n (bs )n n! z and
an improvement of the estimates which were given by Aouf et al. [1,
Theorem 8].
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12
SERAP BULUT
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Serap BULUT
Kocaeli University, Civil Aviation College, Arslanbey Campus, 41285
·
Izmit-Kocaeli, TURKEY
E-mail address: [email protected]