Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 3, Article 117, 2006
FEKETE-SZEGÖ FUNCTIONAL FOR SOME SUBCLASS OF NON-BAZILEVIČ
FUNCTIONS
T.N. SHANMUGAM, M.P. JEYARAMAN, AND S. SIVASUBRAMANIAN
D EPARTMENT OF M ATHEMATICS
A NNA U NIVERSITY
C HENNAI 600025
TAMILNADU , I NDIA
shan@annauniv.edu
D EPARTMENT OF M ATHEMATICS
E ASWARI E NGINEERING C OLLEGE
C HENNAI -600 089
TAMILNADU , I NDIA
jeyaraman mp@yahoo.co.in
sivasaisastha@rediffmail.com
Received 18 November, 2005; accepted 24 March, 2006
Communicated by A. Lupaş
A BSTRACT. 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 for which
α
1+α
z
z
(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.
Key words and phrases: Analytic functions, Starlike functions, Subordination, Coefficient problem, Fekete-Szegö inequality.
2000 Mathematics Subject Classification. Primary 30C45.
1. I NTRODUCTION
Let A denote the class of all analytic functions f (z) of the form
∞
X
ak z k (z ∈ ∆ := {z ∈ C/ |z| < 1})
(1.1)
f (z) = z +
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
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
108-06
2
T.N. S HANMUGAM , M.P. J EYARAMAN , AND S. S IVASUBRAMANIAN
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
zf 00 (z)
≺ φ(z), (z ∈ ∆),
1+ 0
f (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 FeketeSzegö inequality for functions in the class S ∗ (φ). Recently, Shanmugam and Sivasubramanian
[9] obtained Fekete- Szegö inequalities for the class of functions f ∈ A such that
zf 0 (z) + αz 2 f 00 (z)
≺ φ(z) (0 ≤ α < 1).
(1 − α)f (z) + αzf 0 (z)
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
> 0, z ∈ ∆.
< f (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
(
)
α
1+α
z
z
1 + Az
0
− βf (z)
≺
,
N (α, β, A, B) := f ∈ A : (1 + β)
f (z)
f (z)
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 Hadamard product) and in particular we
λ
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 .
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
− βf (z)
≺ φ(z), (β ∈ C, 0 < α < 1).
(1 + β)
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 1.1. Nα,−1 1+z
is the class of Non-Bazilevič functions introduced by Obradovic [4].
1−z
, 0 ≤ γ < 1 is the class of Non-Bazilevič functions of order
Remark 1.2. Nα,−1 1+(1−2γ)z
1−z
γ introduced and studied by Tuneski and Darus [14].
J. Inequal. Pure and Appl. Math., 7(3) Art. 117, 2006
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O N THE F EKETE -S ZEGÖ P ROBLEM
Remark 1.3. We call Nα,β 1 +
2
π2
log
√ 2
1+ z
√
1− z
3
the class of "Non-Bazilevič parabolic star-
like functions".
To prove our main result, we need the following:
Lemma 1.4 ([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.
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 1
1+z
1 1
1−z
p1 (z) =
+ λ
+
− λ
(0 ≤ λ ≤ 1)
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)
and
|c2 − vc21 | + (1 − v)|c1 |2 ≤ 2
(1/2 < v ≤ 1).
2. F EKETE -S ZEGÖ P ROBLEM
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)
if µ ≤ σ1 ;
2 + 2(α+β)2 B1




B1
− (α+2β)
if σ1 ≤ µ ≤ σ2 ;
|a3 − µa22 | ≤




 B2 + µB12 − (1+α) B 2
if µ ≥ σ2 ,
(α+2β)
2(α+β)2
2(α+β)2 1
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)
From (2.1), we obtain
−(α + β)a2 = b1
J. Inequal. Pure and Appl. Math., 7(3) Art. 117, 2006
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4
T.N. S HANMUGAM , M.P. J EYARAMAN , AND S. S IVASUBRAMANIAN
α+1 2
(2β + α)
a2 − a3
2
Since φ(z) is univalent and p ≺ φ, the function
p1 (z) =
= b2 .
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
and from this equation (2.2), we obtain
1
b1 = B1 c1
2
1
1 2
1
b2 = B1 c2 − c1 + B2 c21 .
2
2
4
and
Therefore we have
a3 − µa22 = −
B1
c2 − vc21
2(2β + α)
where
1
B2 (2β + α)(α + 1 − 2µ)
v :=
1−
+
B1 .
2
B1
2(β + α)2
Our result now follows by an application of Lemma 1.4. To show that the bounds are sharp, we
φn
define the functions Kα,β
(n = 2, 3, . . .) by
!α
!1+α
0
z
z
n
(1 + β)
− β K φα,β
(z)
= φ(z n−1 ),
φn
φn
Kα,β (z)
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
!α
z
λ 0
(1 + β)
− β[Fα,β
] (z)
λ
Fα,β
(z)
!1+α
z
λ
Fα,β
(z)
= φ(z n−1 ),
λ
λ
[Fα,β
](0) = 0 = [Fα,β
]]0 (0) − 1
and
(1 + β)
z
Gλα,β
!α
− β[Gλα,β ]0 (z)
z
!1+α
Gλα,β (z)
= φ(z n−1 ),
[Gλα,β ](0) = 0 = [Gλα,β ]0 (0) − 1.
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 of its rotations.
φ3
When σ1 < µ < σ2 , the equality holds if and only if f is Kα,β or one of its rotations. If µ = σ1
λ
then the equality holds if and only if f is Fα,β
or one of its rotations. If µ = σ2 then the equality
λ
holds if and only if f is Gα,β or one of its rotations.
J. Inequal. Pure and Appl. Math., 7(3) Art. 117, 2006
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O N THE F EKETE -S ZEGÖ P ROBLEM
2
π2
5
√ 2
1+ z
√
1− z
Corollary 2.2. Let φ(z) = 1 +
log
. 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,
(1 + α)(2β + α) π164 − 2 3π8 2 −
σ1 :=
2(2β + α) π164
4
π2
(1 + α)(2β + α) π164 − 2 3π8 2 +
σ2 :=
2(2β + α) π164
4
π2
(β + α)2
(β + α)2
.
The result is sharp.
Corollary 2.3. For β = −1, φ(z) = 1+(1−2γ)z
,
1−z
results obtained by Tuneski and Darus [14].
0 ≤ γ < 1 in Theorem 2.1, we get the
Remark 2.4. If σ1 ≤ µ ≤ σ2 , then, in view of Lemma 1.4, Theorem 2.1 can be improved. Let
σ3 be given by
(1 + α)(2β + α)B12 − 2B2 (β + α)2
σ3 :=
.
2(2β + α)B12
If σ1 ≤ µ ≤ σ3 , then
(β + α)2
B1
2
2 (α + 1 − 2µ)(2β + α)
2
B1 − B2 + B1
|a
|
≤
−
.
|a3 − µa2 | −
2
2
(2β + α)B1
2(β + α)2
(2β + α)
If σ3 ≤ µ ≤ σ2 , then
B1
(β + α)2
2 (α + 1 − 2µ)(2β + α)
2
|a3 − µa2 | −
B1 + B2 − B1
|a2 |2 ≤ −
.
2
2
(2β + α)B1
2(β + α)
(2β + α)
3. A PPLICATIONS TO F UNCTIONS D EFINED BY F RACTIONAL D ERIVATIVES
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
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 [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
λ
The class Nα,β
(φ) consists of functions f ∈ A for which Ωλ f ∈ Nα,β (φ). Note that Nα,β
(φ)
g
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
J. Inequal. Pure and Appl. Math., 7(3) Art. 117, 2006
Γ(n + 1 − λ)
zn.
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6
T.N. S HANMUGAM , M.P. J EYARAMAN , AND S. S IVASUBRAMANIAN
P
P
g
n
Let g(z) = z + ∞
gn z n (gn > 0). Since f (z) = z + ∞
n=2
n=2 an z ∈ Nα,β (φ) if and only if
P∞
(f ∗ g) = z + n=2 gn an z n ∈ Nα,β (φ), we obtain the coefficient estimate for functions in the
g
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 + · · · . If f given
g
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
if µ ≥ σ2 ,
+ 2(α+β)
2 g 2 − 2(α+β)2 B1
g3
(α+2β)
2
where
σ1 :=
g3 (1 + α)(2β + α)B12 − 2(B2 − B1 )(β + α)2
g22
2(2β + α)B12
σ2 :=
g3 (1 + α)(2β + α)B12 − 2(B2 + B1 )(β + α)2
.
g22
2(2β + α)B12
The result is sharp.
Since
λ
(Ω f )(z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
n=2
Γ(n + 1 − λ)
an z n ,
we have
g2 :=
(3.2)
Γ(3)Γ(2 − λ)
2
=
Γ(3 − λ)
2−λ
and
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:
g3 :=
(3.3)
Theorem 3.2. Let the function φ be given by φ(z) = 1 + B1 z + B2 z 2 + B3 z 3 + · · · . If f given
g
by (1.1) belongs to Nα,β
(φ), then

n
o
µg3 B12
(2−λ)(3−λ)
(1+α)
B2
2

− (α+2β) − g2 2(α+β)2 + 2(α+β)2 B1
if µ ≤ σ1 ;


6
2


B1
− (2−λ)(3−λ)
if σ1 ≤ µ ≤ σ2 ;
|a3 − µa22 | ≤
6
(α+2β)


n
o

 (2−λ)(3−λ)
µg3 B12
(1+α)
B2
2

+ 2(α+β)
if µ ≥ σ2 ,
2 g 2 − 2(α+β)2 B1
6
(α+2β)
2
where
2(3 − λ) (1 + α)(2β + α)B12 − 2(B2 − B1 )(β + α)2
σ1 :=
,
3(2 − λ)
2(2β + α)B12
σ2 :=
2(3 − λ) (1 + α)(2β + α)B12 − 2(B2 + B1 )(β + α)2
.
3(2 − λ)
2(2β + α)B12
The result is sharp.
J. Inequal. Pure and Appl. Math., 7(3) Art. 117, 2006
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O N THE F EKETE -S ZEGÖ P ROBLEM
7
R EFERENCES
[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] 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.
[6] S. OWA, On the distortion theorems I, Kyungpook Math. J., 18 (1978), 53–58.
[7] V. RAVICHANDRAN, A. GANGADHARAN AND M. DARUS, Fekete-Szegő inequality for certain class of Bazilevic functions, Far East J. Math. Sci. (FJMS), 15(2) (2004), 171–180.
[8] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc.
Amer. Math. Soc., 118 (1993), 189–196.
[9] T.N. SHANMUGAM AND S. SIVASUBRAMANIAN, On the Fekete-Szegö problem for some
subclasses of analytic functions, J. Inequal. Pure and Appl. Math., 6(3) (2005), Art. 71, 6 pp.
[10] 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.
[11] 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.
[12] H.M. SRIVASTAVA
(1984), 383–389.
AND
S. OWA, An application of the fractional derivative, Math. Japon., 29
[13] H.M. SRIVASTAVA AND S. OWA, Univalent functions, Fractional Calculus, and their Applications, Halsted Press/John Wiley and Songs, Chichester/New York, (1989).
[14] N. TUNESKI AND M. DARUS, Fekete-Szegö functional for non-Bazilevic functions, Acta Mathematica Academia Paedagogicae Nyiregyhaziensis, 18 (2002), 63–65.
[15] Z. WANG, C. GAO AND M. LIAO, On certain generalized class of non-Bazilevič functions, Acta
Mathematica Academia Paedagogicae Nyiregyhaziensis, 21 (2005), 147–154.
J. Inequal. Pure and Appl. Math., 7(3) Art. 117, 2006
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