Journal of Inequalities in Pure and
Applied Mathematics
FEKETE-SZEGÖ FUNCTIONAL FOR SOME SUBCLASS OF
NON-BAZILEVIČ FUNCTIONS
volume 7, issue 3, article 117,
2006.
T.N. SHANMUGAM, M.P. JEYARAMAN AND S. SIVASUBRAMANIAN
Department of Mathematics
Anna University
Chennai 600025
Tamilnadu, India
EMail: shan@annauniv.edu
Department of Mathematics
Easwari Engineering College
Chennai-600 089
Tamilnadu, India
EMail: jeyaraman mp@yahoo.co.in
EMail: sivasaisastha@rediffmail.com
Received 18 November, 2005;
accepted 24 March, 2006.
Communicated by: A. Lupaş
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
108-06
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Abstract
In this present investigation, the authors obtain a sharp Fekete-Szegö’s inequality for certain normalized analytic functions f(z) defined on the open unit disk
α
1+α
z
z
for which (1 + β) f(z)
− βf 0 (z) f(z)
, (β ∈ C, 0 < α < 1) lies in a region
starlike with respect to 1 and is symmetric with respect to the real axis. Also,
certain applications of our results for a class of functions defined by convolution
are given. As a special case of this result, Fekete-Szegö’s inequality for a class
of functions defined through fractional derivatives is also obtained.
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Analytic functions, Starlike functions, Subordination, Coefficient problem, Fekete-Szegö inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Fekete-Szegö Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Applications to Functions Defined by Fractional Derivatives 12
References
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
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1.
Introduction
Let A denote the class of all analytic functions f (z) of the form
(1.1)
f (z) = z +
∞
X
ak z k
(z ∈ ∆ := {z ∈ C/ |z| < 1})
k=2
and S be the subclass of A consisting of univalent functions. Let φ(z) be an
analytic function with positive real part on ∆ with φ(0) = 1, φ0 (0) > 0 which
maps the unit disk ∆ onto a region starlike with respect to 1 and is symmetric
with respect to the real axis. Let S ∗ (φ) be the class of functions f ∈ S for
which
zf 0 (z)
≺ φ(z), (z ∈ ∆)
f (z)
and C(φ) be the class of functions f ∈ S for which
00
1+
zf (z)
≺ φ(z),
f 0 (z)
(z ∈ ∆),
where ≺ denotes the subordination between analytic functions. These classes
were introduced and studied by Ma and Minda [3]. They have obtained the
Fekete-Szegö inequality for the functions in the class C(φ). Since f ∈ C(φ) if
and only if zf 0 ∈ S ∗ (φ), we get the Fekete-Szegö inequality for functions in the
class S ∗ (φ). Recently, Shanmugam and Sivasubramanian [9] obtained FeketeSzegö inequalities for the class of functions f ∈ A such that
zf 0 (z) + αz 2 f 00 (z)
≺ φ(z)
(1 − α)f (z) + αzf 0 (z)
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
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(0 ≤ α < 1).
J. Ineq. Pure and Appl. Math. 7(3) Art. 117, 2006
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Also, Ravichandran et al. [7] obtained the Fekete-Szegö inequality for the class
of Bazilevič functions. For a brief history of the Fekete-Szegö problem for the
class of starlike, convex and close-to-convex functions, see the recent paper by
Srivastava et al. [11]. Obradovic [4] introduced a class of functions f ∈ A,
such that, for 0 < α < 1,
α z
0
< f (z)
> 0, z ∈ ∆.
f (z)
He called this class of function as "Non-Bazilevič" type. Tuneski and Darus [14]
obtained the Fekete-Szegö inequality for the non-Bazilevič class of functions.
Using this non-Bazilevič class, Wang et al.[15] studied many subordination results for the class N (α, β, A, B) defined as
N (α, β, A, B)
(
:=
f ∈ A : (1 + β)
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
Title Page
z
f (z)
α
− βf 0 (z)
z
f (z)
1+α
≺
1 + Az
1 + Bz
)
,
where β ∈ C, −1 ≤ B ≤ 1, A 6= B, 0 < α < 1.
In the present paper, we obtain the Fekete-Szegö inequality for functions in
a more general class Nα,β (φ) of functions which we define below. Also we give
applications of our results to certain functions defined through convolution (or
λ
Hadamard product) and in particular we consider a class Nα,β
(φ) of functions
defined by fractional derivatives. The aim of this paper is to give a generalization the Fekete-Szegö inequalities for some subclass of Non-Bazilevič functions
.
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Definition 1.1. Let φ(z) be an univalent starlike function with respect to 1 which
maps the unit disk ∆ onto a region in the right half plane which is symmetric
with respect to the real axis, φ(0) = 1 and φ0 (0) > 0. A function f ∈ A is in
the class Nα,β (φ) if
α
1+α
z
z
0
(1 + β)
− βf (z)
≺ φ(z), (β ∈ C, 0 < α < 1).
f (z)
f (z)
g
For fixed g ∈ A, we define the class Nα,β
(φ) to be the class of functions f ∈ A
for which (f ∗ g) ∈ Nα,β (φ).
Remark 1. Nα,−1 1+z
is the class of Non-Bazilevič functions introduced by
1−z
Obradovic [4].
1+(1−2γ)z
Remark 2. Nα,−1
, 0 ≤ γ < 1 is the class of Non-Bazilevič
1−z
functions of order γ introduced and studied by Tuneski and Darus [14].
√ 2
1+ z
2
Remark 3. We call Nα,β 1 + π2 log 1−√z
the class of "Non-Bazilevič
parabolic starlike functions".
To prove our main result, we need the following:
Lemma 1.1 ([3]). If p1 (z) = 1 + c1 z + c2 z 2 + · · · is an analytic function with
a positive real part in ∆, then

−4v + 2 if v ≤ 0,



2
if 0 ≤ v ≤ 1,
|c2 − vc21 | ≤



4v − 2
if v ≥ 1.
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
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When v < 0 or v > 1, the equality holds if and only if p1 (z) is (1 + z)/(1 − z)
or one of its rotations. If 0 < v < 1, then the equality holds if and only if p1 (z)
is (1 + z 2 )/(1 − z 2 ) or one of its rotations. If v = 0, the equality holds if and
only if
1+z
1 1
1−z
1 1
+ λ
+
− λ
(0 ≤ λ ≤ 1)
p1 (z) =
2 2
1−z
2 2
1+z
or one of its rotations. If v = 1, the equality holds if and only if p1 is the
reciprocal of one of the functions such that equality holds in the case of v = 0.
Also the above upper bound is sharp, and it can be improved as follows when
0 < v < 1:
|c2 − vc21 | + v|c1 |2 ≤ 2 (0 < v ≤ 1/2)
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
and
|c2 − vc21 | + (1 − v)|c1 |2 ≤ 2
(1/2 < v ≤ 1).
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2.
Fekete-Szegö Problem
Our main result is the following:
Theorem 2.1. Let φ(z) = 1 + B1 z + B2 z 2 + B3 z 3 + · · · . If f given by (1.1)
belongs to Nα,β (φ), then

µB12
(1+α)
B2
2

 − (α+2β) − 2(α+β)2 + 2(α+β)2 B1 if µ ≤ σ1 ;



B1
2
− (α+2β)
if σ1 ≤ µ ≤ σ2 ;
|a3 − µa2 | ≤




µB12
 B2
(1+α)
2
− 2(α+β)
if µ ≥ σ2 ,
+
2 B1
(α+2β)
2(α+β)2
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
where,
(1 + α)(2β + α)B12 − 2(B2 − B1 )(β + α)2
,
2(2β + α)B12
(1 + α)(2β + α)B12 − 2(B2 + B1 )(β + α)2
σ2 :=
.
2(2β + α)B12
σ1 :=
The result is sharp.
Proof. For f ∈ Nα,β (φ), let
α
1+α
z
z
0
(2.1) p(z) := (1 + β)
− βf (z)
= 1 + b1 z + b2 z 2 + · · · .
f (z)
f (z)
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From (2.1), we obtain
−(α + β)a2 = b1
J. Ineq. Pure and Appl. Math. 7(3) Art. 117, 2006
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(2β + α)
α+1 2
a2 − a3
2
= b2 .
Since φ(z) is univalent and p ≺ φ, the function
p1 (z) =
1 + φ−1 (p(z))
= 1 + c1 z + c2 z 2 + · · ·
1 − φ−1 (p(z))
is analytic and has a positive real part in ∆. Also we have
p1 (z) − 1
(2.2)
p(z) = φ
p1 (z) + 1
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
and from this equation (2.2), we obtain
1
b1 = B1 c1
2
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and
1
1
b2 = B1 c2 − c21
2
2
1
+ B2 c21 .
4
Therefore we have
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a3 −
µa22
B1
c2 − vc21
=−
2(2β + α)
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where
1
B2 (2β + α)(α + 1 − 2µ)
v :=
1−
+
B1 .
2
B1
2(β + α)2
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Our result now follows by an application of Lemma 1.1. To show that the
φn
bounds are sharp, we define the functions Kα,β
(n = 2, 3, . . .) by
(1 + β)
!α
z
n
− β K φα,β
φn
Kα,β
(z)
0
!1+α
z
(z)
= φ(z n−1 ),
φn
Kα,β
(z)
φn
φn 0
Kα,β
(0) = 0 = [Kα,β
] (0) − 1
λ
and the function Fα,β
and Gλα,β (0 < α < 1) by
(1 + β)
!α
z
λ
Fα,β
(z)
λ 0
− β[Fα,β
] (z)
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
!1+α
z
λ
Fα,β
(z)
= φ(z n−1 ),
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
λ
λ
[Fα,β
](0) = 0 = [Fα,β
]]0 (0) − 1
Title Page
and
(1 + β)
z
Gλα,β
!α
− β[Gλα,β ]0 (z)
z
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!1+α
Gλα,β (z)
= φ(z n−1 ),
[Gλα,β ](0) = 0 = [Gλα,β ]0 (0) − 1.
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φn
λ
Clearly, the functions Kα,β
, [Fα,β
] and [Gλα,β ] ∈ Nα,β (φ)
φ
φ2
Also we write Kα,β
:= Kα,β
.
φ
If µ < σ1 or µ > σ2 , then the equality holds if and only if f is Kα,β
or one
φ3
of its rotations. When σ1 < µ < σ2 , the equality holds if and only if f is Kα,β
λ
or one of its rotations. If µ = σ1 then the equality holds if and only if f is Fα,β
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or one of its rotations. If µ = σ2 then the equality holds if and only if f is Gλα,β
or one of its rotations.
√ 2
1+ z
2
√
Corollary 2.2. Let φ(z) = 1 + π2 log 1− z . If f given by (1.1) belongs to
Nα,β (φ), then

(1+α) 8
8µ
8

− π4 (α+β)
if µ ≤ σ1
− 3π2 (α+2β)
2 + (α+β)2 π 4




4
− π2 (α+2β)
if σ1 ≤ µ ≤ σ2
|a3 − µa22 | ≤




(1+α) 8
8µ

8
+ π4 (α+β)
if µ ≥ σ2
2 − (α+β)2 π 4
3π 2 (α+2β)
where,
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
(1 + α)(2β + α) π164 − 2 3π8 2 −
2(2β + α) π164
4
π2
(1 + α)(2β + α) π164 − 2 3π8 2 +
σ2 :=
2(2β + α) π164
4
π2
σ1 :=
(β + α)2
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(β + α)2
Contents
.
The result is sharp.
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Corollary 2.3. For β = −1, φ(z) = 1+(1−2γ)z
, 0 ≤ γ < 1 in Theorem 2.1,
1−z
we get the results obtained by Tuneski and Darus [14].
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Remark 4. If σ1 ≤ µ ≤ σ2 , then, in view of Lemma 1.1, Theorem 2.1 can be
improved. Let σ3 be given by
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(1 + α)(2β + α)B12 − 2B2 (β + α)2
.
σ3 :=
2(2β + α)B12
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If σ1 ≤ µ ≤ σ3 , then
|a3 −
µa22 |
(β + α)2
2 (α + 1 − 2µ)(2β + α)
−
|a2 |2
B1 − B2 + B1
(2β + α)B12
2(β + α)2
B1
≤−
.
(2β + α)
If σ3 ≤ µ ≤ σ2 , then
(β + α)2
2 (α + 1 − 2µ)(2β + α)
2
|a3 − µa2 | −
B1 + B2 − B1
|a2 |2
(2β + α)B12
2(β + α)2
B1
≤−
.
(2β + α)
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
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3.
Applications to Functions Defined by Fractional
Derivatives
λ
In order to introduce the class Nα,β
(φ), we need the following:
Definition 3.1 (see [5, 6]; see also [12, 13]). Let f be analytic in a simply
connected region of the z-plane containing the origin. The fractional derivative
of f of order λ is defined by
Z z
d
f (ζ)
1
λ
Dz f (z) :=
dζ (0 ≤ λ < 1)
Γ(1 − λ) dz 0 (z − ζ)λ
where the multiplicity of (z − ζ)λ is removed by requiring that log(z − ζ) is real
for z − ζ > 0.
Using the above Definition 3.1 and its known extensions involving fractional
derivatives and fractional integrals, Owa and Srivastava [5] introduced the operator Ωλ : A → A defined by
(Ωλ f )(z) = Γ(2 − λ)z λ Dzλ f (z),
(λ 6= 2, 3, 4, . . .).
λ
The class Nα,β
(φ) consists of functions f ∈ A for which Ωλ f ∈ Nα,β (φ).
g
λ
Note that Nα,β
(φ) is the special case of the class Nα,β
(φ) when
(3.1)
g(z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
n=2
Γ(n + 1 − λ)
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
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zn.
P
P∞
n
n
Let g(z) = z + ∞
n=2 gn z (gn >P0). Since f (z) = z +
n=2 an z ∈
g
n
Nα,β
(φ) if and only if (f ∗ g) = z + ∞
n=2 gn an z ∈ Nα,β (φ), we obtain the
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J. Ineq. Pure and Appl. Math. 7(3) Art. 117, 2006
http://jipam.vu.edu.au
g
coefficient estimate for functions in the class Nα,β
(φ), from the corresponding
estimate for functions in the class Nα,β (φ). Applying Theorem 2.1 for the function (f ∗ g)(z) = z + g2 a2 z 2 + g3 a3 z 3 + · · · , we get the following Theorem 3.1
after an obvious change of the parameter µ:
Theorem 3.1. Let the function φ be given by φ(z) = 1 + B1 z + B2 z 2 + B3 z 3 +
g
· · · . If f given by (1.1) belongs to Nα,β
(φ), then
 n
o
µg3 B12
(1+α)
B2
1
2


−
+
B
if µ ≤ σ1 ;
−

g
(α+2β)
2(α+β)2 1
g22 2(α+β)2

 3
B1
− g13 (α+2β)
if σ1 ≤ µ ≤ σ2 ;
|a3 − µa22 | ≤

n
o


µg3 B12

(1+α)
B2
2
 1
+ 2(α+β)
if µ ≥ σ2 ,
2 g 2 − 2(α+β)2 B1
g3
(α+2β)
2
where
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
Title Page
σ1 :=
σ2 :=
α)B12
2
g3 (1 + α)(2β +
− 2(B2 − B1 )(β + α)
2
g2
2(2β + α)B12
g3 (1 + α)(2β + α)B12 − 2(B2 + B1 )(β + α)2
.
g22
2(2β + α)B12
The result is sharp.
Since
(Ωλ f )(z) = z +
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∞
X
Γ(n + 1)Γ(2 − λ)
n=2
Γ(n + 1 − λ)
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an z n ,
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we have
g2 :=
(3.2)
Γ(3)Γ(2 − λ)
2
=
Γ(3 − λ)
2−λ
and
g3 :=
(3.3)
Γ(4)Γ(2 − λ)
6
=
.
Γ(4 − λ)
(2 − λ)(3 − λ)
For g2 and g3 given by (3.2) and (3.3), Theorem 3.1 reduces to the following:
Theorem 3.2. Let the function φ be given by φ(z) = 1 + B1 z + B2 z 2 + B3 z 3 +
g
· · · . If f given by (1.1) belongs to Nα,β
(φ), then
|a3 − µa22 |

n
o
µg3 B12
(1+α)
(2−λ)(3−λ)
B2
2

+
−
−
B


6
(α+2β)
2(α+β)2 1
g22 2(α+β)2


B1
− (2−λ)(3−λ)
≤
6
(α+2β)


n
o


µg3 B12
(1+α)
B2
2
 (2−λ)(3−λ)
+
−
B
2
2
1
6
(α+2β)
2(α+β)
2(α+β)2 g
2
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
if
µ ≤ σ1 ;
if
σ1 ≤ µ ≤ σ2 ;
if
µ ≥ σ2 ,
Title Page
where
σ1 :=
σ2 :=
Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
Contents
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2(3 − λ) (1 + α)(2β + α)B12 − 2(B2 − B1 )(β + α)2
,
3(2 − λ)
2(2β + α)B12
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α)B12
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2
2(3 − λ) (1 + α)(2β +
− 2(B2 + B1 )(β + α)
.
3(2 − λ)
2(2β + α)B12
Close
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The result is sharp.
J. Ineq. Pure and Appl. Math. 7(3) Art. 117, 2006
http://jipam.vu.edu.au
References
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[2] A.W. GOODMAN, Uniformly convex functions, Ann. Polon. Math., 56
(1991), 87–92.
[3] W. MA AND D. MINDA, A unified treatment of some special classes of
univalent functions, in: Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang (Eds.), Int. Press (1994), 157–
169.
[4] M. OBRADOVIC, A class of univalent functions, Hokkaido Math. J.,
27(2) (1998), 329–335.
[5] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized
bypergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.
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Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
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Fekete-Szegö Functional for
some Subclass of Non-Bazilevič
Functions
T.N. Shanmugam, M.P. Jeyaraman
and S. Sivasubramanian
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Contents
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