General Mathematics Vol. 17, No. 4 (2009), 195–204
Coefficient bounds for some families of
starlike and convex functions of complex
order
Sevtap Sümer Eker, Bilal Şeker , Mugur Acu
Abstract
2000 Mathematics Subject Classification: Primary 30C45, 30C50.
Key words and phrases: Dziok-Srivastava Operator; Coefficient
bounds; Starlike functions of complex order; Convex functions of complex
order; Nonhomogeneous Cauchy-Euler differential equations.
1
Introduction and preliminaries
Let A denote the class of functions of the form
f (z) = z +
∞
X
ak z k ,
k=2
which are analytic in the open unit disk U = {z ∈ C : |z| < 1}.
195
(1.1)
196
Sevtap Sümer Eker, Bilal Şeker, Mugur Acu
Given two functions f, g ∈ A, where f (z) is given by (1.1) and g(z) is
given by
g (z) = z +
∞
X
bk z k ,
k=2
the Hadamard product (or convolution) f ∗ g is defined (as usual) by
(f ∗ g) (z) = z +
∞
X
ak bk z k = (g ∗ f )(z) , z ∈ U.
k=2
−
For αj ∈ C (j = 1, 2, ...q) and βj ∈ C\Z−
0 ; Z0 = {0, −1, −2, ...} (j =
1, 2, ...s) the generalized hypergeometric function q Fs (α1 , ..., αq ; β1 , ..., βs ; z)
is defined by
q Fs (α1 , ..., αq ; β1 , ..., βs ; z) =
∞
X
(α1 )k ...(αq )k z k
k=0
(β1 )k ...(βs )k k!
(q ≤ s + 1, q, s ∈ N0 = {0, 1, 2, ...} = N ∪ {0}) .
Here, and in what follows, (κ)n denotes the Pochhammer symbol (or
shifted factorial) defined, in terms of the Gamma function Γ, by

Γ(κ + n)  1
(κ)n =
=
 κ(κ + 1)...(κ + n − 1)
Γ(κ)
n = 0, κ 6= 0
n∈N
For the function
h(α1 , ..., αq ; β1 , ..., βs ; z) = zq Fs (α1 , ..., αq ; β1 , ..., βs ; z)
the Dziok-Srivastava linear operator [2] (see also [3]) Hsq (α1 , ..., αq ; β1 , ..., βs ; z)
is defined by the following Hadamard product (or convolution) :
Coefficient bounds for some families of starlike and convex functions ... 197
Hsq (α1 , ..., αq ; β1 , ..., βs ; z)f (z) = h(α1 , ..., αq ; β1 , ..., βs ; z) ∗ f (z)
=z+
∞
X
(α1 )k−1 ...(αq )k−1
k=2
(β1 )k−1 ...(βs )k−1
1
ak z k .
(k − 1)!
For notational simplicity, we write
Hsq (α1 , ..., αq ; β1 , ..., βs ; z)f (z) = (Hsq [α1 ]f ) (z)
A function f (z) ∈ A is said to be in the class S ∗ (b) if it satisfies the
following inequality:
·
1
Re 1 +
b
µ
¶¸
zf 0 (z)
−1
>0
f (z)
(z ∈ U, b ∈ C∗ = C \ {0})
Furthermore, a function f (z) ∈ A is said to be in the class C(b) if it also
satisfies the following inequality:
·
1
Re 1 +
b
µ
zf 00 (z)
f 0 (z)
¶¸
>0
(z ∈ U, b ∈ C∗ )
The function classes S ∗ (b) and C(b) were considered earlier by Nasr and
Aouf [4-6] and Wiatrowski [7], respectively.
Furthermore, a function f (z) ∈ A is said to be in the class SC(b, λ, γ)
·
1
Re 1 +
b
µ
¶¸
z[λzf 0 (z) + (1 − λ)f (z)]0
−1
>γ
λzf 0 (z) + (1 − λ)f (z)
(f (z) ∈ A ; 0 ≤ λ ≤ 1 ; 0 ≤ γ < 1 ; b ∈ C ∗ ; z ∈ U) .
(1.2)
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Sevtap Sümer Eker, Bilal Şeker, Mugur Acu
The function class satisfying the inequality (1.2) was considered by
Altıntaş et al. [1].
Let SC q,s
α,β (b, λ, γ) denote the subclass of A consisting of functions f (z)
which satisfy the following condition:
½
µ
¶¾
1 z[λz(Hsq [α1 ]f )0 (z) + (1 − λ)(Hsq [α1 ]f )(z)]0
Re 1 +
−1
> γ, (1.3)
b
λz(Hsq [α1 ]f )0 (z) + (1 − λ)(Hsq [α1 ]f )(z)
(f (z) ∈ A, 0 ≤ λ ≤ 1, 0 ≤ γ < 1, b ∈ C∗ , q ≤ s + 1, q, s ∈ N0 , z ∈ U) .
For q = 2, s = 1, α1 = β1 , α2 = 1, we obtain the class of SC(b, λ, γ).
Furthermore, for q = 2, s = 1, α1 = β1 , α2 = 1, γ = 0 and λ = 0 the
∗
class SC q,s
α,β (b, λ, γ) is coincide the class S (b) and for q = 2, s = 1, α1 = β1 ,
α2 = 1, γ = 0 and λ = 1 we obtain the class C(b).
The main object of the present investigation is to derive some coefficient
q,s
bounds for functions in the subclass Tα,β
(b, λ, γ; µ) of A which consists of
functions f (z) ∈ A satisfying the following nonhomogeneous Cauchy-Euler
differential equation:
z2
dw
d2 w
+
2(1
+
µ)z
+ µ(1 + µ)w = (1 + µ)(2 + µ)g(z)
dz 2
dz
¡
¢
w = f (z) ∈ A, g(z) ∈ SC q,s
α,β (b, λ, γ), µ ∈ R \ (−∞, −1] .
(1.4)
Coefficient bounds for some families of starlike and convex functions ... 199
2
Main results
Theorem 1. Let the function f (z) ∈ A be defined by (1.1). If the function
f (z) is in the class SC q,s
α,β (b, λ, γ), then
k−2
Y
[j + 2|b|(1 − γ)]
|ak | ≤
j=0
[1 + λ(k −
(k ∈ N∗ = N\{1} = {2, 3, 4, ...})
1 )k−1 ...(αq )k−1
1)] (α
(β1 )k−1 ···(βs )k−1
(2.1)
Proof. Let the function f (z) ∈ A be given by (1.1) and let the function
F (z) be defined by
F (z) = λz(Hsq [α1 ]f )0 (z) + (1 − λ)(Hsq [α1 ]f )(z),
(f (z) ∈ A, 0 ≤ λ ≤ 1).
Then from (1.3) and the definition of F (z) above, it is easily seen that
·
1
Re 1 +
b
µ
¶¸
zF 0 (z)
−1
>γ
F (z)
with
F (z) = z +
∞
X
Ak z k ∈ A
k=2
µ
1
(α1 )k−1 · · · (αq )k−1
Ak = [1 + λ(k − 1)]
ak ,
(β1 )k−1 · · · (βs )k−1 (k − 1)!
Thus, by setting
1+
1
b
³
zF 0 (z)
F (z)
´
−1 −γ
1−γ
= p(z)
¶
∗
k∈N
200
Sevtap Sümer Eker, Bilal Şeker, Mugur Acu
or, equivalently,
zF 0 (z) = [1 + b(1 − γ)(p(z) − 1)] F (z),
(2.2)
we get
p(z) = 1 + p1 z + p2 z 2 + · · ·
(z ∈ U).
(2.3)
Since
Re (p(z)) > 0,
0 ≤ γ < 1 ; b ∈ C∗
we conclude that
|pk | ≤ 2
(k ∈ N).
We also find from (2.2) and (2.3) that
(k − 1)Ak = b(1 − γ) (p1 Ak−1 + p2 Ak−2 + · · · + pk−1 ) .
In particular, for k = 2, 3, 4, we have
A2 = b(1 − γ)p1 implies |A2 | ≤ 2 |b| (1 − γ)
2A3 = b(1 − γ)(p1 A2 + p2 ) implies |A3 | ≤
2 |b| (1 − γ)[1 + 2|b|(1 − γ)]
2!
and
3A4 = b(1 − γ)(p1 A3 + p2 A2 + p3 )
Coefficient bounds for some families of starlike and convex functions ... 201
implies
|A4 | ≤
2 |b| (1 − γ)[1 + 2|b|(1 − γ)][2 + 2|b|(1 − γ)]
3!
respectively. Using the principle of mathematical induction, we obtain,
k−2
Y
[j + 2|b|(1 − γ)]
|Ak | ≤
j=0
(k − 1)!
(k ∈ N∗ ).
Moreover, by the relationship between the functions f (z) and F (z), it
is clear that
Ak = [1 + λ(k − 1)]
(α1 )k−1 · · · (αq )k−1
1
ak
(β1 )k−1 · · · (βs )k−1 (k − 1)!
(k ∈ N∗ ).
and then we get
k−2
Y
[j + 2|b|(1 − γ)]
|ak | ≤
j=0
1 )k−1 ...(αq )k−1
[1 + λ(k − 1)] (α
(β1 )k−1 ···(βs )k−1
.
By choosing suitable values of the admissible parameters b, λ, γ, α and
β in Theorem 1 above, we deduce the following corollaries.
Corollary 1. (Altıntaş et al. [1]) If a function f (z) ∈ A is in the class
SC(b, λ, γ), then
k−2
Y
[j + 2|b|(1 − γ)]
|ak | ≤
j=0
(k − 1)![1 + λ(k − 1)]
(k ∈ N∗ ).
202
Sevtap Sümer Eker, Bilal Şeker, Mugur Acu
Corollary 2. (Nasr and Aouf [4]) If a function f (z) ∈ A is in the class
S ∗ (b), then
k−2
Y
[j + 2|b|]
|ak | ≤
j=0
(k ∈ N∗ ).
(k − 1)!
Corollary 3. (Nasr and Aouf [4]) If a function f (z) ∈ A is in the class
C(b), then
k−2
Y
[j + 2|b|]
|ak | ≤
j=0
(k ∈ N∗ ).
k!
Theorem 2. Let the function f (z) ∈ A be defined by (1.1). If the function
q,s
f (z) is in the class Tα,β
(b, λ, γ; µ), then
k−2
Y
(1 + µ)(2 + µ)
|ak | ≤
[j + 2|b|(1 − γ)]
j=0
[1 + λ(k −
1 )k−1 ···(αq )k−1
1)] (α
(k
(β1 )k−1 ···(βs )k−1
+ µ)(k + 1 + µ)
,
(k ∈ N∗ ).
(3.1)
Proof. Let f (z) ∈ A be given by (1.1). Also let
g (z) = z +
∞
X
bk z k ∈ SC q,s
α,β (b, λ, γ),
(3.2)
k=2
so
ak =
(1 + µ)(2 + µ)
bk ,
(k + µ)(k + 1 + µ)
(k ∈ N∗ , µ ∈ R \ (−∞, −1])
(3.3)
Coefficient bounds for some families of starlike and convex functions ... 203
Thus, by using Theorem 1, we readily obtain
k−2
Y
(1 + µ)(2 + µ)
|ak | ≤
[j + 2|b|(1 − γ)]
j=0
[1 + λ(k −
1 )k−1 ···(αq )k−1
1)] (α
(k
(β1 )k−1 ···(βs )k−1
+ µ)(k + 1 + µ)
,
(k ∈ N∗ )
which is precisely the assertion (3.1) of Theorem 2.
References
[1] O. Altintas, H.Irmak, S. Owa, H.M. Srivastava, Coefficient bounds for
some families of starlike and convex functions of complex order, Applied
Mathematics Letters 20(2007) 1218-1222
[2] J. Dziok, H. M. Srivastava, Classes of analytic functions associated
with the generalized hypergeometric function, Appl. Math. Comput.
103(1999), no. 1, 1-13.
[3] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct. 14(2003) 7-18.
[4] M.A. Nasr, M.K. Aouf, Radius of convexity for the class of starlike functions of complex order, Bull. Fac. Sci. Assiut Univ. Sect. A 12(1983),
153-159.
[5] M.A. Nasr, M.K. Aouf, Bounded starlike functions of complex order,
Proc. Indian Acad. Sci. Math. Sci. 92(1983) 97-102.
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Sevtap Sümer Eker, Bilal Şeker, Mugur Acu
[6] M.A. Nasr, M.K. Aouf, Starlike function of complex order, J. Natur.
Sci. Math. 25(1985) 1-12.
[7] P. Wiatrowski, On the coefficients of some family of holomorphic functions, Zeszyty Nauk. Uniw. odz Nauk. Mat.-Przyrod Ser. 39(2) (1970),
75-85.
Sevtap Sümer Eker
Dicle University
Department of Mathematics
21280 - Diyarbakır, Turkey
E-mail: sevtaps@dicle.edu.tr Bilal Şeker
Batman University
Department of Mathematics
72060 - Batman, Turkey
E-mail: bilalseker1980@gmail.com Mugur Alexandru Acu
”Lucian Blaga” University
Department of Mathematics
550012 - Sibiu, Romania
E-mails: acu mugur@yahoo.com