Journal of Inequalities in Pure and
Applied Mathematics
A COEFFICIENT INEQUALITY FOR CERTAIN CLASSES
OF ANALYTIC FUNCTIONS OF COMPLEX ORDER
volume 7, issue 4, article 145,
2006.
K. SUCHITHRA, B. ADOLF STEPHEN AND S. SIVASUBRAMANIAN
Department of Applied Mathematics
Sri Venkateswara College of Engineering
Sriperumbudur, Chennai - 602105, India.
EMail: suchithrak@svce.ac.in
Department of Mathematics
Madras Christian College
Tambaram, Chennai - 600059, India.
EMail: adolfmcc2003@yahoo.co.in
Department of Mathematics
Easwari Engineering College
Ramapuram, Chennai - 600089, India.
EMail: sivasaisastha@rediffmail.com
Received 06 February, 2006;
accepted 25 August, 2006.
Communicated by: G. Kohr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In the present investigation, we obtain the Fekete-Szegö inequality for a certain
normalized
analytic function
f(z) defined on the open unit disk for which 1 +
h
i
1 zf 0 (z)+αz 2 f 00 (z)
− 1 (α ≥ 0 and b 6= 0, a complex number) lies in a region
b
f(z)
starlike with respect to 1 and symmetric with respect to real axis. Also certain
application of the main result for a class of functions of complex order defined
by convolution is given. The motivation of this paper is to give a generalization
of the Fekete-Szegö inequalities for subclasses of starlike functions of complex
order.
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Starlike functions of complex order, Convex functions of complex order,
Subordination, Fekete-Szegö inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Fekete-Szegö Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Application to Functions Defined by Fractional Derivatives . 10
References
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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 f ∈ S for which
00
1+
zf (z)
≺ φ(z), (z ∈ ∆),
f 0 (z)
where ≺ denotes the subordination between analytic functions. These classes
were introduced and studied by Ma and Minda [4]. They have obtained the
Fekete-Szegö inequality for functions in the class C(φ). Since f ∈ C(φ) iff
zf 0 (z) ∈ S ∗ (φ), we get the Fekete-Szegö inequality for functions in the class
S ∗ (φ).
The class Sb∗ (φ) consists of all analytic functions f ∈ A satisfying
1 zf 0 (z)
1+
− 1 ≺ φ(z)
b
f (z)
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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and the class Cb (φ) consists of functions f ∈ A satisfying
1 zf 00 (z)
1+
≺ φ(z).
b f 0 (z)
These classes were defined and studied by Ravichandran et al. [7]. They have
obtained the Fekete-Szegö inequalities for functions in these classes.
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.
[10].
In the present paper, we obtain the Fekete-Szegö inequality for functions in
a more general class Mα,b (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 Mα,b
(φ) of functions
defined by fractional derivatives.
Definition 1.1. Let b 6= 0 be a complex number. 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. A function f ∈ A is in the class Mα,b (φ) if
1 zf 0 (z) + αz 2 f 00 (z)
1+
− 1 ≺ φ(z) (α ≥ 0).
b
f (z)
g
For fixed g ∈ A, we define the class Mα,b
(φ) to be the class of functions f ∈ A
for which (f ∗ g) ∈ Mα,b (φ).
To prove our result, we need the following:
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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Lemma 1.1 ([7]). If p(z) = 1 + c1 z + c2 z 2 + · · · is a function with positive real
part, then for any complex number µ,
|c2 − µc21 | ≤ 2 max{1, |2µ − 1|}
and the result is sharp for the functions given by
p(z) =
1 + z2
,
1 − z2
p(z) =
1+z
.
1−z
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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2.
The 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,b (φ), then
B2
(1
+
2α)
−
2µ(1
+
3α)
B1 |b|
2
.
|a3 − µa2 | ≤
max 1, +
bB
1
2(1 + 3α)
B1
(1 + 2α)2
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
The result is sharp.
Proof. If f (z) ∈ Mα,b (φ), then there is a Schwarz function w(z), analytic in ∆
with w(0) = 0 and |w(z)| < 1 in ∆ such that
1 zf 0 (z) + αz 2 f 00 (z)
(2.1)
1+
− 1 = φ(w(z)).
b
f (z)
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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Define the function p1 (z) by
(2.2)
p1 (z) :=
1 + w(z)
= 1 + c1 z + c2 z 2 + · · · .
1 − w(z)
Since w(z) is a Schwarz function, we see that Rp1 (z) > 0 and p1 (0) = 1.
Define the function p(z) by
1 zf 0 (z) + αz 2 f 00 (z)
− 1 = 1 + b1 z + b2 z 2 + · · · .
(2.3) p(z) := 1 +
b
f (z)
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In view of the equations (2.1), (2.2), (2.3), we have
p1 (z) − 1
(2.4)
p(z) = φ
p1 (z) + 1
and from this equation (2.4), we obtain
1
b1 = B1 c1
2
(2.5)
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
and
(2.6)
1
1
1 2
b2 = B1 c2 − c1 + B2 c21 .
2
2
4
From equation (2.3), we obtain
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(1 + 2α)a2 = bb1 ,
(2 + 6α)a3 = bb2 + (1 + 2α)a22
or equivalently we have
(2.7)
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
a2 =
bb1
,
1 + 2α
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(2.8)
1
b2 b21
a3 =
bb2 +
.
2 + 6α
1 + 2α
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Applying (2.5) in (2.7) and (2.5), (2.6) in (2.8), we have
bB1 c1
,
2(1 + 2α)
2 2
bB1 c2
c21
b B1
a3 =
+
− b(B1 − B2 ) .
4(1 + 3α) 8(1 + 3α) 1 + 2α
a2 =
Therefore we have
(2.9)
where
a3 − µa22 =
bB1
c2 − vc21 ,
4(1 + 3α)
1
B2
2µ(1 + 3α) − (1 + 2α)
v :=
1−
+
bB1 .
2
B1
(1 + 2α)2
Our result now follows by an application of Lemma 1.1. The result is sharp for
the function defined by
1 zf 0 (z) + αz 2 f 00 (z)
1+
− 1 = φ(z 2 )
b
f (z)
and
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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1
1+
b
zf 0 (z) + αz 2 f 00 (z)
− 1 = φ(z).
f (z)
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Example 2.1. By taking b = (1 − β)e−iλ cos λ, φ(z) =
following sharp inequality
1+z
,
1−z
we obtain the
(1 − β) cos λ
1+ 3α
iλ
2µ(1 + 3α) − (1 + 2α)
× max 1, e − 2
(1 − β) cos λ .
2
(1 + 2α)
|a3 − µa22 | ≤
Remark 1. When α = 0, Example 2.1 reduces to a result of [7] for λ-spirallike
function f (z) of order β.
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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3.
Application to Functions Defined by Fractional
Derivatives
λ
In order to introduce the class Mα,b
(φ), we need the following:
Definition 3.1. (See [5, 6]; see also [11, 12]). Let the function 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 log(z − ζ) to be real
when 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
λ
(Ω f )(z) = Γ(2 − λ)z
λ
Dzλ f (z),
(λ 6= 2, 3, 4, . . . ).
λ
The class Mα,b
(φ) consists of functions f ∈ A for which Ωλ f ∈ Mα,b (φ). Note
0
0
λ
that M0,b
(φ) = Sb∗ (φ) and M0,1
(φ) = S ∗ (φ). Also Mα,b
(φ) is the special case
g
of the class Mα,b (φ) when
(3.1)
g(z) = z +
∞
X
n=2
Γ(n + 1)Γ(2 − λ) n
z .
Γ(n + 1 − λ)
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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Let
g(z) = z +
∞
X
gn z n (gn > 0).
n=2
Since
f (z) = z +
∞
X
g
an z n ∈ Mα,b
(φ)
n=2
if and only if
(f ∗ g)(z) = z +
∞
X
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
gn an z n ∈ Mα,b (φ),
n=2
g
we obtain the coefficient estimate for functions in the class Mα,b
(φ), from the
corresponding estimate for functions in the class Mα,b (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 after an obvious change of the parameter µ:
2
Theorem 3.1. Let the function φ(z) be given by φ(z) = 1 + B1 z + B2 z +
g
B3 z 3 + · · · . If f (z) given by (1.1) belongs to Mα,b
(φ), then
µa22 |
|a3 −
B2
B1 |b|
(1 + 2α)g22 − 2µ(1 + 3α)g3
≤
max 1, +
bB1 .
2
2
2g3 (1 + 3α)
B1
(1 + 2α) g2
The result is sharp.
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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Since
λ
(Ω f )(z) = z +
∞
X
Γ(n + 1)Γ(2 − λ)
n=2
Γ(n + 1 − λ)
Page 11 of 14
n
an z ,
J. Ineq. Pure and Appl. Math. 7(4) Art. 145, 2006
http://jipam.vu.edu.au
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(z) be given by φ(z) = 1 + B1 z + B2 z 2 +
λ
B3 z 3 + · · · . If f (z) given by (1.1) belongs to Mα,b
(φ), then
B1 |b|(2 − λ)(3 − λ)
12(1 + 3α)
B2
(1 + 2α)(3 − λ) − 3µ(1 + 3α)(2 − λ)
.
× max 1, +
bB
1
B1
(3 − λ)(1 + 2α)2
|a3 − µa22 | ≤
The result is sharp.
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
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] F.R. KEOGH AND E.P. MERKES, A coefficient inquality for certain
classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8–12.
[4] 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.
[5] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized hypergeometric 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, METIN BOLCAL, YASAR POLATOGLU AND
A. SEN, Certian subclasses of Starlike and Convex functions of Complex
Order, Hacettepe Journal of Mathematics and Statistics, 34 (2005), 9-15.
[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, S. SIVASUBRAMANIAN AND M. DARUS,
Fekete-Szegö inequality for a certain class of analytic functions, Preprint.
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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[10] 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.
[11] H.M. SRIVASTAVA AND S. OWA, An application of the fractional derivative, Math. Japon., 29 (1984), 383–389.
[12] H.M. SRIVASTAVA AND S. OWA, Univalent functions, Fractional Calculus, and their Applications, Halsted Press / John Wiley and Sons, Chichester / New York, (1989).
A Coefficient Inequality For
Certain Classes Of Analytic
Functions Of Complex Order
K. Suchithra, B. Adolf Stephen
and S. Sivasubramanian
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