Gen. Math. Notes, Vol. 22, No. 2, June 2014, pp.59-66
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Properties for a Subclass of Convex Functions
Puoi Choo Chuah1 and Aini Janteng2
1,2
School of Science and Technology
Universiti Malaysia Sabah, Jalan UMS
88400 Kota Kinabalu, Sabah, Malaysia
1
E-mail: colorwind.cpc@gmail.com
2
E-mail: aini− jg@ums.edu.my
(Received: 14-2-14 / Accepted: 1-4-14)
Abstract
Let S be the class of functions which are analytic and univalent in the open
∞
P
unit disc D = {z : |z| < 1} given by f (z) = z +
an z n and an a complex
n=2
number. Let T denote the class consisting of functions f of the form f (z) =
∞
P
z−
an z n where an is a non negative real number. In this paper, we introduce
n=2
a new subclass of S by adopting the idea of Ramesha et al. [3] and Sudharsan
et al. [5]. We also determine coefficient estimates, growth and extreme points
for f belonging to this class.
Keywords: Analytic, Univalent, Coefficient Estimates.
1
Introduction
Let S be the class of functions f which are analytic and univalent in the open
unit disc D = {z : |z| < 1} given by
f (z) = z +
∞
X
an z n
(1)
n=2
where an is a complex number. In [3], Ramesha et al. introduced the class
H(α) as follows:
60
Puoi Choo Chuah et al.
Definition 1.1 Let f be given by (1). Then, f ∈ A for which
2 00
αz f (z) zf 0 (z)
Re
+
>0
f (z)
f (z)
for α ≥ 0 and
f (z)
z
(2)
6= 0, z ∈ D.
analytic
Sudharsan et al. [5] introduced Ss∗ (α, β) of functions
and
univalent in
zf 0 (z)
αzf 0 (z)
D given by (1) and satisfying the condition f (z)−f (−z) − 1 < β f (z)−f
+
1
(−z)
for some 0 ≤ α ≤ 1, 0 < β ≤ 1 and z ∈ D. By developing the idea of Ramesha
et al. [3], we now introduce a new subclass of S denote as C(α) as follows:
Definition 1.2 Let f be given by (1). Then, f ∈ A for which
α(z 2 f 00 (z))0 (zf 0 (z))0
+
> 0,
Re
f 0 (z)
f 0 (z)
(3)
for 0 ≤ α < 1.
However, for this paper we consider a subclass of T where T denotes the class
consisting of functions f of the form as
f (z) = z −
∞
X
an z n
(4)
n=2
where an is a non negative real number. For f ∈ T, we define the class CT (α)
with α satisfying the condition 0 ≤ α < 1.
Definition 1.3 A function f ∈ CT (α) if and only if it satisfies
α(z 2 f 00 (z))0 (zf 0 (z))0
α(z 2 f 00 (z))0 (zf 0 (z))0
+
−
1
<
+
+
1
f 0 (z)
f 0 (z)
f 0 (z)
f 0 (z)
(5)
for z ∈ D.
We note that the above condition on α, is necessary to ensure this class form
subclass of S.
2
Preliminary Result
The following preliminary lemma is required for proving the main results.
61
Properties for a Subclass of Convex Functions
Lemma 2.1 Let f ∈ T , then
∞
X
n(n(n − 1)α + (n + 1))an |z|n−1 < 2
(6)
n=2
Proof: Since f ∈ T ,
∞
P
∞
P
n|an ||z|n−1 < 1,
n=2
∞
P
n2 |an ||z|n−1 < 1 and
n=2
n3 |an ||z|n−1 <
n=2
1 (see Silverman [4]). Thus, we obtain
∞
X
n(n(n − 1)α + (n + 1))an |z|n−1
n=2
=
∞
X
n(n(n − 1)α)an |z|n−1 +
n=2
∞
X
=α
=α
n=2
∞
X
∞
X
n(n + 1)an |z|n−1
n=2
3
2
n−1
(n − n )an |z|
+
∞
X
2
n−1
n an |z|
+
3
n−1
n an |z|
n=2
−α
2
nan |z|n−1
n=2
n=2
∞
X
∞
X
n−1
n an |z|
n=2
+
∞
X
2
n−1
n an |z|
n=2
+
∞
X
nan |z|n−1
n=2
<α−α+1+1
=2
3
Main Results
In this section, we give results concerning the coefficient estimates, growth and
extreme points for the functions f ∈ CT (α).
Theorem 3.1 f ∈ CT (α) if and only if
∞
X
n=2
n2 ((n − 1)α + 1)an ≤ 1
(7)
62
Puoi Choo Chuah et al.
Proof: We adopt the method used by Clunie and Keogh [1] and also by Owa
[2]. First we prove the ‘if’ part. According to Definition 1.3
α(z 2 f 00 (z))0 + (zf 0 (z))0 − f 0 (z) − α(z 2 f 00 (z))0 + (zf 0 (z))0 + f 0 (z)
∞
∞
X
X
= −
(n(n − 1)(nα + 1))an z n−1 − 2 −
n(n(n − 1)α + (n + 1))an z n−1 n=2
n=2
≤
<
∞
X
n=2
∞
X
n(n − 1)(nα + 1)an r
n−1
−2+
∞
X
n(n(n − 1)α + (n + 1))an rn−1
n=2
n(n − 1)(nα + 1)an − 2 +
n=2
∞
X
n(n(n − 1)α + (n + 1))an
n=2
∞ X
=
n(n − 1)(nα + 1) + n(n(n − 1)α + (n + 1)) an − 2
=
n=2
∞
X
2n2 ((n − 1)α + 1)an − 2
n=2
≤ 0 by (7)
Thus,
2 00 0
α(z f (z)) +(zf 0 (z))0
−
1
2 00 f 0 (z)
<1
α(z f (z))0 +(zf 0 (z))0
+ 1
f 0 (z)
and hence f ∈ CT (α).
To prove the ’only if’ part, let
2 00 0
α(z f (z)) +(zf 0 (z))0
P
−
1
− ∞
n(n − 1)(nα + 1)an z n−1
n=2
2 00 f 0 (z)
α(z f (z))0 +(zf 0 (z))0
= 2 − P∞ n(n(n − 1)α + (n + 1))an z n−1 < 1.
+ 1
n=2
f 0 (z)
We note that since f is analytic, continuous and non constant in D, the maximum modulus principle gives
P∞
n−1
−
n(n
−
1)(nα
+
1)a
z
n
n=2
2 − P∞ n(n(n − 1)α + (n + 1))an z n−1 n=2
P
| ∞
n(n − 1)(nα + 1)an z n−1 |
P∞n=2
=
|2 − n=2 n(n(n − 1)α + (n + 1))an z n−1 |
P∞
n(n − 1)(nα + 1)an |z|n−1
P∞n=2
≤
2 − n=2 n(n(n − 1)α + (n + 1))an |z|n−1
P∞
n−1
n=2 n(n − 1)(nα + 1)an r
P
≤
n−1
2− ∞
n=2 n(n(n − 1)α + (n + 1))an r
63
Properties for a Subclass of Convex Functions
P∞
n−1
where
< 2 from Lemma 2.1. Since
n=2 n(n(n − 1)α + (n + 1))an |z|
f ∈ CT (α) and 0 < r < 1, we obtain
P∞
n(n − 1)(nα + 1)an rn−1
n=2
P
<1
(8)
n−1
2− ∞
n=2 n(n(n − 1)α + (n + 1))an r
Now letting r → 1, in (8) and using Lemma 2.1, we obtain
∞
X
n(n − 1)(nα + 1)an ≤ 2 −
n=2
∞
X
n(n(n − 1)α + (n + 1))an
n=2
and hence
∞
X
n2 ((n − 1)α + 1)an ≤ 1
n=2
as required. Thus the proof of the Theorem 3.1 is completed.
The result is sharp for functions given by
fn (z) = z −
n2 ((n
1
z n , n ≥ 2.
− 1)α + 1)
Corollary 3.1 If f ∈ CT (α) then
an ≤
1
,
n2 ((n − 1)α + 1)
n ≥ 2.
Next, we give the result for growth of the class CT (α).
Theorem 3.2 Let the functions f be defined by (4) and belongs to the class
CT (α). Then for {z : 0 < |z| = r < 1},
r−
1
1
r2 ≤ |f (z)| ≤ r +
r2
4(α + 1)
4(α + 1)
The result is sharp.
Proof: First, it is obvious that
4(α + 1)
∞
X
n=2
an ≤
∞
X
n=2
n2 ((n − 1)α + 1)an
64
Puoi Choo Chuah et al.
and as f ∈ CT (α), using inequality in Theorem 3.1 yields
∞
X
an ≤
n=2
1
4(α + 1)
(9)
From (4) with |z| = r(r < 1), we have
|f (z)| ≤ r +
∞
X
n
an r ≤ r +
n=2
and
|f (z)| ≥ r −
∞
X
∞
X
an r 2
n=2
n
an r ≥ r −
n=2
∞
X
an r 2
n=2
Finally, using (9) in the above inequalities, gives the result in Theorem 3.2.
We note that result in Theorem 3.2 is sharp for the following function,
f2 (z) = z −
1
z2
4(α + 1)
at z = ±r.
Here, we consider extreme points for functions f ∈ CT (α).
1
Theorem 3.3 Let f1 (z) = z and fn (z) = z − n2 ((n−1)α+1)
z n for n ≥ 2. Then
P∞
f ∈ CT (α) if and only
if
it
can
be
expressed
in
the
form
f
(z)
=
n=1 λn fn (z)
P∞
where λn ≥ 0 and n=1 λn = 1.
Proof: Adopting the same technique used by Silverman [4], we first assume
f (z) =
∞
X
λn fn (z)
n=1
=z−
∞
X
n=2
λn
1
2
n ((n − 1)α + 1)
zn.
Next, since
2
∞
X
1
n ((n − 1)α + 1)
λn
·
2 ((n − 1)α + 1)
n
1
n=2
=
∞
X
λn
n=2
= 1 − λ1
≤ 1,
65
Properties for a Subclass of Convex Functions
therefore by Theorem 3.3, f ∈ CT (α).
Conversely, suppose f ∈ CT (α). Since
an ≤
n2 ((n
1
, n ≥ 2,
− 1)α + 1)
we may set λn = n2 ((n − 1)α + 1)an , (n ≥ 2) and λ1 = 1 −
∞
X
λn fn (z) = λ1 f1 (z) +
∞
X
n=1
P∞
n=2
λn . Then
λn fn (z)
n=2
= λ1 z +
=
1−
∞
X
λn
n=2
∞
X
1
zn
z− 2
n ((n − 1)α + 1)
∞
∞
X
X
λn z +
λn z −
λn
n=2
=z−
∞
X
n=2
=z−
∞
X
λn
n=2
n2 ((n
n=2
1
zn
n2 ((n − 1)α + 1)
1
zn
− 1)α + 1)
an z n
n=2
= f (z)
Hence, we complete the proof of Theorem 3.3.
Acknowledgements: This work was supported by FRG0268-ST-2/2010
Grant, Malaysia. The authors express their gratitude to the referee for his
valuable comments.
References
[1] J. Clunie and F.R. Keogh, On starlike and convex schlicht functions, J.
London Math. Soc., 35(1960), 229-233.
[2] S. Owa, On the special classes of univalent functions, Tamkang Journal,
15(1984), 123-136.
[3] C. Ramesha, S. Kumar and K.S. Padmanabhan, A sufficient condition for
starlikeness, Chinese J. Math., 23(1995), 167-171.
[4] H. Silverman, Univalent function with negative coefficients, Proc. Amer.
Math. Soc., 51(1) (1975), 109-116.
66
Puoi Choo Chuah et al.
[5] T.V. Sudharsan, P. Balasubrahmananyam and K.G. Subramanian, On
functions starlike with respect to symmetric and conjugate points, Taiwanese Journal of Mathematics, 2(1) (1998), 57-68.