Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 751721, 11 pages
doi:10.1155/2010/751721
Research Article
On Certain Multivalent Starlike or
Convex Functions with Negative Coefficients
Neslihan Uyanik,1 Erhan Deniz,2 Ekrem Kadioǧlu,2
and Shigeyoshi Owa3
1
Department of Mathematics, Kazim Karabekir Faculty of Education, Atatürk University,
Erzurum 25240, Turkey
2
Department of Mathematics, Science and Art Faculty, Atatürk University, Erzurum 25240, Turkey
3
Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
Correspondence should be addressed to Shigeyoshi Owa, shige21@ican.zaq.ne.jp
Received 2 April 2010; Accepted 3 June 2010
Academic Editor: N. Govil
Copyright q 2010 Neslihan Uyanik et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
By means of a differential operator, we introduce and investigate some new subclasses of pvalently analytic functions with negative coefficients, which are starlike or convex of complex
order. Relevant connections of the definitions and results presented in this paper with those
obtained in several earlier works on the subject are also pointed out.
1. Introduction
Let Am p denote the class of functions of the following form:
fz zp −
∞
ak zk
ak ≥ 0; m, p ∈ N : {1, 2, 3, . . .} ,
1.1
kpm
which are analytic and multivalent in the open unit dics U {z : z ∈ C and |z| < 1}.
Let f q denote the qth-order ordinary differential operator for a function f ∈ Am p,
that is,
∞
p!
k!
zp−q −
ak zk−q ,
p−q !
k
−
q
!
kpm
f q z where p > q; p ∈ N; q ∈ N0 N ∪ {0}, z ∈ U.
1.2
2
Journal of Inequalities and Applications
Next, we define the differential operator Dn f q as
∞
p!
k−q n
k!
ak zk−q
zp−q −
p
−
q
p−q !
k
−
q
!
kpm
Dn f q z m ∈ N; z ∈ U.
1.3
In view of 1.3, it is clear that
D0 f q z f q z,
D1 f q z Df q z n q
D f
z D
n−1
Df
q
1 q z f z ,
p−q
1.4
z .
If we take p 1 and q 0 for Dn f q , then Dn f q become the differential operator defined by
Sălăgean 1.
n
q
Finally, in terms of a differential operator Dn f q defined by 1.3 above, let Em,p
denote the subclass of Am p consisting of functions f which satisfy the following
inequality:
n
Em,p
⎫
⎧
∞
⎬
⎨
Dn f q z
p
k
ak z , ak ≥ 0 ,
q f ∈ Am p :
/ 0, z ∈ C − {0}, fz z −
p−q
⎭
⎩
z
kpm
1.5
where k ∈ N, n ∈ N0 , k > n, m ∈ N; k − q/p − q ≥ p − q − n − 1 ≥ 0; z ∈ U.
n
q.
For m ∈ N, n ∈ N0 , and γ ∈ C − {0}, we define the next subclasses of Em,p
n
Em,p
n
Nm,p
q, γ q, γ f∈
⎧
⎨
f∈
⎩
n
Em,p
1 Dn1 f q z
−pqn
> 0, z ∈ U ,
q : Re 1 γ
Dn f q z
n
Em,p
∞
Re γ k−q n
k−q
k!
− p q n γ ak
q :
γ p−q
p−q
kpm k − q !
Re γ
p!
,
≤
γ −p q n 1 γ p−q !
⎧
∞
⎨
k−q n k−q
k!
n
n
Km,p q, γ f ∈ Em,p q :
− p q n γ ak
⎩
p−q
p−q
kpm k − q !
p! ≤
γ −pqn1 ,
p−q !
1.6
where γ ∈ C − {0}; m ∈ N; k − q/p − q ≥ p − q − n − 1 ≥ 0; z ∈ U.
Journal of Inequalities and Applications
3
0
0, γ S∗ b was studied by Nasr and Aouf 2 also see Bulboacă et al.
Remark 1.1. 1 Em,1
3.
0
1
0, 1 − α Tα m and Em,1
0, 1 − α Cα m, α ∈ 0, 1 were introduced by
2 Em,1
Srivastava et al. 4.
0
1
0, 1 − α T ∗ α and E1,1
0, 1 − α Cα, α ∈ 0, 1 were introduced by
3 E1,1
Silverman 5.
0
1
0
1
0, γ Om
γ and Km,1
0, γ Om
γ were introduced by Parvathan and
4 Km,1
Ponnusanny 6, pages 163-164.
n
n
n
q, γ, Nm,p
q, γ, and Km,p
q, γ are closely
5 For p 1 and q 0, the classes Em,p
related with Tn,m γ, On,m γ, and Pn,m γ which are defined by Owa and Sălăgean in 7.
n
n
In this paper we give relationships between the classes of Em,p
q, γ, Nm,p
q, γ, and
In the particular case when m ∈ N and n 0, p 1, and q 0, we obtain the same
results as in 8.
n
q, γ.
Km,p
2. Main Results
Our main results are contained in
Theorem 2.1. Let m ∈ N, n ∈ N0 and let γ ∈ C − {0}; then
n
n
1 Km,p
q, γ ⊆ Em,p
q, γ;
n
n
2 Em,p
q, γ ⊆ Nm,p
q, γ;
3 if γ ∈ 0, ∞, then
n
n
n
Km,p
q, γ Em,p
q, γ Nm,p
q, γ ;
2.1
n
n
q, γ ⊆
q, γ;
4 if γ ∈ −∞, 0, then Nm,p
/ Em,p
n
n
5 if γ ∈ −∞, 0, then Em,p
q, γ /
⊆ Km,p
q, γ.
n
Proof. 1 Let f ∈ Km,p
q, γ. We prove that
Dn1 f q z
− p q n < γ ,
n q
D f z
z ∈ U.
2.2
If f has the series expansion
fz zp −
∞
ak zk ,
kpm
ak ≥ 0,
2.3
4
Journal of Inequalities and Applications
then
Dn1 f q z
− p q n − γ n q
D f z
− p!/ p − q ! 1 − p q n
≤
n
k−p
p!/ p − q ! − ∞
kpm k!/ k − q ! k − q/p − q ak |z|
∞
kpm
n k!/ k − q ! k − q/p − q ak k − q / p − q − p q n |z|k−p − γ .
n
k−p
p!/ p − q ! − ∞
a
k!/
k
−
q
!
k
−
q/p
−
q
|z|
k
kpm
2.4
We use the fact that Dn f q z/zp−q /
0 for z ∈ U − {0} and limz → o Dn f q z/zp−q p!/p − q!; these imply
∞
p!
k−q n
k!
ak |z|k−p > 0
−
p − q ! kpm k − q ! p − q
2.5
for z ∈ U.
From 2.4 and 2.5, we deduce
Dn1 f q z
− p q n − γ n q
D f z
− p!/ p − q ! 1 − p q n γ <
n .
p!/ p − q ! − ∞
kpm k!/ k − q ! k − q/p − q ak
n ∞
k − q / p − q − p q n γ kpm k!/ k − q ! k − q/p − q ak
.
n
p!/ p − q ! − ∞
kpm k!/ k − q ! k − q/p − q ak
2.6
n
q, γ from this last inequality we, obtain 2.2 which
By using the definition of Km,p
implies
1 Dn1 f q z
Re
−pqn
> −1 z ∈ U,
γ
Dn f q z
n
q, γ.
hence f ∈ Em,p
2.7
Journal of Inequalities and Applications
5
n
q, γ. Then 2.7 holds and, by using 2.3, this is equivalent to
2 Let f be in Em,p
⎧ ⎛ ⎞⎫
n1
⎨1
⎬
ak zk−q
p!/ p − q ! zp−q − ∞
kpm k!/ k − q ! k − q/p − q
⎝ Re
− p q n⎠
p−q ∞
n
⎩γ
⎭
p!/ p − q ! z − kpm k!/ k − q ! k − q/p − q ak zk−q
> −1 z ∈ U.
2.8
For z t ∈ 0, 1 if t → 1− , from 2.8 we obtain
⎧⎛ n1 ⎞⎫ 2
⎨ − p!/ p − q ! ∞
ak ⎬ γ kpm k!/ k − q ! k − q/p − q
⎝
⎠ ≤
−pqn
∞
n
⎭ Re γ
⎩
p!/ p − q ! − kpm k!/ k − q ! k − q/p − q ak
2.9
which is equivalent to
2
2
γ p!
k − q n k − q γ k!
− p q n ak ≤ −pqn1 .
p−q
p − q Re γ
p − q ! Re γ
kpm k − q !
∞
2.10
n
Then multiplying the relation last inequality with Re γ/|γ|, we obtain f ∈ Nm,p
q, γ.
n
n
q, γ are
3 if γ is a real positive number, then the definitions of Nm,p q, γ and Km,p
n
n
equivalent, hence Nm,p q, γ Km,p q, γ. By using 1 and 2 from this theorem, we obtain
3.
4 We have the following two cases.
Case 1. γ ∈ p − q − n − 1 − m/p − q, 0.
Let fm,α p, q, n; z be defined by
fm,α
p m − q ! pm
p!
p m − q −n
p
p, q, n; z z − α
z
p−q
pm !
p−q !
2.11
and let α > 0. We have
Re γ k−q
k−q n
k!
− p q n γ ak
γ p−q
p−q
kpm k − q !
∞
pm !
Re γ pm−q n
pm−q
γ
−pqn
≤
γ p−q
p−q
pm−q !
pm−q !
p m − q −n
p!
×α
p−q
pm !
p−q !
γ
pm−q
p!
−pqn
−γ
α
p−q
−γ
p−q !
2.12
6
Journal of Inequalities and Applications
or
Re γ k−q n
k−q
k!
− p q n γ ak
γ p−q
p−q
kpm k − q !
∞
p!
m
γ ≤0
≤ −α −p q n 1 p−q
p−q !
p!
<
p−q !
2.13
Re γ −p q n 1 γ ,
γ n
n
and then fm,α p, q, n; z ∈ Nm,p
q, γ see the definition of Nm,p
q, γ.
Let now
⎛
⎞
q 1 ⎝ Dn1 fm,α p, q, n; z
⎠ z ∈ U.
Fz 1 −pqn
q γ
Dn f
p, q, n; z
2.14
m,α
Then, by a simple computation and by using the fact that
q q
fm,α p, q, n; z fm,α z
pm !
p m − q ! pm−q
p!
p!
p m − q −n
zp−q − α z
p−q
p−q !
pm−q !
pm !
p−q !
p!
p!
p m − q −n pm−q
z
.
zp−q − α p−q
p−q !
p−q !
p!
p m − q −n p m − q n pm−q
p!
z
zp−q − α p−q
p−q
p−q !
p−q !
q
Dn fm,α z p!
zp−q 1 − αzm ,
p−q !
q
pm−q m
p!
z ,
zp−q 1 − α
p−q
p−q !
Dn1 fm,α z 2.15
Journal of Inequalities and Applications
7
we obtain
1
Fz 1 γ
1
1
γ
q
Dn1 fm,α z
q
Dn fm,α z
−pqn
p!/ p − q ! zp−q 1 − α p m − q / p − q zm
−pqn
p!/ p − q ! zp−q 1 − αzm −p q n 1 − αzm −p q n p m − q / p − q
1
γ1 − αzm 1
2.16
a − αbζ
1 ϕζ,
γ1 − αζ
where ζ zm , a −p q n 1, b −p q n 1 m/p − q, and
ϕζ a − αbζ
.
γ1 − αζ
2.17
For α > 1 we, have ϕU C∞ − Dc, d, where D is the disc with the center
c
α2 b − a
γα2 − 1
2.18
d
αb − a
.
γ1 − α2 2.19
and the radius
We have FU C∞ − Dc 1, d where Dc, d {w : |w − c| < d} and we deduce that
Re Fz > 0 for all z ∈ U does not hold.
n
n
∈ Em,p
q, γ, but fm,α /
q, γ and in this case
We have obtained that for α > 1, fm,α ∈ Nm,p
n
n
⊆ Em,p q, γ.
Nm,p q, γ /
Case 2. γ ∈ −∞, p − q − n − 1 − m/p − q.
We consider the function fm,α defined by 2.11 for α ∈ 1, −p q n 1 γ/−p q n 1 m/p − q. In this case, the inequality 2.13 holds too and this implies that
n
q, γ.
fm,α ∈ Nm,p
n
q, γ like in Case 1.
We also obtain that f /
∈ Em,p
8
Journal of Inequalities and Applications
5 Let f fm,α be given by 2.11, where α > |γ| − p q n 1/|γ| − p q n 1 m/p − q and |γ| − p q n 1 m/p − q > 0. Then
∞
kpm
k−q n k−q
k!
− p q n γ ak
p−q
k−q ! p−q
pm !
pm−q n pm−q
−pqn γ
p−q
p−q
pm−q !
pm−q !
p m − q −n
p!
×α
· p−q
pm !
p−q !
p!
pm−q
− p q n γ α
p−q
p−q !
2.20
p! γ −p q n 1
p−q !
>
which implies that
n
fm,α /
∈ Km,p
q, γ
for m ∈ N, n ∈ N0 , γ ∈ −∞, 0.
2.21
We have
1
Fz 1 γ
q
Dn1 fm,α z
q
Dn fm,α z
−pqn
1
a − αbζ
1 ϕζ,
γ1 − αζ
2.22
where ϕ is given by 2.17.
From ϕU Dc, d where c and d are given by 2.18 and 2.19, we obtain
Re Fz ≥ 1 αb a
.
γα 1
2.23
If γ ∈ −∞, p − q − n − 1 − m/p − q and α ∈ |γ| − p q n 1/|γ| − p q n 1 m/p − q, 1, then
α γ b γ a
> 0,
γα 1
2.24
Journal of Inequalities and Applications
9
and if
m
,0 ,
p−q
γ − p q n 1
γ − p q n 1
α∈ , ∩ 0, 1,
γ − p q n 1 m/ p − q
−p q n 1 m/ p − q − γ 2.25
γ∈
p−q−n−1−
n
then 2.24 also holds. By combining 2.24 with 2.23 and the definition of Em,p
q, γ, we
obtain that
fm,α ∈
n
Em,p
q, γ
for α ∈
γ − p q n 1
,
γ − p q n 1 m/ p − q
γ − p q n 1
∩ 0, 1, γ ∈ −∞, 0.
−p q n 1 m/ p − q − γ 2.26
Appendix
n
In this paper, we discuss the class Em,p
q, γ of analytic functions with negative coefficients.
Let us consider the functions f given by
fz zp ∞
A.1
ak zk
kp1
which are analytic in U. For such a function f, we say that f ∈ Gn1,p q, γ if it satisfies
1
Re 1 γ
Dn1 f q z
Dn f q z
−pqn
>0
z ∈ U
A.2
for some complex number γ with 0 < Re1/γ < 1/p − q − n − 1.
If we define the function F for f ∈ Gn1,p q, γ by
1 1/γ Dn1 f q z/Dn f q z − p q n − i 1 − p q n Im 1/γ
Fz ,
1 1 − p q n Re 1/γ
A.3
10
Journal of Inequalities and Applications
then we know that F is analytic in U, F0 1, and Re fz > 0 z ∈ U. Thus F is the
Carathéodory function. Since the extremal function for the Carathéodory function F is given
by
Fz 1z
,
1−z
A.4
we can write
1 1/γ Dn1 f q z/Dn f q z − p q n − i 1 − p q n Im 1/γ
1z
.
1−z
1 1 − p q n Re 1/γ
A.5
This shows us that
1
1z
1
γ
1
1
−
p
q
n
Re
.
−
p
q
n
γ
−
iγ
1
−
p
q
n
Im
q
n
γ
γ
1−z
D f z
A.6
Dn1 f q z
Noting that
Dn1 f q z 1 n q z D f z ,
p−q
A.7
we see that
n q D f z
1
1
1
−
p
−
q
−
n
−
γ
iγ
1
−
p
q
n
Im
p − q Dn f q z
z
γ
1
1
2
γ 1 1 − p q n Re
,
γ
1−z z
A.8
n q D f z
1
1
1
1
2γ
1
1
−
p
q
n
Re
.
−
q
n
p − q D f z
z
γ
1−z
A.9
that is,
It follows from the above that
! z
0
n q ! z
D f t
1
1
1
1
dt
2γ
1
1
−
p
q
n
Re
dt.
−
p − q Dn f q t
t
γ
1
−t
0
A.10
Calculating the above integrations, we have that
1
1
n q
log D f z − log z −2γ 1 1 − p q n Re
log1 − z.
p−q
γ
A.11
Journal of Inequalities and Applications
11
Therefore, we obtain that
Dn f q z
1/p−q
z
1 − z
2γ11−pqn Re1/γ
,
A.12
that is,
n q
D f
z z
1 − z2γ11−pqn Re1/γ
p−q
.
A.13
Consequently, the function f defined by the above is the extremal function for the class
n
Gn1,p q, γ. But our class Em,p
q, γ is defined with analytic functions f with negative
coefficients. Thus we do not know how we can consider the extremal function for this class.
References
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