Journal of Inequalities in Pure and
Applied Mathematics
ON THE FEKETE-SZEGÖ PROBLEM FOR SOME SUBCLASSES
OF ANALYTIC FUNCTIONS
volume 6, issue 3, article 71,
2005.
T.N. SHANMUGAM AND S. SIVASUBRAMANIAN
Department of Mathematics
College of Engineering
Anna University, Chennai-600 025
Tamilnadu, India.
EMail: shan@annauniv.edu
Department of Mathematics
Easwari Engineering College
Ramapuram, Chennai-600 089
Tamilnadu, India.
EMail: sivasaisastha@rediffmail.com
Received 12 May, 2005;
accepted 24 May, 2005.
Communicated by: S. Saitoh
Abstract
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Abstract
In this present investigation, the authors obtain Fekete-Szegö’s inequality for
certain normalized analytic functions f(z) defined on the open unit disk for
zf 0 (z)+αz 2 f 00 (z)
which (1−α)f(z)+αzf
0 (z) (α ≥ 0) lies in a region starlike with respect to 1 and
is symmetric with respect to the real axis. Also certain applications of the main
result 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 obtained. The Motivation of this paper is to give
a generalization of the Fekete-Szegö inequalities obtained by Srivastava and
Mishra .
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Applications to Functions Defined by Fractional Derivatives 10
References
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam 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 which is symmetric
with respect to the real axis. Let S ∗ (φ) be the class of functions in f ∈ S for
which
zf 0 (z)
≺ φ(z), (z ∈ ∆)
f (z)
and C(φ) be the class of functions in f ∈ S for which
zf 00 (z)
1+ 0
≺ φ(z),
f (z)
(z ∈ ∆),
where ≺ denotes the subordination between analytic functions. These classes
were introduced and studied by Ma and Minda [10]. 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 (z) ∈ S ∗ (φ), we get the Fekete-Szegö inequality for functions
in the class S ∗ (φ). 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. [7].
In the present paper, we obtain the Fekete-Szegö inequality for functions in
a more general class Mα (φ) of functions which we define below. Also we give
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam and
S. Sivasubramanian
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applications of our results to certain functions defined through convolution (or
the Hadamard product) and in particular we consider a class Mαλ (φ) of functions defined by fractional derivatives. The motivation of this paper is to give a
generalization of the Fekete-Szegö inequalities of Srivastava and Mishra [6].
Definition 1.1. Let φ(z) be a 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 Mα (φ) if
zf 0 (z) + αz 2 f 00 (z)
≺ φ(z)
(1 − α)f (z) + αzf 0 (z)
(α ≥ 0).
For fixed g ∈ A, we define the class Mαg (φ) to be the class of functions f ∈ A
for which (f ∗ g) ∈ Mα (φ).
To prove our main result, we need the following:
2
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam and
S. Sivasubramanian
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Lemma 1.1. [10] If p1 (z) = 1 + c1 z + c2 z + · · · is an analytic function with
positive real part in ∆, then

−4v + 2 if v ≤ 0;



2
if 0 ≤ v ≤ 1;
|c2 − vc21 | ≤



4v − 2
if v ≥ 1.
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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)
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is (1 + z 2 )/(1 − z 2 ) or one of its rotations.
only if
1+z
1
1 1
p1 (z) =
+ λ
+
−
2 2
1−z
2
If v = 0, the equality holds if and
1
λ
2
1−z
1+z
(0 ≤ λ ≤ 1)
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 the 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:
1
2
2
|c2 − vc1 | + v|c1 | ≤ 2
0<v≤
2
and
|c2 −
vc21 |
2
+ (1 − v)|c1 | ≤ 2
1
<v≤1 .
2
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam and
S. Sivasubramanian
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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 (z) given by (1.1)
belongs to Mα (φ), then
 B
µ
1
2
2
2
− (1+α)
if µ ≤ σ1 ;

2 B1 + 2(1+2α)(1+α)2 B1
2(1+2α)



B1
if σ1 ≤ µ ≤ σ2 ;
|a3 − µa22 | ≤
2(1+2α)



 − B2 + µ B 2 −
1
B 2 if µ ≥ σ2 ,
2(1+2α)
(1+α)2 1
2(1+2α)(1+α)2 1
T.N. Shanmugam and
S. Sivasubramanian
where
(1 + α)2 (B2 − B1 ) + (1 + α2 )B12
,
2(1 + 2α)B12
(1 + α)2 (B2 + B1 ) + (1 + α2 )B12
σ2 :=
.
2(1 + 2α)B12
σ1 :=
The result is sharp.
Proof. For f (z) ∈ Mα (φ), let
p(z) :=
2 00
zf (z) + αz f (z)
= 1 + b1 z + b2 z 2 + · · · .
(1 − α)f (z) + αzf 0 (z)
From (2.1), we obtain
(1 + α)a2 = b1
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(2.1)
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and
(2 + 4α)a3 = b2 + (1 + α2 )a22 .
J. Ineq. Pure and Appl. Math. 6(3) Art. 71, 2005
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Since φ(z) is univalent and p ≺ φ, the function
p1 (z) =
1 + φ−1 (p(z))
= 1 + c1 z + c2 z 2 + · · ·
−1
1 − φ (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
and from this equation (2.2), we obtain
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
1
b1 = B1 c1
2
and
T.N. Shanmugam and
S. Sivasubramanian
1
1
1
b2 = B1 (c2 − c21 ) + B2 c21 .
2
2
4
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Therefore we have
(2.3)
Contents
a3 −
µa22
B1
=
c2 − vc21 ,
4(1 + 2α)
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where
1
B2 (2µ − 1) + α(4µ − α)
v :=
1−
+
B1 .
2
B1
(1 + α)2
Our result now follows by an application of Lemma 1.1. To show that the
bounds are sharp, we define the functions Kαφn (n = 2, 3, . . .) by
z[Kαφn ]0 (z) + αz 2 [Kαφn ]00 (z)
(1 − α)[Kαφn ](z) + αz[Kαφn ]0 (z)
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= φ(z
n−1
),
Kαφn (0)
=0=
[Kαφn ]0 (0)
−1
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and the function Fαλ and Gλα (0 ≤ λ ≤ 1) by
z[Fαλ ]0 (z) + αz 2 [Fαλ ]00 (z)
z(z + λ)
=φ
,
(1 − α)[Fαλ ](z) + αz[Fαλ ]0 (z)
1 + λz
0
F λ (0) = 0 = (F λ ) (0) − 1
and
z[Gλα ]0 (z) + αz 2 [Gλα ]00 (z)
z(z + λ)
=φ −
,
(1 − α)[Gλα ](z) + αz[Gλα ]0 (z)
1 + λz
0
Gλ (0) = 0 = (Gλ ) (0).
Clearly the functions Kαφn , Fαλ , Gλα ∈ Mα (φ). Also we write Kαφ := Kαφ2 .
If µ < σ1 or µ > σ2 , then the equality holds if and only if f is Kαφ or one of
its rotations. When σ1 < µ < σ2 , the equality holds if and only if f is Kαφ3 or
one of its rotations. If µ = σ1 then the equality holds if and only if f is Fαλ or
one of its rotations. If µ = σ2 then the equality holds if and only if f is Gλα or
one of its rotations.
Remark 1. If σ1 ≤ µ ≤ σ2 , then, in view of Lemma 1.1, Theorem 2.1 can be
improved. Let σ3 be given by
(1 + α)2 B2 + (1 + α2 )B12
.
σ3 :=
2(1 + 2α)B12
If σ1 ≤ µ ≤ σ3 , then
|a3 −
µa22 |
(1 + α)2
(2µ − 1) + α(4µ − α) 2
B1 |a2 |2
+
B1 − B2 +
2
2(1 + 2α)B1
(1 + 2α)
B1
≤
.
2(1 + 2α)
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam and
S. Sivasubramanian
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If σ3 ≤ µ ≤ σ2 , then
|a3 −
µa22 |
(1 + α)2
(2µ − 1) + α(4µ − α) 2
+
B1 |a2 |2
B1 + B2 −
2(1 + 2α)B12
(1 + 2α)2
B1
≤
.
2(1 + 2α)
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam and
S. Sivasubramanian
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3.
Applications to Functions Defined by Fractional
Derivatives
In order to introduce the class Mαλ (φ), we need the following:
Definition 3.1 (see [3, 4]; see also [8, 9]). Let f (z) 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
1
d
f (ζ)
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 [3] introduced the operator Ωλ : A → A defined by
λ
(Ω f )(z) = Γ(2 − λ)z
λ
Dzλ f (z),
(λ 6= 2, 3, 4, . . .).
The class Mαλ (φ) consists of functions f ∈ A for which Ωλ f ∈ Mα (φ).
Note that M00 (φ) ≡ S ∗ (φ) and Mαλ (φ) is the special case of the class Mαg (φ)
when
(3.1)
g(z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
n=2
Γ(n + 1 − λ)
zn.
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam and
S. Sivasubramanian
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Let
g(z) = z +
∞
X
gn z n
(gn > 0).
n=2
Since
f (z) = z +
∞
X
an z n ∈ Mαg (φ)
n=2
if and only if
(f ∗ g) = z +
∞
X
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
gn an z n ∈ Mα (φ),
n=2
we obtain the coefficient estimate for functions in the class Mαg (φ), from the
corresponding estimate for functions in the class Mα (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:
T.N. Shanmugam and
S. Sivasubramanian
Theorem 3.1. Let the function φ(z) be given by φ(z) = 1 + B1 z + B2 z 2 +
B3 z 3 + · · · . If f (z) given by (1.1) belongs to Mαg (φ), then
Contents
|a3 − µa22 |
i
 h
µg3
B2
1
1
2
2

−
B
+
B
 g3 2(1+2α) (1+α)2 g22 1 2(1+2α)(1+α)2 1


 h
i
B1
1
≤
g3 2(1+2α)


h
i


1
2
 1 − B2 + µg3 2 B 2 −
B
1
g3
2(1+2α)
2(1+2α)(1+α)2 1
(1+α)2 g
2
if
µ ≤ σ1 ;
if
σ1 ≤ µ ≤ σ2 ;
if
µ ≥ σ2 ,
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where
g22 (1 + α)2 (B2 − B1 ) + (1 + α2 )B12
g3
2(1 + 2α)B12
g22 (1 + α)2 (B2 + B1 ) + (1 + α2 )B12
σ2 :=
.
g3
2(1 + 2α)B12
σ1 :=
The result is sharp.
Since
(Ωλ f )(z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
n=2
Γ(n + 1 − λ)
an z n ,
we have
(3.2)
g2 :=
Γ(3)Γ(2 − λ)
2
=
Γ(3 − λ)
2−λ
T.N. Shanmugam and
S. Sivasubramanian
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and
(3.3)
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
g3 :=
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(z) be given by φ(z) = 1 + B1 z + B2 z 2 +
B3 z 3 + · · · . If f (z) given by (1.1) belongs to Mαλ (φ), then
 (2−λ)(3−λ)
γ
if µ ≤ σ1 ;

6



(2−λ)(3−λ)
B1
|a3 − µa22 | ≤
· 2(1+2α)
if σ1 ≤ µ ≤ σ2 ;
6



 (2−λ)(3−λ)
γ
if µ ≥ σ2 ,
6
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where
γ :=
B2
3(2 − λ)
µ
1
−
B12 +
B12 ,
2
2
2(1 + 2α) 2(3 − λ) (1 + α)
2(1 + 2α)(1 + α)
2(3 − λ) (1 + α)2 (B2 − B1 ) + (1 + α2 )B12
.
3(2 − λ)
2(1 + 2α)B12
2(3 − λ) (1 + α)2 (B2 + B1 ) + (1 + α2 )B12
σ2 :=
.
.
3(2 − λ)
2(1 + 2α)B12
σ1 :=
The result is sharp.
2
2
Remark 2. When α = 0, B1 = 8/π and B2 = 16/(3π ), the above Theorem 3.1 reduces to a recent result of Srivastava and Mishra[6, Theorem 8, p.
64] for a class of functions for which Ωλ f (z) is a parabolic starlike function
[2, 5].
On the Fekete-Szegö Problem
for Some Subclasses of
Analytic Functions
T.N. Shanmugam and
S. Sivasubramanian
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References
[1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.
[2] A.W. GOODMAN, Uniformly convex functions, Ann. Polon. Math., 56
(1991), 87–92.
[3] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized
bypergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.
[4] S. OWA, On the distortion theorems I, Kyungpook Math. J., 18 (1978),
53–58.
[5] F. RØNNING, Uniformly convex functions and a corresponding class of
starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196.
[6] H.M. SRIVASTAVA AND A.K. MISHRA, Applications of fractional calculus to parabolic starlike and uniformly convex functions, Computer
Math. Appl., 39 (2000), 57–69.
[7] H.M. SRIVASTAVA, A.K. MISHRA AND M.K. DAS, The Fekete-Szegö
problem for a subclass of close-to-convex functions, Complex Variables,
Theory Appl., 44 (2001), 145–163.
[8] H.M. SRIVASTAVA AND S. OWA, An application of the fractional derivative, Math. Japon., 29 (1984), 383–389.
[9] H.M. SRIVASTAVA AND S. OWA, Univalent functions, Fractional Calculus, and their Applications, Halsted Press/John Wiley and Songs, Chichester/New York, (1989).
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for Some Subclasses of
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[10] 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.
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for Some Subclasses of
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