General Mathematics Vol. 17, No. 3 (2009), 99–104
A general coefficient inequality 1
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
Abstract
In the present note, we obtain a general coefficient inequality regarding a multiplier transformation in the open unit disc E = {z :
|z| < 1}. As special case to our main result, we obtain a coefficient inequality for n-starlikeness of analytic functions. Also certain
known coefficient inequalities for starlikeness and convexity of analytic functions appear as particular cases of our main result.
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words and phrases: Analytic function, Starlike function, Convex
function, Multiplier transformation.
1
Received 4 September, 2008
Accepted for publication (in revised form) 16 September, 2008
99
100
1
S. Singh, S. Gupta and S. Singh
Introduction
Let Ap denote the class of functions of the form
∞
X
p
f (z) = z +
ak z k , p ∈ N = {1, 2, · · · },
k=p+1
which are analytic in the open unit disc E = {z : |z| < 1}. We write
A1 = A.
Denote by S ∗ (α) and K(α), the classes of starlike functions of order α
and convex functions of order α respectively, which are analytically defined
as follows:
¾
½
zf 0 (z)
> α, z ∈ E
S (α) = f (z) ∈ A : <
f (z)
∗
and
½
µ
¶
¾
zf 00 (z)
K(α) = f (z) ∈ A : < 1 + 0
> α, z ∈ E
f (z)
where α is a real number such that 0 ≤ α < 1.
We shall use S ∗ and K to denote S ∗ (0) and K(0), respectively which
are the classes of univalent starlike (w.r.t. the origin) and univalent convex
functions.
For f ∈ Ap , we define the multiplier transformation Ip (n, λ) as
¶n
∞ µ
X
k+λ
Ip (n, λ)f (z) = z +
ak z k , (λ ≥ 0, n ∈ Z).
p
+
λ
k=p+1
p
I1 (n, 0) is the well-known Sălăgean [2] derivative operator Dn , defined
as
n
D f (z) = z +
∞
X
k=2
and for f ∈ A.
k n ak z k , n ∈ N0 = N ∪ {0},
A general coefficient inequality
101
Denote by Sn∗ (α), the class of n-starlike functions of order α, which is
analytically defined as follows
¾
½
Dn+1 f (z)
∗
> α, z ∈ E
Sn (α) = f (z) ∈ A : <
Dn f (z)
where α is a real number such that 0 ≤ α < 1.
In the present note, we obtain a general coefficient inequality regarding
multiplier transformation Ip (n, λ) in the open unit disc E = {z : |z| < 1}.
As special case to our main result, we obtain a coefficient inequality for nstarlikeness of analytic functions. Also certain known coefficient inequalities
for starlikeness and convexity of analytic functions appear as particular cases
of our main result.
2
Main Result
Theorem 1 If f ∈ Ap satisfies
¶n µ
¶
∞ µ
X
k+λ
k+λ
− 1 + p − α |ak | ≤ p − α,
p+λ
p+λ
k=p+1
then
¯
¯
¯ Ip (n + 1, λ)f (z)
¯
¯
¯ < p − α, 0 ≤ α < p, z ∈ E.
−
1
¯ Ip (n, λ)f (z)
¯
Proof. To prove the required result, we prove that
|Ip (n + 1, λ)f (z) − Ip (n, λ)f (z)| − (p − α) |Ip (n, λ)f (z)| ≤ 0
Indeed, we have
|Ip (n + 1, λ)f (z) − Ip (n, λ)f (z)| − (p − α) |Ip (n, λ)f (z)|
102
S. Singh, S. Gupta and S. Singh
¯ ∞ µ
¯
¶
¯
¯ X k + λ ¶n µ k + λ
¯
¯
= ¯
− 1 ak z k ¯
¯
¯
p+λ
p+λ
k=p+1
¯
¯
¶n
∞ µ
¯
¯
X
k
+
λ
¯
¯
−(p − α) ¯z p +
ak z k ¯
¯
¯
p
+
λ
k=p+1
¶n µ
¶
∞ µ
X
¯ ¯
k+λ
k+λ
≤
− 1 |ak | ¯z k ¯
p+λ
p+λ
k=p+1
Ã
!
¶n
∞ µ
X
¯
¯
k
+
λ
−(p − α) |z p | −
|ak | ¯z k ¯
p
+
λ
k=p+1
µ
¶
µ
¶
∞
n
X k+λ
¯ ¯
k+λ
=
− 1 + p − α |ak | ¯z k ¯ − (p − α) |z p |
p+λ
p+λ
k=p+1
¶n µ
¶
∞ µ
X
k+λ
k+λ
− 1 + p − α |ak | − (p − α) ≤ 0.
<
p
+
λ
p
+
λ
k=p+1
Hence
3
¯
¯
¯ Ip (n + 1, λ)f (z)
¯
¯
¯
¯ Ip (n, λ)f (z) − 1¯ < p − α, 0 ≤ α < p, z ∈ E.
Applications
For λ = 0 and p = 1 in Theorem 1, we obtain the following result.
Corollary 1 If f ∈ A satisfies
∞
X
k n (k − α)|ak | ≤ 1 − α,
k=2
then f ∈ Sn∗ (α).
For λ = 0, p = 1 and n = 0 in Theorem 1, we obtain the following result.
A general coefficient inequality
103
Corollary 2 If f ∈ A satisfies
∞
X
(k − α)|ak | ≤ 1 − α,
k=2
then f ∈ S ∗ (α).
For λ = 0, p = 1 and n = 1 in Theorem 1, we obtain the following result.
Corollary 3 If f ∈ A satisfies
∞
X
k(k − α)|ak | ≤ 1 − α,
k=2
then f ∈ K(α).
Remark 1 For α = 0, Corollary 2 and Corollary 3 are the results of Clunie
and Keogh [1] and Silverman [3].
References
[1] Clunie, J. and Keogh, F.R. , On starlike and convex schlicht functions,
J. londan Math. Soc., 35(1960), 229-233.
[2] Sălăgean, G. S. , Subclasses of univalent functions, Lecture Notes in
Math., 1013, 362-372, Springer-Verlag, Heideberg,1983.
104
S. Singh, S. Gupta and S. Singh
[3] Silverman, H. , Univalent functions with negative coefficient, Proc.
Amer. Math. Soc., 51(1975), 109-116.
Sukhwinder Singh
Department of Applied Sciences
Baba Banda Singh Bahadur Engineering College
Fatehgarh Sahib -140407 (Punjab), INDIA
e-mail : ss billing@yahoo.co.in
Sushma Gupta, Sukhjit Singh
Department of Mathematics
Sant Longowal Institute of Engineering & Technology
Longowal-148106 (Punjab), INDIA
e-mail : sushmagupta1@yahoo.com, sukhjit d@yahoo.com