Gen. Math. Notes, Vol. 3, No. 2, April 2011, pp. 88-96
ISSN 2219-7184; Copyright © ICSRS Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Coefficient Inequality for Certain
Sub-Classes of Analytic Functions with
Respect to Symmetric Points
B.S. Mehrok1, Harjinder Singh2 and Deepak Gupta3
1
Ex. Prof, Department of Mathematics, Khalsa College, Amritsar (Punjab)
E-mail: beantsingh.mehrok@gmail.com
2
Department of Mathematics, Govt. Rajindra College, Bathinda (Punjab)
E-mail: harjindpreet@gmail.com
3
Department of Mathematics, MM Engg. College, Mullana (Haryana)
E-mail: guptadeepak2003@yahoo.co.in
(Received: 8-2-11/ Accepted: 23-3-11)
Abstract
Let be the class of analytic functions = + ∑
in the unit disc
E = z: |z| < 1. Denote by Sα; δ, A, B the class of analytic functions in satisfying the
! condition 1 − ! #!# + "
$ ! " %
"
&! #!# '
"
≺ $)*+ % , (0 ≤ α ≤ 1 and 0 < 4 ≤ 1. We
)*,
-
obtain sharp upper bounds for the functional |5 − 6 |.
Keywords: Analytic functions, Starlike functions with respect to symmetrical points,
Subordination.
1.
Introduction
Let denote the class of functions
= + 7 1.1
Coefficient Inequality for Certain…
89
which are analytic in the unit disc E = z: |z| < 1.
Denote by U, the class of functions
wz = 7 c; z ;
1.2
;)
which are analytic in the unit disc E = z: |z| < 1 and satisfying there the conditions
w0 = 0 and |wz| < 1. It is known [5] that
|c) | ≤ 1,
|c | ≤ 1 − |c) | .
1.3
Let z and Fz be two analytic functions in E. Then z is subordinate to Fz if
there is an analytic function wz ∈ U such that z = F&wz'. Symbolically, we write
z ≺ Fz.
1.4
2 F CD E
G > 0.
− −
1.5
To avoid repetitions, it is admitted once that 0 ≤ α ≤ 1 , −1 ≤ B < A ≤ 1 , 0 < 4 ≤ 1
and E = z: |z| < 1.
Sakaguchi [6] introduced the class SA ∗ of functions in satisfying the condition
These functions are called starlike with respect to symmetrical points in E.
The class K K of convex functions with respect to symmetrical points in E was defined and
studied by Das and Singh [2]. A function ∈ K K if and only if
CD L
&2 F '
F
& − −'
FM
> 0.
Denote by OK the class of functions in satisfying the following condition
CD E
2z F z
G > 0,
gz − g−z
g ∈ SA ∗ .
1.6
1.7
Goel and Mehrok[3] introduced and studied the class SA ∗ A, B of functions of the form
(1.1) satisfying the condition
2 F 1 + Az
≺
.
− − 1 + Bz
1.8
Janteng and Halim [4] considered the class K A A, B of functions of the form (1.1)
satisfying the condition
2 F F
1 + Az
≺
.
1.9
F
− −
1 + Bz
90
B. S. Mehrok et al.
Denote by CA A, B and CA F A, B the classes of functions of the form (1.1) satisfying
respectively the conditions
and
1 + Az
2z F z
≺
,
gz − g−z 1 + Bz
2z F zF
1 + Az
≺
,
F
gz − g−z
1 + Bz
g ∈ SA ∗
g ∈ SA ∗ .
1.10
1.11
Let FA A, B and FA F A, B denote the classes of functions of the form (1.1) satisfying
respectively the conditions
and
2z F z
1 + Az
≺
,
hz − h−z 1 + Bz
2z F zF
1 + Az
≺
,
hz − h−zF 1 + Bz
h ∈ KK
h ∈ KK .
1.12
1.13
It is easy to verify that SA ∗ 1, −1 ≡ SA ∗, K A 1, −1 ≡ K A and CA 1, −1 ≡ CA .
Also SA ∗ 1 − 2β, −1 ≡ SA ∗ β is the class of starlike functions with respect to symmetrical
points of order defined by Das and Singh [1].
Let us denote by SA ∗ δ; A , B, K A δ; A , B, CA δ; A , B, CA F δ; A, B, FA δ; A, B and
FA F δ; A, B the subclasses of functions z in which satisfy respectively the following
conditions
2 F 1 + Az Z
≺X
Y ,
1.14
− −
1 + Bz
&2 F '
1 + Az Z
Y ,
F ≺X
1 + Bz
& − −'
F
2z F z
1 + Az Z
≺X
Y ; g ∈ SA ∗ ,
gz − g−z
1 + Bz
&2z F z'
1.16
1 + Az Z
Y ; g ∈ SA ∗ ,
1 + Bz
1.17
2z F zF
1 + Az Z
≺X
Y ; h ∈ KK .
hz − h−zF
1 + Bz
1.19
F
&gz − g−z'
and
1.15
F
≺X
2z F z
1 + Az Z
≺X
Y ; h ∈ KK ,
hz − h−z
1 + Bz
We also introduce the following classes
1.18
Coefficient Inequality for Certain…
Sα; δ; A, B = L ∈ ; 1 − α
91
Z
&2 F '
2 F ≺ X1 + AzY M
+α
F
− −
1 + Bz
& − −'
F
2z F z
1 + Az Z
&2z F z'
Tα; δ; A, B = L ∈ ; 1 − α
+α
Y , g ∈ SA ∗ M
F ≺X
gz − g−z
1 + Bz
&gz − g−z'
F
2z F z
1 + Az Z
&2z F z'
Hα; δ; A, B = L ∈ ; 1 − α
+α
≺
X
Y , h ∈ KAM
hz − h−zF
1 + Bz
hz − h−z
F
2.
1.20
1.21
1.22
Coefficient Inequality
]^__` a. b [6]. eDf gz = z + 7 bh z h ∈ . iℎDk, lm k ≥ 2,
h
1 if g ∈ SA ∗
|bh | ≤ o1
if g ∈ K A
n
Theorem 2.1. Let ∈ Sα; δ; A, B . Then
where
u δA − B , if |λ + μ| ≤ ν,
s21 + 2α
|5 − 6 | ≤
t δ A − B
s 41 + α |λ + μ|, if |λ + μ| ≥ ν,
r
1 + α [2B + 1 − δA − B]
,
δ1 + 2αA − B
21 + α
ν=
.
δ1 + 2αA − B
λ=
Proof. Since ∈ Sα; δ; A, B, it follows that
&2 F '
1 + Awz Z
2 F 1 − α
+α
Y
F =X
− −
1 + Bwz
& − −'
F
2.1
2.3
2.4
2.5
2.6
2.7
By expanding (2.7), we obtain
1 + 21 + α z + 21 + 2α5 z + ⋯
1
= 1 + δA − Bc) z + XδA − Bc − δA − B zB + 1 − δA − B{ c) Y z + ⋯
2
Identifying the terms and solving, we get
=
δA − B
c
21 + α )
2.2
2.8
92
B. S. Mehrok et al.
5 =
δA − B
δA − B[2B + 1 − δA − B] c −
c)
21 + 2α
41 + 2α
From (2.8) and (2.9), we have
5 − 6 =
where λ is defined by (2.5).
δA − B
δ A − B
λ + μc) ,
c −
21 + 2α
41 + α
Applying triangular inequality,
|5 − 6 | ≤
δA − B
δ A − B
|c | +
|λ + μ||c) |
21 + 2α
41 + α
From (1.3) using |c | ≤ 1 − |c) | , (2.10) leads us to
|5 −
6 |
where ν is defined by (2.6).
If |λ + μ| ≤ ν, then
δA − B δ A − B
|λ + μ| − ν|c) | ,
≤
+
21 + 2α 41 + α
|5 − 6 | ≤
The bound is sharp for wz = z .
δA − B
.
21 + 2α
If |λ + μ| ≥ ν, then from (1.3) using |c) | ≤ 1, we get
|5 − 6 | ≤
This bound is sharp for wz = z.
δ A − B
|λ + μ|.
41 + α
Corollary 2.1. If ∈ SA ∗ δ; A , B then
where
δA − B
if |λ) + μ| ≤ ν) ,
|5 − 6 | ≤ | 2
δ A − B
|λ) + μ| if |λ) + μ| ≥ ν) ,
4
λ) =
2.9
2B + 1 − δA − B
δA − B
Corollary 2.2. If ∈ SA ∗ A , B, then
k} ν) =
2
.
δA − B
2.10
Coefficient Inequality for Certain…
where
93
A − B
if |λ + μ| ≤ ν ,
2
|5 − 6 | ≤ |
A − B
|λ + μ| if |λ + μ| ≥ ν ,
4
λ =
Corollary 2.3. If ∈ SA ∗ , then
2B
A − B
k} ν =
2
.
A − B
|5 − 6 | ≤ max. 1, |μ − 1|.
Corollary 2.4. If ∈ K A δ; A , B , then
δA − B
6
|5 − 6 | ≤
tδ A − B
|λ5 + μ|
r
16
u
where
λ5 =
if |λ5 + μ| ≤ ν5 ,
if |λ5 + μ| ≥ ν5 ,
42B + 1 − δA − B
8
and ν5 =
.
3δA − B
3δA − B
Corollary 2.5. If ∈ K A A, B, then
A − B
if |λ€ + μ| ≤ ν€ ,
6
|5 − 6 | ≤ |
A − B
|λ€ + μ| if |λ€ + μ| ≥ ν€ ,
16
where
λ€ =
Corollary 2.6. If ∈ K K , then
8B
8
k} ν€ =
.
3A − B
3A − B
4
1 1
|5 − 6 | ≤ Max. ‚ , ƒμ − ƒ„.
3 4
3
Theorem 2.2. If f ∈ Tα; δ; A, B and µ is a complex number, then
|5 −
where
6 |
≤
u δA − B + 1 + 2α
s
31 + 2α
1
tδ A − B
s 41 + α |λ + μ| + 3
r
if |λ + μ| ≤ ν,
if |λ + μ| ≥ ν,
2.11
2.12
94
B. S. Mehrok et al.
21 + α [2B + 1 − δA − B]
λ=
,
3δ1 + 2αA − B
41 + α
ν=
.
3δ1 + 2αA − B
Proof. Since ∈ Tα; δ; A, B, by definition of subordination
&2z F z'
2z F z
1 + Awz Z
1 − α
+α
=
X
Y
F
gz − g−z
1 + Bwz
&gz − g−z'
F
2.13
2.14
2.15
for some gz = z + 7 b; z ; ∈ SA∗ .
;
By expanding (2.15), we obtain
1 + 21 + α z + 1 + 2α35 − b5 z + ⋯
1
= 1 + δA − Bc) z + zδA − Bc − δA − B ‚B + 1 − δA − B„ c) { z + ⋯
2
After equating the terms and solving, we get
=
δA − B
c
21 + α )
δA − B
δA − B[2B + 1 − δA − B] 1
c −
c) + b5
31 + 2α
61 + 2α
3
From (2.16) and (2.17), we have
5 =
5 − 6 =
where λ is defined by (2.13).
δA − B
δ A − B
1
λ
c −
+
μc
+
b ,
)
41 + α
31 + 2α
3 5
2.16
2.17
Applying triangular inequality, we obtain
|5 − 6 | ≤
δA − B
δ A − B
1
|c | +
|λ
|
|b |.
+
μ||c
+
)
31 + 2α
41 + α
3 5
Using lemma 2.1 and |c | ≤ 1 − |c) |, (2.18) leads us to
1
δA − B δ A − B
|5 − 6 | ≤
|λ + μ| − ν|c) | + ,
+
31 + 2α 41 + α
3
where ν is defined by (2.14).
If |λ + μ| ≤ ν, then
|5 − 6 | ≤
δA − B + 1 + 2α
.
31 + 2α
2.18
Coefficient Inequality for Certain…
95
This is the sharp bound for wz = z .
If |λ + μ| ≥ ν, then from (1.3) using |c) | ≤ 1, we get
|5 − 6 | ≤
This bound is sharp for wz = z.
δ A − B
1
|λ + μ| + .
41 + α
3
Corollary 2.7. If ∈ CA δ; A , B then
δA − B + 1
3
|5 − 6 | ≤
A
− B |λ‰ + μ| + 4
3δ
t
r
12
u
where
λ‰ =
22B + 1 − δA − B
3δA − B
Corollary 2.8. If ∈ CA A , B, then
where
if |λ‰ + μ| ≤ ν‰ ,
if |λ‰ + μ| ≥ ν‰ ,
k} ν‰ =
4
.
3δA − B
A−B+1
if |λŠ + μ| ≤ νŠ ,
3
|5 − 6 | ≤ |
3A − B |λŠ + μ| + 4
if |λŠ + μ| ≥ νŠ ,
12
λŠ =
Corollary 2.9. If ∈ CA , then
4B
4
k} νŠ =
.
3A − B
3A − B
1
2
|5 − 6 | ≤ max. ‚1, + ƒμ − ƒ „.
3
3
Corollary 2.10. If ∈ CA F δ; A, B, then
δA − B + 3
if |λ‹ + μ| ≤ ν‹ ,
9
|5 − 6 | ≤
t3δ A − B |λ‹ + μ| + 16
if |λ‹ + μ| ≥ ν‹ ,
r
48
u
where
λ‹ =
82B + 1 − δA − B
16
k} ν‹ =
.
9δA − B
9δA − B
96
B. S. Mehrok et al.
Corollary 2.11. If ∈ CA F A, B then
A−B+3
if |λŒ + μ| ≤ νŒ ,
9
|a5 − μa | ≤ |
3A − B |λŒ + μ| + 16
,
if |λŒ + μ| ≥ νŒ ,
48
16B
16
k} νŒ =
.
9A − B
9A − B
Corollary 2.12. If ∈ CA F , then
where
λŒ =
5 1 1
8
|5 − 6 | ≤ max. ‚ , + ƒμ − ƒ„.
9 3 4
9
On the same lines as in Theorem 2.2, we have the following:
Theorem 2.3. If f ∈ Hα; δ; A, B and µ is a complex number, then
where
u 3δA − B + 2α + 1 ,
s
91 + 2α
|5 − 6 | ≤
1
t δ A − B
s 41 + α |λ + μ| + 9 ,
r
21 + α [2B + 1 − δA − B]
λ=
3δ1 + 2αA − B
References
[1]
if |λ + μ| ≤ ν,
if |λ + μ| ≥ ν,
41 + α
k} ν =
.
3δ1 + 2αA − B
R.N. Das and P. Singh, On properties of certain subclasses of close-to-convex
functions, Ann. Univ. Mariae Curie SklodowskaI, 2(1976), 15-22.
[2]
R.N. Das and P. Singh, Radius of convexity for certain subclass of close-to-convex
functions, J. Indian Math. Soc., 41(1977), 363-369.
[3]
R.M. Goel and B.S. Mehrok, A subclass of starlike functions with respect to
symmetric points, Tamkang J. of Math., 13(1) (1982), 11-24.
[4]
A. Janteng and S.A. Halim, A subclass of convex functions with respect to symmetric
points, Proc. 16th National Symp. on Sc. Math., (2008).
[5]
Z. Nehari, Conformal Mapping, McGraw-Hill, Comp., Inc., New York, (1952).
[6]
K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11(1959), 72-80.