Hacettepe Journal of Mathematics and Statistics
Volume 34 (2005), 9 – 15
CERTAIN SUBCLASSES OF
STARLIKE AND CONVEX FUNCTIONS
OF COMPLEX ORDER
V Ravichandran∗, Yasar Polatoglu†, Metin Bolcal† , and Arzu Sen†
Received 24 : 12 : 2004 : Accepted 05 : 12 : 2005
Abstract
In the present investigation, we consider certain subclasses of starlike
and convex functions of complex order, giving necessary and sufficient
conditions for functions to belong to these classes.
Keywords: Starlike functions, Convex functions, Starlike functions of complex order,
Convex functions of complex order.
2000 AMS Classification: 30 C 45
1. Introduction
Let A be the class of all analytic functions
(1)
f (z) = z + a2 z 2 + a3 z 3 + · · ·
in the open unit disk ∆ = {z ∈ C; |z| < 1}. A function f ∈ A is subordinate to an
univalent function g ∈ A, written f (z) ≺ g(z), if f (0) = g(0) and f (∆) ⊆ g(∆).
Let Ω be the family of analytic functions ω(z) in the unit disc ∆ satisfying the conditions ω(0) = 0, |ω(z)| < 1 for z ∈ ∆. Note that f (z) ≺ g(z) if there is a function
w(z) ∈ Ω such that f (z) = g(ω(z)).
Let S be the subclass of A consisting of univalent functions. The class S ∗ (φ), introduced and studied by Ma and Minda [5], consists of functions in f ∈ S for which
zf 0 (z)
≺ φ(z), (z ∈ ∆).
f (z)
∗
Department of Computer Applications, Sri Venkateswara College of Engineering, Sriperumbudur 602 105, India. E-mail: [email protected]
†
˙
Department of Mathematics, Faculty of Sciences and Arts, K¨
ult¨
ur University, Istanbul.
E-mail: (Y. Polatoglu) [email protected] (M. Bolcal) [email protected] (A. Sen)
[email protected]
10
V Ravichandran, Y. Polatoglu, M. Bolcal, A. Sen
The functions hφn (n = 2, 3, . . .) are defined by
zh0φn (z)
= φ(z n−1 ), hφn (0) = 0 = h0φn (0) − 1.
hφn (z)
The functions hφn are all functions in S ∗ (φ). We write hφ2 simply as hφ . Clearly,
¶
µZ z
φ(x) − 1
dx .
(2)
hφ (z) = z exp
x
0
Following Ma and Minda [5], we define a more general class related to the class of
starlike functions of complex order as follows.
1.1. Definition. Let b 6= 0 be a complex number. Let φ(z) be an analytic function with
positive real part on ∆, which satisfies φ(0) = 1, φ0 (0) > 0, and which maps the unit
disk ∆ onto a region starlike with respect to 1 and symmetric with respect to the real
axis. Then the class Sb∗ (φ) consists of all analytic functions f ∈ A satisfying
¶
µ
1 zf 0 (z)
− 1 ≺ φ(z).
1+
b
f (z)
The class Cb (φ) consists of the functions f ∈ A satisfying
1+
1 zf 00 (z)
≺ φ(z).
b f 0 (z)
Moreover, we let S ∗ (A, B, b) and C(A, B, b) (b 6= 0, complex) denote the classes Sb∗ (φ)
and Cb (φ) respectively, where
φ(z) =
1 + Az
, (−1 ≤ B < A ≤ 1).
1 + Bz
The class S ∗ (A, B, b), and therefore the class Sb∗ (φ), specialize to several well-known
classes of univalent functions for suitable choices of A, B and b.
The class S ∗ (A, B, 1) is denoted by S ∗ (A, B). Some of these classes are listed below:
(1) S ∗ (1, −1, 1) is the class S ∗ of starlike functions [1, 2, 7].
(2) S ∗ (1, −1, b) is the class of starlike functions of complex order introduced by
Wiatrowski [12].
(3) S ∗ (1, −1, 1 − β), 0 ≤ β < 1, is the class S ∗ (β) of starlike functions of order β.
This class was introduced by Robertson [8].
(4) S ∗ (1, −1, e−iλ cos λ), |λ| <
Spacek [11].
π
2
is the class of λ-spirallike functions introduced by
(5) S ∗ (1, −1, (1 − β)e−iλ cos λ), 0 ≤ β < 1, |λ| < π2 , is the class of λ-spirallike
functions of order β. This class was introduced by Libera [4].
³ 0
´
(z)
Let ST (b) denote 1 + 1b zff (z)
− 1 . Then we have the following:
(6) S ∗ (1, 0, b) is the set defined by |ST (b) − 1| < 1.
(7) S ∗ (β, 0, b) is the set defined by |ST (b) − 1| < β, 0 ≤ β < 1.
¯
¯
¯ ST (b)−1 ¯
< β, 0 ≤ β < 1.
(8) S ∗ (β, −β, b) is the set defined by ¯ (ST
(b)+1) ¯
(9) S ∗ (1, (−1 +
1
M
), b) is the set defined by |ST (b) − M | < M .
∗
(10) S (1 − 2β, −1, b) is the set defined by ReST (b) > β, 0 ≤ β < 1.
To prove our main result, we need the following Lemma due to Miller and Mocanu:
Starlike and Convex Functions of Complex Order
11
1.2. Lemma. [6, Corollary 3.4h.1, p.135] Let q(z) be univalent in ∆ and let ϕ(z) be
analytic in a domain containing q(∆). If zq 0 (z)/ϕ(q(z)) is starlike, then
zp0 (z)ϕ(p(z)) ≺ zq 0 (z)ϕ(q(z))
implies that p(z) ≺ q(z), and q(z) is the best dominant.
Let C be the class of convex analytic functions in ∆. We will also need the following
result:
1.3. Lemma. [10, Theorem 2.36, p. 86] For f, h ∈ C and g ≺ h, we have f ∗ g ≺ f ∗ h.
2. A necessary and Sufficient Condition
We begin with the following:
2.1. Lemma. Let φ be a convex
on ∆ and satisfying φ(0) = 1. As in
³R function defined
´
z φ(x)−1
Equation (1) let hφ (z) = z exp 0
dx , and let q(z) = 1 + c1 z + · · · be analytic in
x
∆. Then
zq 0 (z)
≺ φ(z)
(3)
1+
q(z)
if and only if for all |s| ≤ 1 and |t| ≤ 1, we have
(4)
q(tz)
shφ (tz)
≺
.
q(sz)
thφ (sz)
Proof. Our result and its proof are motivated by a similar result of Ruscheweyh [rus] for
functions in the class S ∗ (φ). Also see Ruscheweyh [10, Theorem 2.37, pages 86-88].
Let q(z) satisfy (3). Since the function
¶
Z zµ
t
s
−
dx
p(z) =
1 − sx
1 − tx
0
is convex and univalent in ∆ for s, t ∈ ∆ := ∆ ∪ {z ∈ C : |z| = 1}, s 6= t, by Lemma 1.2
we have:
µ 0
¶
zq (z)
(5)
∗ p(z) ≺ (φ(z) − 1) ∗ p(z).
q(z)
For an analytic function h(z) with h(0) = 0, we have
Z tz
dx
(6)
(h ∗ p)(z) =
h(x) ,
x
sz
and using (6), we see that (5) is equivalent to
¶
¶
Z tz µ 0
Z tz µ
q (x)
φ(x) − 1
dx ≺
dx,
q(x)
x
sz
sz
which gives the desired assertion (4) upon exponentiation.
To prove the converse, let us assume that (4) holds. By taking t = 1 in (4), we have
(7)
q(z)
shφ (z)
≺
,
q(sz)
hφ (sz)
and therefore we have
q(z)
shφ (φs (z))
(8)
=
,
q(sz)
hφ (sφs (z))
12
V Ravichandran, Y. Polatoglu, M. Bolcal, A. Sen
where φs (z) are analytic in ∆ and satisfy |φs (z)| ≤ |z|. Thus we can find a sequence
sk → 1 such that φsk → φ∗ locally uniformly in ∆, where |φ∗ (z)| ≤ |z| (z ∈ ∆).
Therefore, by making use of (8), we have for any fixed z ∈ ∆,
¸
·
zq 0 (z)
sk q(sk z) − q(z)
1+
= lim
k→∞
q(z)
(sk − 1)q(z)
¸
·
hφ (sk φsk (z)) − hφ (φsk (z))
φsk (z)
= lim
k→∞ hφ (φsk (z))
sk φsk (z) − φsk (z)
φ∗ (z)h0φ (φ∗ (z))
.
=
hφ (φ∗ (z))
This shows that
1+
zq 0 (z)
∈
q(z)
µ
zh0φ
hφ
¶
(∆) = φ(∆), (z ∈ ∆),
which completes the proof of our Lemma 2.1.
¤
By making use of Lemma 2.1, we now have the following:
2.2. Theorem.
³R Let φ be a´convex function defined on ∆ which satisfies φ(0) = 1, and
z φ(x)−1
dx be as in Equation‘(1). The the function f belongs to Sb∗ (φ)
x
0
hφ (z) = z exp
if and only if for all |s| ≤ 1 and |t| ≤ 1, we have
¶1
µ
shφ (tz)
sf (tz) b
≺
.
(9)
tf (sz)
thφ (sz)
Proof. Define the function q(z) by
¶1/b
µ
f (z)
(10)
q(z) :=
.
z
Then a computation show that
µ
¶
zq 0 (z)
1 zf 0 (z)
=1+
−1 .
1+
q(z)
b
f (z)
The result now follows from Lemma 2.1.
¤
As an immediate consequence of Theorem 2.2, we have:
2.3. Corollary. Let φ(z) and hφ (z) be as in Theorem 2.2. If f ∈ Sb∗ (φ), then we have
µ
¶1
hφ (z)
f (z) b
≺
(11)
.
z
z
3. Another Subordination Result
In this section, we prove the following without the assumption that the function φ is
convex. We only require that the function φ be starlike with respect to the origin.
3.1. Corollary. If f ∈ Sb∗ (φ), then we have
¶1
µ
f (z) b
hφ (z)
≺
(12)
,
z
z
where hφ (z) is given by (2).
Starlike and Convex Functions of Complex Order
13
Proof. Define the functions p(z) and q(z) by
¶1/b
µ
f (z)
hφ (z)
.
p(z) :=
, q(z) :=
z
z
Then a computation yields
1+
zp0 (z)
1
=1+
p(z)
b
µ
zf 0 (z)
−1
f (z)
¶
and
zh0φ (z)
zq 0 (z)
=
− 1 = φ(z) − 1.
q(z)
hφ (z)
Since f ∈ Sb∗ (φ), we have
¶
µ
zp0 (z)
zq 0 (z)
1 zf 0 (z)
=
− 1 ≺ φ(z) − 1 =
.
p(z)
b
f (z)
q(z)
The result now follows by an application of Lemma 1.1.
¤
4. The Fekete-Szeg¨
o inequality
In this section, we obtain the Fekete-Szeg¨
o inequality for functions in the class S b∗ (φ).
4.1. Theorem. Let φ(z) = 1 + B1 z + B2 z 2 + B3 z 3 + · · · . If f (z) given by Equation (1)
belongs to Sb∗ (φ), then
½
¾
B2
|a3 − µa22 | ≤ 2 max 1; |
+ (1 − 2µ)bB1 | .
B1
The result is sharp.
Proof. If f (z) ∈ Sb∗ (φ), then there is a Schwarz function w(z), analytic in ∆, with
w(0) = 0 and |w(z)| < 1 in ∆ and such that
µ
¶
1 zf 0 (z)
− 1 = φ(w(z)).
(13)
1+
b
f (z)
Define the function p1 (z) by
(14)
p1 (z) :=
1 + w(z)
= 1 + c 1 z + c2 z 2 + · · · .
1 − w(z)
Since w(z) is a Schwarz function, we see that <p1 (z) > 0 and p1 (0) = 1. Define the
function p(z) by
¶
µ
1 zf 0 (z)
− 1 = 1 + b 1 z + b2 z 2 + · · · .
(15)
p(z) := 1 +
b
f (z)
In view of the equations (13), (14) and (15), we have
µ
¶
p1 (z) − 1
(16)
p(z) = φ
.
p1 (z) + 1
Since
·
¸
p1 (z) − 1
c2
1
c3
c1 z + (c2 − 1 )z 2 + (c3 + 1 − c1 c2 )z 3 + · · ·
=
p1 (z) + 1
2
2
4
and therefore
¶
µ
·
¸
p1 (z) − 1
1
1
1
1
= 1 + B 1 c1 z +
φ
B1 (c2 − c21 ) + B2 c21 z 2 + · · · ,
p1 (z) + 1
2
2
2
4
14
V Ravichandran, Y. Polatoglu, M. Bolcal, A. Sen
from this equation and (16), we obtain
b1 =
1
B 1 c1
2
b2 =
1
1
1
B1 (c2 − c21 ) + B2 c21 .
2
2
4
and
Since
zf 0 (z)
= 1 + a2 z + (2a3 − a22 )z 2 + (3a4 + a32 − 3a3 a2 )z 3 + · · · ,
f (z)
from Equation (15), we see that
(17)
bb1 = a2 ,
(18)
bb2 = 2a3 − a22 ,
or equivalently we have
a2 = bb1 =
bB1 c1
,
2
ª
1©
bb2 + b2 b21
2
ª
b
c2 ©
= B1 c1 + 1 b2 B12 − b(B1 − B2 ) .
4
8
a3 =
Therefore we have
(19)
a3 − µa22 =
where
v :=
ª
bB1 ©
c2 − vc21 ,
4
¸
·
B2
1
1−
+ (2µ − 1)bB1 .
2
B1
We recall from [5] that if p(z) = 1 + c1 z + c2 z 2 + · · · is a function with positive real part,
then
|c2 − µc21 | ≤ 2 max{1, |2µ − 1|},
the result being sharp for the functions given by
p(z) =
1 + z2
,
1 − z2
p(z) =
1+z
.
1−z
Our result now follows from an application of the above inequality, and we see that he
result is sharp for the functions defined by
µ
¶
1 zf 0 (z)
1+
− 1 = φ(z 2 )
b
f (z)
and
1+
1
b
µ
zf 0 (z)
−1
f (z)
¶
= φ(z).
This completes the proof of the theorem.
¤
Starlike and Convex Functions of Complex Order
15
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