Gen. Math. Notes, Vol. 21, No. 1, March 2014, pp. 86-96
ISSN 2219-7184; Copyright © ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Fekete-Szegö Inequality for a New Class and
Its Certain Subclasses of Analytic Functions
Gurmeet Singh
Khalsa College, Patiala
E-mail: meetgur111@gmail.com
(Received: 2-12-13 / Accepted: 25-1-14)
Abstract
We introduce some classes of analytic functions, its subclasses and obtain
sharp upper bounds of the functional | − µ | for the analytic function
= + ∑∞
, | | < 1 belonging to these classes and subclasses.
Keywords: Univalent functions, Starlike functions, Close to convex functions
and bounded functions.
1
Introduction
Let
denote the class of functions of the form
=
+ ∑∞
which are analytic in the unit disc = { : | | < 1|}. Let
functions of the form (1.1), which are analytic univalent in .
(1.1)
be the class of
In 1916, Bieber Bach ( [7], [8] ) proved that | | ≤ 2 for the functions
1923, Löwner [5] proved that | | ≤ 3 for the functions
∈ .
∈ . In
Fekete-Szegö Inequality for a New Class and…
87
With the known estimates | | ≤ 2 and | | ≤ 3, it was natural to seek some
relation between
and
for the class , Fekete and Szegö [9] used Löwner’s
method to prove the following well known result for the class .
Let ( ) ∈ , then
|
−µ
µ ≤ 0;
3 − 4µ,
1 + 2 exp $
|≤
4µ − 3,
% µ
&%µ
0 ≤ µ ≤ 1; *
',
µ ≥ 1.
(1.2)
The inequality (1.2) plays a very important role in determining estimates of higher
coefficients for some sub classes (See Chhichra [1], Babalola [6]).
Let us define some subclasses of .
We denote by S*, the class of univalent starlike functions
+( ) = + , -
∈
∞
;< $ >(=) ' > 0, ∈ .
=>(=)
We
denote
ℎ( ) =
;<
+,B
C(=D′ (=)E
D′ (=)
by
∞
,
> 0, ∈ .
A function ( ) ∈
that
and satisfying the condition
@,
∈
(1.3)
the
class
of
univalent
and satisfying the condition
convex
(1.4)
is said to be close to convex if there exists +( ) ∈ F ∗ such
;< $ >(=) ' > 0, ∈ .
=H′(=)
functions
(1.5)
The class of close to convex functions is denoted by C and was introduced by
Kaplan [3] and it was shown by him that all close to convex functions are
univalent.
F ∗ (I, J) = K ( ) ∈
@(I, J) = Q ( ) ∈
;
;
=H ′ (=)
H(=)
$=H ′ (=)'′
H′(=)
≺
≺
&MN=
&MO=
&MN=
&MO=
, −1 ≤ J < I ≤ 1, ∈ P
, −1 ≤ J < I ≤ 1, ∈ R
(1.6)
(1.7)
88
Gurmeet Singh
It is obvious that F ∗ I, J is a subclass of F ∗ and @ I, J is a subclass of @.
We introduce a new class as S
will denote this class as F ∗
∈
, ′,
′′
;
.
U
=T$H ′ = ' MH = H ′′ = V
H = H′ =
We will also deal with two subclasses of F ∗
F
∗
F∗
, ,
′
, ′,
′′
′′
; I, J = S
∈
; I, J, X = S
;
U
, ′,
=T$H ′ = ' MH = H ′′ = V
∈
H = H′ =
;
U
′′
≺
=T$H ′ = ' MH = H ′′ = V
H = H′ =
≺
&M=
&%=
; ∈ W and we
defined as follows:
&MN=
&MO=
; ∈ W
&MN= Y
≺ $&MO=' ; ∈ W
(1.8)
(1.9)
Symbol ≺ stands for subordination, which we define as follows:
Principle of Subordination
and Z
be two functions analytic in . Then
is called subordinate
Let
to F(z) in if there exists a function [
analytic in satisfying the conditions
[ 0 = 0 and |[ | < 1 such that
=Z [
; ∈
and we write
≺ Z .
By \, we denote the class of analytic bounded functions of the form
[
= ∑∞
&B
, [ 0 = 0, |[
| < 1.
It is known that |B& | ≤ 1, |B | ≤ 1 − |B& | .
2
&%^ =
3
(1.11)
Preliminary Lemmas
For 0 < B < 1, we write [
&M^ =
(1.10)
= 1 + 2B + 2
+ ⋯.
Main Results
Theorem 3.1: Let
∈ F∗
= $&M]=' so that
]M=
(2.1)
, ′,
′′
, then
Fekete-Szegö Inequality for a New Class and…
|
−` |≤
g
h
d f − e `, ` ≤ i ;
b&
h
j
,
≤`≤ ;
i
g
cgg
j
b ` − &e ,
`≥ .
ae
f
g
89
(3.1)
&e
(3.2)*
(3.3)
The results are sharp.
Proof: By definition of F ∗ ( , ′ ,
U
=T$H ′ (=)' MH(=)H ′′ (=)V
H(=)H ′ (=)
=
&M^(=)
&%^(=)
′′ ),
we have
; [( ) ∈ \.
(3.4)
Expanding the series (3.4), we get
(1 + 2
+3
+ − − −) + ( +
+
+ − − −)(2
+
+ − − −)(1 + 2
+3
12 g + − − −) = (1 +
−)(1 + 2B& + 2(B + B& ) + − − −).
1+4
(1 + 3
+ (6 +4
+ (4 +2
)
)
+6
+
+−−
+−−− + 2
+ (6 +2 ) + − − − =
+ − − −)(1 + 2B& + 2(B + B& ) + − − −).
1+6
+ 6(2 + )
2B + 2B& ) + − − −
+ − − −= 1 + (3
+ 2B& ) + (4
+2
+6
B& +
(3.5)
Identifying terms in (3.5), we get
=
=
&
g
B&
(3.6)
B +
&e
f
B& .
(3.7)
From (3.6) and (3.7), we obtain
−`
= B +l
&
g
&e
f
− `m B& .
g
e
(3.8)
Taking absolute value, (3.8) can be rewritten as
|
−`
| ≤ g |B | + n
&
&e
f
− e `n |B& |.
g
Using (1.9) in (3.9), we get
|
−`
| ≤ (1 − |B& | ) + n
g
&
&e
f
(3.9)
− e `n |B& | = g + Kn
g
&
&e
f
− e `n − gP |B& | . (3.10)
g
&
90
Gurmeet Singh
Case I: ` ≤
&e
f
. (3.10) can be rewritten as
|
−`
|≤
&
|
−`
|≤
&
|
−`
| ≤ − K ` − P |B& | ≤ .
g
e
&i
g
+ K$
g
&e
f
− `' − P |B& | = + K
g
&
e
&
g
g
h
&i
Subcase I (a): ` ≤ . Using (1.9), (3.11) becomes
+K
g
h
i
− `P =
h
g
&i
e
&e
f
− `P |B& | .
g
e
(3.11)
− `.
g
e
(3.12)
Subcase I (b): ` ≥ . We obtain from (3.11)
Case II: ` ≥
|
−`
|≤
&
&e
&
g
g
f
h
i
h
&
(3.13)
. Preceding as in case I, we get
+ Ke ` − eP |B& | .
g
Subcase II (a): ` ≤ g.
Takes the form |
j
−`
j
(3.14)
(3.14)
|≤ .
g
&
(3.15)
Combining subcase I (b) and subcase II (a), we obtain
|
−`
|≤
|
−`
|≤ `−
e
&
g
≤ ` ≤ g.
i
h
j
(3.16)
Subcase II (b): ` ≥ g. Preceding as in subcase I (a), we get
g
j
&e
.
f
(3.17)
Combining (3.12), (3.16) and (3.17), the theorem is proved.
Extremal
&(
) = o2 K
function
&
&%=
for
− log (1 − )P.
(3.1)
Extremal function for (3.2) is defined by
Theorem 3.2: Let F ∗ ( , ′ ,
′′
; I, J), then
and
(3.3)
is
( ) = olog $ U ' .
&%=
&
defined
by
Fekete-Szegö Inequality for a New Class and…
−`
q
q≤
d
b
b
c
b
b
a
I − J 5I − 14J
I−J
5I − 14J − 9
−
`, ` ≤
; 3.18
72
9
8 I−J
I−J
5I − 14J − 9
5I − 14J + 9
,
≤`≤
;
3.19 *.
8
8 I−J
8 I−J
I−J
I − J 5I − 14J
5I − 14J + 9
`−
,
`≥
. 3.20
9
72
8 I−J
The results are sharp.
Proof: By definition of F ∗
U
=T$H′ = ' MH = H′′ = V
H =
H′
91
=
=
, ′,
′′
; I, J , we have
;[
&MN^ =
&MO^ =
∈ \.
(3.21)
Expanding the series (3.21), we get
1+6
+6 2 +
+ − − −= 1 + {3 + I − J B& } + 4
+−−−
3 I − J B& + I − J B − JB&
Identifying terms in (3.22), we get
And
=
B +
N%O
i
N%O hN%&gO
j
=
N%O
B& .
B&
+2
+
(3.22)
(3.23)
(3.24)
From (3.23) and (3.24), we obtain
−`
=
B +l
N%O
i
N%O hN%&gO
j
−
N%O U
e
`m B& .
Taking absolute value, (3.25) can be rewritten as
|
−`
|≤
(N%O)
i
|B | + n
(N%O)(hN%&gO)
j
−
(N%O)U
e
Using (1.9) in (3.26), we get
|
−`
|≤
Case I: ` ≤
|
−`
(N%O)
=
i
(1 − |B& | ) + n
(N%O)
i
+ Kn
(hN%&gO)
|≤
i(N%O)
(N%O)
i
(N%O)(hN%&gO)
(N%O)(hN%&gO)
j
j
−
−
(3.25)
`n B& .
(N%O)U
e
(3.26)
`n |B& |
`n −
(N%O)
`' −
(N%O)
(N%O)U
e
i
P |B& |.
. (3.27) can be rewritten as
+ K$
(N%O)(hN%&gO)
j
−
(N%O)U
e
i
P |B& |
(3.27)
92
Gurmeet Singh
=
+$
N%O
i
N%O hN%&gO%e
j
Subcase I (a): ` ≤
(hN%&gO%e)
Subcase I (b): ` ≥
(hN%&gO%e)
q
−`
|
−`
q≤
(N%O)
|≤
Case II: ` ≥
|
−`
|≤
≤
i
+$
(N%O)
i
j
i(N%O)
−$
(N%O)U
e
(hN%&gO)
i(N%O)
(N%O)
i
i
−`
|≤
(N%O)
|
−`
|≤
(N%O)
|
−`
|≤
(N%O)U
i
+$
.
e
(3.28)
−
(N%O)U
e
`' =
(N%O)(hN%&gO)
. We obtain from (3.28)
`−
j
' |B& | ≤
(N%O)(hN%&gO%e)
j
−
(N%O)U
e
.
(N%O)
i
`. (3.29)
(3.30)
. Preceding as in case I, we get
+ K$
(N%O)
`' |B& |.
N%O U
Using (1.9), (3.28) becomes
(N%O)(hN%&gO%e)
Subcase II (a): ` ≤
|
i(N%O)
−
(N%O)U
e
(N%O)U
e
`−
`−
(hN%&gOMe)
i(N%O)
'−
(N%O)(hN%&gO)
j
j
i
' |B& |.
(N%O)(hN%&gOMe)
P |B& |
(N%O)
(3.31)
. (3.31) takes the form
(3.32)
Combining subcase I (b) and subcase II (a), we obtain
i
Subcase II (b): ` ≥
e
(hN%&gO%e)
i(N%O)
`−
(hN%&gOMe)
i(N%O)
.
(N%O)(hN%&gO)
j
(3.33)
. Preceding as in subcase I (a), we get
(hN%&gOMe)
i(N%O)
≤` ≤
Combining (3.29), (3.33) and (3.34), the theorem is proved.
Extremal function for (3.18) and (3.20) is defined by
N
2
O
(
)
)
v
=
T(1
+
J
(I − 1) − 1V.
&
I(I + J)
(3.34)
Fekete-Szegö Inequality for a New Class and…
93
=o
Extremal function for (3.19) is defined by
NMO
K 1+J
Corollary 3.3: Putting I = 1, J = −1 in the theorem 3.2, we get
|
&e
theorem (3.1).
g
Theorem 3.4: Let F ∗ ( , ′ ,
q
−`
′′
h
; X), then
gY U
`,
%Y)U
&i( %Y)(
&
eY { %&gY z % &YU Mj Y% f
,
( %Y)
iY U ( %Y)
YCeY z %&gYU %& YM fE
q≤
− 1P .
d f − e `, ` ≤ i ;
b&
h
j
|≤
,
≤
`
≤
,* which is the result obtained in
g
i
g
cg
j
b ` − &e ,
`≥ .
ae
f
g
−`
d
b
wxy
Uy
%Y)U
− e(
c
b gY U ` − YCeY z %&gYU %& YM
ae( %Y)U
&i( %Y)( %Y)U
The results are sharp.
Proof: By definition of F ∗ ( , ′ ,
U
=T$H ′ (=)' MH(=)H ′′ (=)V
′′
&M^(=) Y
fE
,
`≤
≤`
(3.35)
eY { %&gYz % &YU Mj Y% f
;
iY U ( %Y)
eY { %&gY z % YU M f
≤
;
iY U ( %Y)
`≥
(3.36)*.
(3.37)
eY { %&gYz % Y U M f
.
iY U ( %Y)
; X), we have
= $&%^(=)' ; [( ) ∈ \.
H(=)H ′ (=)
(3.38)
Expanding the series (3.38), we get
1+6
+ 6(2 + ) + − − −= 1 + (3 + 2B& ) + (4 +2
6 B& + 2B + 2B& ) + − − − Y
eY%h
= K1 + X(3 + 2B& ) + X(4 +
+ 6X B& + 2B + 2XB& )
+
+ − − −P
(3.39)
Identifying terms in (3.39), we get
=
=
Y
( %Y)
&
( %Y)
B&
(3.40)
B +
Y(eY z %&gY U %& YM f)
&i( %Y)( %Y)U
From (3.40) and (3.41), we obtain
−`
=
&
( %Y)
B +l
B& .
Y(eY z %&gY U %& YM f)
&i( %Y)(
%Y)U
(3.41)
− e(
gY U
%Y)U
`m B& .
(3.42)
94
Gurmeet Singh
Taking absolute value, (3.42) can be rewritten as
|
−`
|≤
|B | + n
&
Y(eY z %&gY U %& YM f)
( %Y)
&i( %Y)(
%Y)U
−
e( %Y)U
Using (1.9) in (3.43), we get
|
−`
|≤
Case I: ` ≤
|
−`
=
&
( %Y)
(
−`
−`
|≤
=
|≤
=
|≤
Case II: ` ≥
|
−`
Y(eY z %&gY U %& YM f)
−`
&i( %Y)(
(
iY( %Y)
+ K$
%Y)
&
+$
%Y)
&
(
%Y)U
(
+$
%Y)
&
|≤
≤
|≤
Y(eY z %&gY U %& YM f)
&i( %Y)( %Y)U
&i( %Y)(
&i( %Y)(
(
iY U ( %Y)
`−
%Y)U
+ K$e(
%Y)
+ $e(
%Y)
&
%Y)U
− e(
gY U
iY( %Y)
&
( %Y)
gY U
%Y)U
gY U
`n −
%Y)U
&
`n |B& |
P |B& |.
( %Y)
(3.44)
gY U
iY U (
%Y)U
− e(
`' −
gY U
%Y)U
&
P |B& |
( %Y)
`' |B& |.
(3.45)
%Y)U
`.
− e(
`'
gY U
%Y)U
(3.46)
. We obtain from (3.45)
' |B& | ≤
eY { %&gY z % &Y U Mj Y% f
&i( %Y)( %Y)U
`−
`−
%Y)U
gY U
gY U
&
.
( %Y)
(3.47)
. Preceding as in case I, we get
%Y)U
eY { %&gY z %
.
gY U
− e(
Using (1.9), (3.45) becomes
eY { %&gY z % &Y U Mj Y% f
(eY z %&gY U %& YM f)
(
− e(
eY { %&gY z % &Y U Mj Y% f
− $e(
%Y)
&
%Y)U
iY U ( %Y)
&i( %Y)( %Y)U
(
− e(
eY { %&gY z % &Y U Mj Y% f
Y(eY z %&gY U %& YM f)
&
%Y)U
(3.43)
. (3.44) can be rewritten as
eY { %&gY z % &Y U Mj Y% f
Subcase II (a): ` ≤
|
&i( %Y)(
(eY z %&gY U %& YM f)
Subcase I (b): ` ≥
|
Y(eY z %&gY U %& YM f)
+ Kn
%Y)
&
Subcase I (a): ` ≤
|
(1 − |B& | ) + n
`n B& .
gY U
YUM
%Y)
'−
Y(eY z %&gY U %& YM f)
&i( %Y)( %Y)U
' |B& |.
eY { %&gY z % Y U M f
f
&i( %Y)( %Y)U
&
P |B& |
( %Y)
(3.48)
. (3.48) takes the form
(3.49)
Fekete-Szegö Inequality for a New Class and…
95
Combining subcase I (b) and subcase II (a), we obtain
|
−`
|≤
&
( %Y)
Subcase II (b): ` ≥
eY { %&gY z % &Y U Mj Y% f
iY U (
%Y)
eY { %&gY z % Y U M f
iY U ( %Y)
Proceeding as in subcase I (a), we get
|
−`
|≤
gY U
U` −
e( %Y)
≤` ≤
eY { %&gY z % Y U M f
iY U ( %Y)
.
Y(eY z %&gY U %& YM f)
&i( %Y)( %Y)U
(3.50)
(3.51)
Combining (3.46), (3.50) and (3.51), the theorem is proved.
Extremal function for (3.35) and (3.37) is defined by
&(
=
)
=
d
b |2 } <
b
„
c
=
b
|2 } <
b
„
a
Q
Q
& &M= ~•€ & &M= ~•z & &M= ~••
&
$
'
M
$
'
M
$
'
M%%%M ‚ƒ>=R
Y%& &%=
Y% &%=
Y%h &%=
&M= ~•€
&
$
'
Y%& &%=
M
&M= ~•z
&
$
'
Y% &%=
M
&M= ~••
&
$
'
Y%h &%=
&
M%%%M ‚ƒ>
Extremal function for (3.36) is defined by
… ,
g=
R
(&%=)U
… ,
−`
†‡ˆ<‰ Š‹‰ ‹ X Œ ‹……
( ) = olog $ U ' .
&%=
Corollary 3.5: Putting X = 1, in the theorem 3.4, we get
|
†‡ˆ<‰ Š‹‰ ‹ X Œ <•<†
&
5
19 4
d − `, ` ≤ ;
8
b 36 9
1
5
7
*
|≤
,
≤`≤
8
4
c4
19
7
b4
`
−
,
`
≥
.
a9
36
4
which is the result obtained in theorem (3.1).
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[1]
[2]
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Journal of Mathematics, 8(1982), 343-357.
*.
96
Gurmeet Singh
[3]
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K.O. Babalola, The fifth and sixth coefficients of α-close-to-convex
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abbildung, Math. Ann., 77(1916), 153-172.
L. Bieberbach, Uber die koeffizientem derjenigen potenzreihen, welche
eine schlithe abbildung des einheitskreises vermitteln, Preuss. Akad. Wiss.
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funktionen, J. London Math. Soc., 8(1933), 85-89.
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