http://scienceasia.asia
_______________
2010 Mathematics Subject Classification: 30C45.
Key words and phrases: Subordination, Hankel determinant, functions with positive real part, univalent functions and close to star functions.
©2014 Science Asia 1 / 13
SECOND HANKEL DETERMINANT FOR SUBCLASSES OF PASCU CLASSES OF ANALYTIC FUNCTIONS
M. S. SAROA, GURMEET SINGH AND GAGANDEEP SINGH
Abstract: We define two subclasses of the class of Pascu functions. For any real 𝜇, we are interested in determining the upper bound of 2
2 4 3
a a −µa for an analytic function
( )
2 32 3 ...
f z = +z a z +a z +
(
z <1)
belonging to these classes.
1. INTRODUCTION AND DEFINITION:
PRINCIPLES OF SUBORDINATION: Let f z
( )
and F z( )
be two analytic functions in the unit disc { : 1}E= z z < . Then, f z
( )
is said to be subordinate toF z( )
in the unit disc E if there exists an analytic function w z( )
in E satisfying the conditionw( )
0 =0, w z( )
<1 such that f z( )
=F w z( )
( )
and we write asf z
( )
F z( )
. In particular if F z( )
is univalent in D, the above definition is equivalent to f( ) ( )
0 =F 0 and f E( )
⊂F E( )
.FUNCTIONS WITH POSITIVE REAL PART: Let Ƥ denotes the class of analytic functions of the form
(1.1)
( )
21 2
1 ...
P z = +p z p z+ +
with ReP z
( )
>0, z D∈ .Let Adenote the class of functions of the form (1.2)
( )
2
k k k
f z z ∞ a z =
= +
∑
which are analytic in the open unit disc D={ :z z <1}.
Sis the class of functions of the form (1.2) which are univalent. The Hankel determinant: ([9],[10])
Let
( )
0
n n n
f z ∞ a z =
=
∑
be analytic in D. For q≥1, the qth Hankel determinant is defined by( )
1 12 11 2 2
... ... ...
n n n q
q n n n q
n q n q n q
a a a
H n a a a
a a a
+ + −
+ + +
+ − + + −
The Hankel determinant was studied by various authors including Hayman[3] and Ch. Pommerenke ([13],[14]). For q= 2 and n = 2 , the second Hankel determinant for the analytic function f z
( )
is defined by( )
2 3(
2)
2 2 4 3
3 4
2 a a
H a a a
a a
= = −
0
R represents the class of functions f z
( )
∈A and satisfying the condition(1.3) Re f z
( )
0, z
>
z D∈ .
0
R is a particular case of the class of close to star function defined by Reade[17]. The class R0 and its subclasses were vastly studied by several authors including Mac-Gregor[7].
Let Rbe the class of functions f z
( )
∈Aand satisfying (1.4) Ref z′( )
>0,z D∈ .The class R was introduced by Noshiro [11] and Warschawski[18] ( known as N-W class) and it was shown by them thatRis a class of univalent functions. The class Rand its subclasses were investigated by various authors including Goel and the author ([1], [2]). For α ≥0, R1
( )
α and R2( )
α denote the classes of functions in Awhich satisfy, respectively, the conditions(1.5) Re 1
(
) ( )
f z f z( )
0, zα α
− + ′ >
z D∈
and
(1.6) Ref z′
( )
+αzf z′′( )
>0, z D∈ .The classes R1
( )
α and R2( )
α were introduced by Pascu [12] and are called Pascu classes offunctions. It is obvious that f z
( )
∈ R1( )
α implies that zf z′( )
∈R2( )
α .We shall deal with the following classes
(1.7) 1
(
)
(
) ( )
( )
1
; , : 1 , 0, 1 1,
1
f z Az
R A B f A f z B A z D
z Bz
α = ∈ −α +α ′ + α ≥ − ≤ < ≤ ∈
+
and
(1.8) R2
(
α; ,A B)
f A f z:( )
αzf z( )
11 AzBz,α 0, 1 B A 1,z D +
′ ′′
= ∈ + ≥ − ≤ < ≤ ∈
+
.
(
)
1 ;1, 1
R α − ≡ R1
( )
α and R2(
α − ≡;1, 1)
R2( )
α .R1(
α; ,A B)
is a subclass of R1( )
α andR2(
α; ,A B)
is asublass of R2
( )
α . The classes R1(
α; ,A B)
and R2(
α; ,A B)
were studied by theauthor[8].Througout the paper, we assume that α≥ − ≤0, 1 B A< ≤1 and z D∈ .
Lemma 2.1 [15]. Let P z
( )
∈ Ƥ( )
z , then2
n
p ≤ (n=1,2,3,..)
Lemma 2.2 [5]. Let P z
( )
∈ Ƥ( )
z , then(
)
2 2
2 1 1
2p = p + 4−p x,
(
)
(
)
(
)
23 2 2 2 2
3 1 1 1 1 1 1
4p = p +2p 4− p x p− 4−p x +2 4− p 1− x z
for some x and z with x ≤1,z ≤1 and p1∈ 0,2.
3. MAIN RESULTS
Theorem 3.1: Let f ∈ R1
(
α; ,A B)
, then2
2 4 3
a a −µa ≤
(3.1)
(3.2)
(3.3)
(3.4)
Proof. By definition of subordination,
(
1) ( )
( )
1( )
( )
1
f z Aw z
f z
z Bw z
α α ′ +
− + =
+ ,
Taking real parts,
(
) ( )
( )
1( )
( )
Re 1 Re
1
f z Aw z
f z
z Bw z
α α
+
− + ′ =
+
(
)
(
)(
)
{
}
(
)(
)(
) (
{
)
(
)(
)
}
(
)
(
)
(
)(
)
{
}
(
)(
)(
) (
{
)
(
)(
)
}
(
)
(
)
(
)(
)
2 2
2
2
2 2
2 2
2
2
2 2
2
2
3 1 2 2 1 1 3
1 4 0;
1 2
2 1 1 3 1 2 1 2 1 1 3
3 1 2 4 1 1 3
1 4
1 2
2 1 1 3 1 2 1 2 1 1 3
3 1 2
0 ;
4 1 1 3
1 4
1
B if
A B
B A B
if
B A B
α µ α α µ
µ α
α α α α µ α α
α µ α α µ
α
α α α α µ α α
α µ
α α
µ
+ − + +
− − ≤
−
+ + + + − + + +
+ − + +
−
+
− + + + + − + + +
+ ≤ ≤
+ +
−
−
(
+)
(
)
(
)(
)
(
(
)(
)
)
(
)(
) (
)
{
}
(
)(
)(
) (
{
)(
) (
)
}
(
)
(
)
(
)(
)
2 2
2
2 2 2
2
2 2
2
3 1 2 3 1 2
;
4 1 1 3 2 1 1 3
2
2 1 1 3 3 1 2
1 4
1 2
2 1 1 3 1 2 1 1 3 1 2
3 1 2 . 2 1 1 3
if
B A B
if
α α
µ
α α α α
α
µ α α α µ
α
α α α µ α α α
α µ
α α
+ +
≤ ≤
+ + + +
+ + − +
− +
−
+ + + + + − + +
+
≥
+ +
≥11−BrAr >11−AB
− − (z r= )
which implies that
(3.5)
(
)
(
)
2(
)
3( )
2 3 4
1
1 B 1 a z 1 2 a z 1 3 a z ... P z
A B α α α
−
+ + + + + + + =
− .
Equating the coefficients in (3.5), we get
(3.6)
(
)
(
)
(
)
1 2
2 3
3 4
1
1 1
1 2 1
1 3 p B a
A B p B a
A B p B a
A B
α
α
α
−
=
− +
−
=
− +
−
=
− +
System (3.6) ensures that
(3.7)
( )
(
2)
(
)
2(
)(
)( )
22 4 3 1 2 1(4 )3 1 1 3 2 2
C α a a −µa = + α p p −µ +α + α p ,
(3.8)
( )
(
)(
)(
)
2
2
4 1 1 3 1 2
1 A B C
B
α = − +α + α + α
−
.
Using Lemma 2.2 in (3.7), we obtain
( )
(
2)
(
)
2 3(
2)
(
2)
2(
2)
22 4 3 1 2 1 1 2 1 4 1 1 4 1 2 4 1 1
C α a a −µa = + α p p + p −p x p− − p x + −p − x z
(
)(
)
2(
2)
21 1
1 1 3 p 4 p x
µ α α
− + + + −
for some x and z with x ≤1 , z ≤1.
or
(3.9)
( )
(
2)
(
)
2(
)(
)
42 4 3 1 2 1 1 3 1
C α a a −µa = + α −µ +α + α p
(
)
(
)(
)
(
)
(
)
{
(
)
(
)(
)
}
(
)(
)
(
)
(
)
2 2 2
1 1
2
2 2 2
1 1
2 2 2
1 1
2 1 2 1 1 3 4
4 1 2 1 1 3 4 1 1 3
2 1 2 4 1
p p x
p p x
p p x z
α µ α α
α µ α α µ α α
α
+ + − + + −
− − + − + + + + +
+ + − −
Replacing p1by p∈[0,2] and applying triangular inequality to (3.9), we get
( )
(
)
(
)(
)
(
)
(
)(
)
(
)
(
)
(
)
(
)(
)
(
)(
)
(
)
(
)(
)
(
)
2 4 2 2 2
2
2 2 2 2
2 4 3
2 2 2
1 2 1 1 3 2 1 2 1 1 3 4
4 1 2 1 1 3 4 1 1 3
2 1 2 4 1 , 1
p p p
C a a a p p
p p x
α µ α α α µ α α δ
α µ α µ α α µ α α δ
α δ δ
+ − + + + + − + + −
− ≤ + − + − + + + + +
+ + − − = ≤
( )
(
)
(
)(
)
(
)
(
)
(
)
(
)(
)
(
)
(
)
{
(
)
(
)(
)
}
(
)(
)
(
)
(
)
(
)(
)
(
)
(
)
(
)
(
)(
)
(
)
(
)
(
)
2 4 2 2
2 2 2
2 2
2 2 2
2 4 2 2
2 2 2
2 2
2 4 3
1 2 1 1 3 2 1 2 4
2 1 2 1 1 3 4
4 1 2 1 1 3 4 1 1 3 2 1 2
0;
1 2 1 1 3 2 1 2 4
2 1 2 1 1 3 4
4 1 2
p p p
p p
p p p
if
p p p
p p
p
C a a a
α µ α α α
α µ α α δ
α µ α α µ α α α δ
µ
α µ α α α
α µ α α δ
α α µ + − + + + + − + + − + + − + − + − + + − + + − + ≤ + − + + + + − + + − + + − + − + − ≤
{
(
)(
)
}
(
)(
)
(
)
(
)(
)
(
)(
) (
)
(
)
(
)
(
)(
) (
)
(
)
(
)
{
(
)(
) (
)
}
(
)(
)
(
)
(
)
(
)(
)
2 2 2 2
2
2 4 2 2
2 2 2
2 2
2 2 2
2
1 1 3 4 1 1 3 2 1 2
(1 2 )
0 ;
1 1 3
1 1 3 1 2 2 1 2 4
2 1 1 3 1 2 4
4 1 1 3 1 2 4 1 1 3 2 1 2
1 2
1 1 3
p p
if
p p p
p p
p p p
if
µ α α µ α α α δ
α µ
α α
µ α α α α
µ α α α δ
µ α α α µ α α α δ
α µ α α − + + + + + − + + ≤ ≤ + + + + − + + + − + + + − + − + − + + − + + + + − + + ≥ + +
= F
( )
δ .( )
0F′ δ > and therefore F
( )
δ is increasing in [0,1]. F( )
δ attains its maximum value atδ =1. (3.10) reduces to( )
(
)
(
)(
)
(
)
(
)(
)
(
)
(
)
{
(
)
(
)(
)
}
(
)(
)
(
)
(
)(
)
(
)
(
)(
)
(
)
(
)
{
(
)
(
)(
)
}
(
)(
)
2 4 2 2 2
2
2 2
2 4 2 2 2
2
2 2 2
2 4 3
1 2 1 1 3 2 1 2 1 1 3 4
4 1 2 1 1 3 4 1 1 3 0;
1 2 1 1 3 2 1 2 1 1 3 4
(1
4 1 2 1 1 3 4 1 1 3 0
p p p
p p if
p p p
C a a a p p if
α µ α α α µ α α
α µ α α µ α α µ
α µ α α α µ α α
α µ α µ α α µ α α µ
+ − + + + + − + + − + − + − + + − + + ≤ + − + + + + − + + − − ≤ + − + − + + + + + ≤ ≤
(
)(
)
(
)(
) (
)
(
)(
) (
)
(
)
(
)
{
(
)(
) (
)
}
(
)(
)
(
(
)(
)
)
22 4 2 2 2
2 2
2 2
2 ) ;
1 1 3
1 1 3 1 2 2 1 1 3 1 2 4
1 2
4 1 1 3 1 2 4 1 1 3 ,
1 1 3
p p p
p p if
α
α α
µ α α α µ α α α
α
µ α α α µ α α µ
= G p
( )
, or(3.11)
( )
22 4 3
C α a a −µa ≤ max G p
( )
, Case (i) µ ≤0( )
(
)
2(
)(
)
4(
)
2(
)(
)
2(
)(
)
2 1 2 1 1 3 4 3 1 2 2 1 1 3 16 1 1 3
G p = − + α −µ +α + α p + + α − µ +α + α p − +α + α µ
( )
G p is maximum for
G p′
( )
= −8 1 2(
+ α)
2−µ(
1+α)(
1 3+ α)
p3+8 3 1 2(
+ α)
2−2 1µ(
+α)(
1 3+ α)
p=0
which implies that
(
)
(
)(
)
(
)
(
)(
)
2 2
3 1 2 2 1 1 3
1 2 1 1 3
p
α µ α α
α µ α α
+ − + +
=
+ − + +
.
Putting the corresponding value of G p
( )
along with C( )
α from (3.8) in (3.11), we get(3.1)
Case (ii) 0
(
(1 2 ))(
2)
1 1 3
α µ
α α
+
≤ ≤
+ +
( )
(
)
2(
)(
)
4(
)
2(
)(
)
2(
)(
)
2 1 2 1 1 3 4 3 1 2 4 1 1 3 16 1 1 3
G p = − + α −µ +α + α p + + α − µ +α + α p + +α + α µ
Sub-case (a) 0
(
3(1 2 ))(
2)
4 1 1 3α µ
α α
+ ≤ ≤
+ +
It is easy to see that G p
( )
is maximum at(
)
(
)(
)
(
)
(
)(
)
2 2
3 1 2 4 1 1 3
1 2 1 1 3
p α µ α α
α µ α α
+ − + +
=
+ − + +
.
Substituting the corresponding value of G p
( )
and the value of C( )
α in (3.11), (3.2) followsSub-case (b)
(
3(1 2 ))(
2)
4 1 1 3α µ
α α
+
≤ ≤
+ +
(
)(
)
2
(1 2 )
1 1 3
α
α α
+
+ +
( )
0G p′ < and G p
( )
is maximum at p=0 In this sub-case maxG p( )
=16 1(
+α)(
1 3+ α µ)
.Case (iii)
(
(1 2 ))(
2)
1 1 3
α µ
α α
+ ≥
+ +
( )
(
)(
) (
)
2 4(
)(
) (
)
2 2(
)(
)
2 1 1 3 1 2 4 2 1 1 3 3 1 2 16 1 1 3
G p = − µ +α + α − + α p + µ +α + α − + α p + +α + α µ
Sub-case (a)
(
(1 2 ))(
2)
1 1 3
α µ
α α
+
≤ ≤
+ +
(
)(
)
2
3(1 2 ) 2 1 1 3
α
α α
+
( )
0G p′ < and maximum G p
( ) ( )
=G 0 =16 1(
+α)(
1 3+ α µ)
Combining the cases (ii)-(b) and (iii)-(a) we arrive at (3.3) Sub-case (b)
(
3(1 2 ))(
2)
2 1 1 3
α µ
α α
+ ≥
+ +
A simple calculus shows that G p
( )
is maximum at(
)(
) (
)
(
)(
) (
)
2 2
2 1 1 3 3 1 2
1 1 3 1 2
p µ α α α
µ α α α
+ + − +
=
+ + − +
Substituting the corresponding value of G p
( )
and the value of C( )
α in (3.11), (3.4) followsRemark 3.1 Put A=1 and B= -1 in the theorem we get the estimates for the classR1
( )
α .Taking A = 1, B = -1 and α =0 in the theorem we have Corollary 3.1 Iff R∈ 0, then
(
)
(
)
(
)
(
)
(
)
(
)
2 2 2
2 4 3
2
3 2
4 , 0; 2 1
3 4 3
4 ,0 ;
4 2 1
3 3
4 , ;
4 2
2 3 3
4 , . 2
2 1
a a a
µ
µ µ µ
µ
µ µ
µ µ
µ µ
µ
µ µ µ
−
− ≤
−
−
+ ≤ ≤
−
− ≤
≤ ≤
−
+ ≥
−
Letting A = 1,B = -1 and α =1 we get Corollary 3.2 If f R∈ , then
(
)
(
)
(
)
(
)
(
)
(
)
2 2 2
2 4 3
2
27 16 4
0; 9 144 9 8
27 32 4 27
,0 ;
9 32
144 9 8
4 27, 27;;
9 32 16
16 27 4 27
, .
9 16
144 8 9 a a a
µ µ µ
µ
µ µ
µ µ
µ
µ µ
µ µ µ
µ
−
− ≤
−
−
+ ≤ ≤
−
− ≤
≤ ≤
−
+ ≥
−
This results was proved by Janteng et al [4] Theorem 3.2 Let f R∈ 2
(
α; ,A B)
, then2
2 4 3
(3.12)
(3.13)
(3.14)
(3.15)
Proof. We have
( )
( )
1( )
( )
1
Aw z f z zf z
Bw z
α +
′ + ′′ =
+
Taking real parts,
( )
( )
1( )
( )
1 1Re Re
1 1
1
Aw z Ar A
f z zf z
Br B
Bw z
α + − −
′ + ′′ = ≥ >
+ − −
(z r= )
This implies that
(3.16)
(
)
(
)
2(
)
3( )
2 3 4
1
1 B 2 1 a z 3 1 2 a z 4 1 3 a z ... P z
A B α α α
−
+ + + + + + + =
−
Identifying the terms in (3.16), we get
(3.17)
(
)
(
)
(
)
1 2 2 3 3 41 2 1
1 3 1 2
1 4 1 3
p A B a B p A B a B p A B a B α α α − = − + − = − + − = − +
System (3.17) yields (3.18)
( )
(
2)
2 4 3
C α a a −µa =9
(
1 2+ α) ( )
2p1 4p3 −8 1µ(
+α)(
1 3+ α)( )
2p2 2,(
)
(
)(
)
{
}
(
)(
)(
) (
{
)
(
)(
) }
(
)
(
)
(
)(
)
{
}
(
)(
)(
) (
{
)
(
)(
)
}
(
)
(
)
(
)(
)
2 2 2 2 2 2 2 2 2 2 2 2 227 1 2 16 1 1 3
1 4
9 1 2 144 1 1 3 1 2 9 1 2 8 1 1 3
0;
27 1 2 32 1 1 3
1 4
9 1 2 144 1 1 3 1 2 9 1 2 8 1 1 3
27 1 2
0 ;
32 1 1 3
1 B A B if B A B if B
α µ α α µ
α
α α α α µ α α
µ
α µ α α µ
α
α α α α µ α α
α µ α α + − + + − − − + + + + − + + + ≤ + − + + − + − + + + + − + + + + ≤ ≤ + + −
(
)
(
)
(
)(
)
(
(
)(
)
)
(
)(
)
(
)
{
}
(
)(
)(
)
{
(
)(
) (
)
}
(
)
(
)
(
)(
)
2 2 2 2 2 2 2 2 2 2 2 4 9 1 227 1 2 27 1 2
; 32 1 1 3 16 1 1 3
16 1 1 3 27 1 2
1 4
9 1 2 144 1 1 3 1 2 8 1 1 3 9 1 2
27 1 2 . 16 1 1 3 A B if B A B if µ α α α µ
α α α α
µ α α α µ
α
α α α µ α α α
(3.19)
( )
(
)(
)(
)
2
2
288 1 1 3 1 2 1
A B C
B
α = − +α + α + α
−
By Lemma 2.2, (3.19) can be written as
(
)(
)
2(
2)
21 1
8 1µ α 1 3α p 4 p x
− + + + −
for some x and z with x ≤1,z ≤1. or
(3.20)
( )
(
2)
2 4 3
C α a a −µa
(
)
2(
)(
)
4 19 1 2α 8 1µ α 1 3α p
= + − + +
(
)
(
)(
)
(
)
2 2 2
1 1
2 9 1 2 α 8 1µ α 1 3α p 4 p x
+ + − + + −
(
2)
{
(
)
2(
)(
)
(
)(
)
}
21
4 p 9 1 2α 8 1µ α 1 3α 32 1µ α 1 3α x
− − + − + + + + +
(
)
2(
2)
21 1
18 1 2α p 4 p 1 x z
+ + − −
.
Replacing p1by p∈ 0,2and applying triangular inequality to (3.20), we get
( )
22 4 3
C α a a −µa ≤ 9 1 2
(
+ α)
2−8 1µ(
+α)(
1 3+ α)
p4
(
)
(
)(
)
(
)
2 2 2
2 9 1 2+ α −8 1µ +α 1 3+ α p 4− p δ
(
)
(
)
(
)(
)
(
)(
)
2
2 2 2
4 p 9 1 2α 8 1µ α 1 3α p 32 1µ α 1 3α δ
+ − + − + + + + +
+18 1 2
(
+ α)
2p(
4−p2)
1−δ2 . (δ = x ≤1)
which can be put in the form
( )
(
a a a)
(
)
p[
p p(
p)
x p(
p)
x(
p)
(
x)
z]
( )
(
)
(
)(
)
(
)
(
)
(
)
(
)(
)
(
)
(
)
{
(
)
(
)(
)
}
(
)(
)
(
)
(
)
(
)(
)
(
)
(
)
(
)
(
)(
)
2 4 2 2
2 2 2
2 2
2 2
2
2 4 2 2
2 2
2 4 3
9 1 2 8 1 1 3 18 1 2 4
2 9 1 2 8 1 1 3 4
9 1 2 8 1 1 3 32 1 1 3
4 0;
18 1 2
9 1 2 8 1 1 3 18 1 2 4
2 9 1 2 8 1 1 3
p p p
p p
p
p if
p
p p p
C a a a
α µ α α α
α µ α α δ
α µ α α µ α α
δ µ
α
α µ α α α
α µ α α
α µ
+ − + + + + −
+ + − + + −
+ − + + − + +
+ − ≤
− +
+ − + + + + −
+ + − + +
− ≤
(
)
(
)
{
(
)
(
)(
)
}
(
)(
)
(
)
(
)
(
)(
)
(
)(
) (
)
(
)
(
)
(
)(
) (
)
(
)
(
)
{
(
)(
) (
)
}
(
)(
)
(
)
2 2
2 2
2 2
2 2
2 4 2 2
2 2 2
2 2 2
2
4
9 1 2 8 1 1 3 32 1 1 3
4
18 1 2 9 1 2
0 ;
8 1 1 3
8 1 1 3 9 1 2 18 1 2 4
2 8 1 1 3 9 1 2 4
8 1 1 3 9 1 2 32 1 1 3
4
18 1 2
p p
p p
p
if
p p p
p p
p p
p
δ
α µ α α µ α α
δ α
α µ
α α
µ α α α α
µ α α α δ
µ α α α µ α α
α
−
+ − + + + + +
−
− +
+ ≤ ≤
+ +
+ + − + + + −
+ + + − + −
+ + − + + + +
−
− +
(
)
(
)(
)
2 2
9 1 2 . 8 1 1 3 if
δ
α µ
α α
+
≥
+ +
= F
( )
δ( )
0F′ δ > which means that F
( )
δ is increasing in [0 , 1] and F( )
δ( )
(
)
(
)
(
)(
)
(
)
(
)(
)
(
)
(
)
{
(
)
(
)(
)
}
(
)(
)
(
)
(
)(
)
(
)
(
)(
)
(
)
(
)
(
)(
)
{
}
(
)
2 4 2 2 2
2
2 2
2 4 2 2 2
2
2 2
2
2 4 3
9 1 2 8 1 1 3 2 9 1 2 8 1 1 3 4
4 9 1 2 8 1 1 3 32 1 1 3 , 0;
9 1 2 8 1 1 3 2 9 1 2 8 1 1 3 4
(4 ) 9 1 2 8 1 1 3 32 1 1
p p p
p p if
p p p
p p
C a a a
α µ α α α µ α α
α µ α α µ α α µ
α µ α α α µ α α
α µ α α µ α
α µ
+ − + + + + − + + −
+ − + − + + − + + ≤
+ − + + + + − + + −
+ − + − + + + +
− ≤
(
)
(
)
(
)(
)
(
)(
) (
)
(
)(
) (
)
(
)
(
)
{
(
)(
) (
)
}
(
)(
)
(
)
(
)(
)
2
2 4 2 2 2
2
2 2
2
3 9 1 2
0 ;
8 1 1 3
8 1 1 3 9 1 2 2 8 1 1 3 9 1 2 4
4 8 1 1 3 9 1 2 32 1 1 3
9 1 2 . 8 1 1 3 if
p p p
p p
if
α
α µ
α α
µ α α α µ α α α
µ α α α µ α α
α µ
α α
+
+
≤ ≤
+ +
+ + − + + + + − + −
+ − + + − + + + +
+
≥
+ +
Or
(3.22)
( )
(
2)
2 4 3
C α a a −µa ≤G p
( )
. Case (i) µ ≤0( )
(
)
2(
)(
)
4(
)
2(
)(
)
22 9 1 2 8 1 1 3 4 27 1 2 16 1 1 3
G p = − + α − µ +α + α p + + α − µ +α + α p
(
)(
)
128 1 α 1 3α µ
+ + + .
( )
G p is maximum for
which gives
(
)
(
)(
)
(
)
(
)(
)
2 2
27 1 2 16 1 1 3
9 1 2 8 1 1 3
p α µ α α
α µ α α
+ − + +
=
+ − + + .
Putting the corresponding value of G p
( )
along with the value of C( )
α from (3.19) in (3.22),we get (3.12).
Case (ii)
(
(
)(
)
)
2
9 1 2 0
8 1 1 3
α µ
α α
+ ≤ ≤
+ +
( )
(
)
2(
)(
)
4(
)
2(
)(
)
22 9 1 2 8 1 1 3 4 27 1 2 32 1 1 3
G p = − + α − µ +α + α p + + α − µ +α + α p
(
)(
)
128 1 α 1 3α µ
+ + +
( )
=−8[
9(
1+2)
2 −8(
1+)(
1+3)
]
3 +8[
27(
1+2)
2 −16(
1+)(
1+3)
]
=0′ p p p
Sub-case (a)
(
(
)(
)
)
2
27 1 2 0
32 1 1 3
α µ
α α
+ ≤ ≤
+ +
An elementary calculus shows that G p
( )
is maximum at(
)
(
)(
)
(
)
(
)(
)
2 2
27 1 2 32 1 1 3
9 1 2 8 1 1 3
p α µ α α
α µ α α
+ − + +
=
+ − + + .
With the corresponding value of G p
( )
along with the value of C( )
α in (3.22) , we arrive at(3.13).
Sub-case (b)
(
(
)(
)
)
(
(
)(
)
)
2 2
27 1 2 9 1 2
32 1 1 3 8 1 1 3
α α
µ
α α α α
+ +
≤ ≤
+ + + +
G p′
( )
<0 and G p( )
is maximum at p=0 . maxG p( ) ( )
=G 0 =128 1(
+α)(
1 3+ α µ)
.Case (iii)
(
(
)(
)
)
2
9 1 2 8 1 1 3
α µ
α α
+ ≥
+ +
( )
(
)(
) (
)
2 4(
)(
)
(
)
2 22 8 1 1 3 9 1 2 4 16 1 1 3 27 1 2
G p = − µ +α + α − + α p + µ +α + α − + α p
(
)(
)
128 1 α 1 3α µ
+ + + .
Sub-case (a)
(
(
)(
)
)
(
(
)(
)
)
2 2
9 1 2 27 1 2
8 1 1 3 16 1 1 3
α α
µ
α α α α
+ +
≤ ≤
+ + + +
( )
0G p′ < and MaxG p
( ) ( )
=G 0 =128 1(
+α)(
1 3+ α µ)
.Combining the cases (ii-b) and (iii-a), (3.14) follows.
Sub-case (b)
(
(
)(
)
)
2
27 1 2 16 1 1 3
α µ
α α
+ ≥
+ +
An easy calculation shows thatG p
( )
is maximum at(
)(
)
(
)
(
)(
) (
)
2 2
16 1 1 3 27 1 2
8 1 1 3 9 1 2
p µ α α α
µ α α α
+ + − +
=
+ + − + .
Substituting the corresponding value of G p
( )
along with the value of C( )
α in (3.22), weobtain (3.15).
Remark 3.2 Putting A=1 and B= -1 in the theorem we get the estimates for the class R2
( )
α .REFERENCES
[1] R.M.Goel and B.S.Mehrok, A subclass of univalent functions, Houston J. Math., 8(1982), 343-357.
[2] R.M.Goel and B.S.Mehrok, A subclass of univalent functions, J.Austral.Math.Soc. (Series A), 35(1983), 1-17.
[3] W.K. Hayman, Multivalent functions, Cambridge Tracts in math. and math-physics, No. 48 Cambridge university press, Cambridge (1958).
[4] A.Janteng , S.A.Halim and M.Darus, Estimates on the second Hankel functional for functions whose derivative has a positive real part, J. Quality Measurement and analysis, JQMA(1) (2008) 189-195.
[5] R.J.Libera and E.J.Zlotkiewics, Early coefficients of the inverse of a regular convex function, Proceeding Amer. Math.Soc., 85(1982) , 225-230.
[6] J.E. Littlewood , On inequalties in the theory of functions. Procc. London Math. Soc. 23 (1925) ,481-519 . [7] T.H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14 (1963) 514-520.
[8] B S Mehrok. Two subclasses of certain analytic functions (to appear) .
[9] J W Noonan and D K Thomas , Coefficient differences and Hankel Determinant of really p-val;ent functions , Proc. London Math Soc.[ 3],25 (1972), 503-524.
[10] J W Noonan and D K Thomas , On the second Hankel determinant of a really mean p-valent functions. Trans. American Math Society (223) 1976, 337-346.
[11] K.Noshiro, On the theory of schlicht functions,J. Fac. Soc. Hokkaido univ.Ser.1,2 (1934-1935), 129-155. [12] N N Pascu, Alpha- starlike functions, Bull. Uni Brasov CXIX, 1977.
[13] C H Pommerenke, On the coefficients and Hankel determinant of univalent functions, J. London math. Soc. 41, 1966, 111-122.
[14] C H Pommerenke , On the Hankel determinant of univalent functions, Mathematica 14 (C) , 1967, 108-112.
[15] C.H. Pommerenke , Univalent functions wadenheoch and Ruprech, Gotingen, 1975
[16] W.W. Rogosinski, On coefficients of subordinate functions, Proc. London Math. Zeith (1932), 92-123. [17] M O Reade, On close to convex univalent functions, Michhigan Math. J. 1955, 59-62.
[18] S.S.Warschawski, On the higher derivative at the boundary in conformal mapping, Tran. Amer. Math.Soc., 38(1935), 310-340.
