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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
ˇ
Neslihan Uyanik,1 Erhan Deniz,2 Ekrem Kadioglu,2
and Shigeyoshi Owa3
1

Department of Mathematics, Kazim Karabekir Faculty of Education, Ataturk University,
ă
Erzurum 25240, Turkey
2
Department of Mathematics, Science and Art Faculty, Ataturk University, Erzurum 25240, Turkey
ă
3
Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
Correspondence should be addressed to Shigeyoshi Owa,
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:
f z

∞

zp −

ak zk

ak ≥ 0; m, p ∈ N : {1, 2, 3, . . .} ,

1.1

k p m

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,
f

where p > q; p ∈ N; q ∈ N0

q

z

p!
zp−q −
p−q !

k

N ∪ {0}, z ∈ U.

∞
p m

k!
ak zk−q ,
k−q !

1.2


2

Journal of Inequalities and Applications
q

as

k−q
k!
k−q ! p−q

n

Next, we define the differential operator Dn f
Dn f

q

p!
zp−q −
p−q !
k

z

∞
p m

m ∈ N; z ∈ U .

ak zk−q

1.3

In view of 1.3 , it is clear that
D0 f

q

z

f

q

D1 f

z,
n

D f

q

q

z

Df

q

Df

q

z
D

n−1

1
z f
p−q

z

q

z

,
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
S˘ l˘ gean 1 .
aa
n
Finally, in terms of a differential operator Dn f q defined by 1.3 above, let Em,p q
denote the subclass of Am p consisting of functions f which satisfy the following
inequality:

n
Em,p

⎧
⎨
q

f ∈ Am

⎩

Dn f q z

p :
/ 0, z ∈ C − {0} , f z
zp−q

∞

z −
p

k p

⎫
⎬
ak z , ak ≥ 0 ,
⎭
m
k

1.5
where k ∈ N, n ∈ N0 , k > n, m ∈ N; k − q / p − q ≥ p − q − n − 1 ≥ 0; z ∈ U.
n
For m ∈ N, n ∈ N0 , and γ ∈ C − {0}, we define the next subclasses of Em,p q .
n
Em,p q, γ

n
Nm,p q, γ

n
f ∈ Em,p q : Re 1

⎧
⎨
⎩

k p m

≤

n
Km,p q, γ

∞

n
f ∈ Em,p q :

⎧
⎨

p!
p−q !

n
f ∈ Em,p q :

⎩

≤

∞
k p m

p!
γ −p
p−q !

q

Dn f q

z

q

n

1

n

1

n

Re γ
γ

k−q
k!

k−q ! p−q
q

−p

z

k−q
k!
k−q ! p−q
−p

γ

Dn 1 f

1
γ

n

q

> 0, z ∈ U

n

k−q
−p
p−q

q

n

Re γ
γ

,

γ

ak

,
k−q
−p
p−q

q

n

γ ak

,
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
S∗ b was studied by Nasr and Aouf 2 also see Bulboac˘ et al.
a
Remark 1.1. 1 Em,1 0, γ
3 .
0
1
Tα m and Em,1 0, 1 − α
Cα m , α ∈ 0, 1 were introduced by
2 Em,1 0, 1 − α
Srivastava et al. 4 .
0
1
T ∗ α and E1,1 0, 1 − α
C α , α ∈ 0, 1 were introduced by
3 E1,1 0, 1 − α
Silverman 5 .
0
1
0
1
Om γ and Km,1 0, γ
Om γ were introduced by Parvathan and
4 Km,1 0, γ
Ponnusanny 6, pages 163-164 .

n
n
n
5 For p
1 and q
0, the classes Em,p q, γ , Nm,p q, γ , and Km,p q, γ are closely
aa
related with Tn,m γ , On,m γ , and Pn,m γ which are defined by Owa and S˘ l˘ gean in 7 .
n
n
In this paper we give relationships between the classes of Em,p q, γ , Nm,p q, γ , and
q, γ . In the particular case when m ∈ N and n 0, p 1, and q 0, we obtain the same
results as in 8 .

n
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
Km,p q, γ

n
Em,p q, γ

n
Nm,p q, γ ;

2.1

n
4 if γ ∈ −∞, 0 , then Nm,p q, γ ⊆ Em,p q, γ ;
/ n
n
5 if γ ∈ −∞, 0 , then Em,p q, γ / Km,p q, γ .
⊆ n
n
Proof. 1 Let f ∈ Km,p q, γ . We prove that

Dn 1 f

q

Dn f q

z
z

−p

n < γ ,

q

z∈U .

2.2

If f has the series expansion

f z

zp −

∞

ak zk ,

k p m

ak ≥ 0,

2.3


4

Journal of Inequalities and Applications

then
Dn 1 f

Dn f
≤

q

z

q

−p

z

n − γ

q

− p!/ p − q ! 1 − p
∞
k p m

p!/ p − q ! −
∞
k p m

k!/ k − q !

k!/ k − q !

q

k−q / p−q
n

k−q / p−q
∞
k p m

p!/ p − q ! −

n

ak

k!/ k − q !

n

ak |z|k−p

k−q / p−q −p
k−q / p−q

n

n |z|k−p

q

ak |z|k−p

− γ .
2.4

We use the fact that Dn f
p!/ p − q !; these imply

q

z /zp−q / 0 for z ∈ U − {0} and limz → o Dn f

p!
−
p−q ! k

∞
p m

n

k−q
k!
k−q ! p−q

q

z /zp−q

ak |z|k−p > 0

2.5

for z ∈ U.
From 2.4 and 2.5 , we deduce

Dn 1 f

q

Dn f q
<

z
z

−p

n − γ

q

− p!/ p − q ! 1 − p
∞
k p m

p!/ p − q ! −
∞
k p m

q

γ

n

k!/ k − q ! k − q / p − q

k!/ k − q !

k−q / p−q
∞
k p m

p!/ p − q ! −

n

ak

n

ak

.

k−q / p−q −p

k!/ k − q !

k−q / p−q

q
n

ak

n

γ

.
2.6

n
By using the definition of Km,p q, γ from this last inequality we, obtain 2.2 which
implies

Re

n
hence f ∈ Em,p q, γ .

1
γ

Dn 1 f
Dn f q

q

z
z

−p

q

n

> −1

z∈U ,

2.7


Journal of Inequalities and Applications

5

n
2 Let f be in Em,p q, γ . Then 2.7 holds and, by using 2.3 , this is equivalent to

⎧ ⎛
⎨1
p!/ p − q ! zp−q −
⎝
Re
⎩γ
p!/ p − q ! zp−q −

> −1

∞
k p m k!/ k − q !
∞
k p m k!/ k − q !

k−q / p−q
k−q / p−q

n 1
n

ak zk−q

ak zk−q

−p

q

⎞⎫
⎬
n⎠
⎭

z∈U .
2.8
t ∈ 0, 1 if t → 1− , from 2.8 we obtain

For z

⎧⎛
⎨ − p!/ p − q !
⎝
⎩
p!/ p − q ! −

∞
k p m k!/ k − q !
∞
k p m k!/ k − q !

n 1

k−q / p−q
k−q / p−q

n

ak

ak

⎞⎫
2
⎬
γ
⎠ ≤
−p

⎭ Re γ

q

n
2.9

which is equivalent to
∞
k p m

k−q
k!
k−q ! p−q

n

k−q
p−q

2

γ
−p
Re γ

n ak ≤

q

p!
p−q !

2

γ
−p
Re γ

q

n

1 .
2.10

n
Then multiplying the relation last inequality with Re γ/|γ|, we obtain f ∈ Nm,p q, γ .
n
n
3 if γ is a real positive number, then the definitions of Nm,p q, γ and Km,p q, γ are
n
n
Km,p q, γ . By using 1 and 2 from this theorem, we obtain
equivalent, hence Nm,p q, γ
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

zp − α

fm,α p, q, n; z

m−q
p−q

p

−n

p m−q ! p
p!
z
p m !
p−q !

m

2.11

and let α > 0. We have
∞
k p m

n

k−q
k!
k−q ! p−q

p

≤

p

m−q !

p
×α
α

m !

p

m−q
p−q

p!
p−q !

p

k−q
−p
p−q
m−q
p−q
−n

q

n

p

n

Re γ
γ

m−q
−p
p−q

p m−q !
p!
p m !
p−q !

m−q
−p
p−q

q

n

γ

−γ
−γ

γ

q

ak

n

Re γ
γ

γ
2.12


6

Journal of Inequalities and Applications

or

∞
k p m

≤ −α

−p

n
and then fm,α p, q, n; z ∈ Nm,p q, γ
Let now

1

k−q
−p
p−q

p!
−p
p−q !

p!
p−q !

<

F z

n

k−q
k!
k−q ! p−q

q

q

n

n

Re γ
γ

Re γ
γ

n

m
p−q

1

1

q

γ

ak

≤0

γ

2.13

,

γ

n
see the definition of Nm,p q, γ .

⎛
q
1 ⎝ Dn 1 fm,α p, q, n; z
−p
q
γ
Dn f
p, q, n; z

⎞
n⎠

q

z∈U .

2.14

m,α

Then, by a simple computation and by using the fact that

q

fm,α p, q, n; z

q

fm,α z
−n

p m !
p!
p m−q
zp−q − α
p−q
p−q !
p m−q !
p!
p!
p m−q
zp−q − α
p−q
p−q !
p−q !
q

Dn fm,α z

−n

p!
p m−q
p!
zp−q − α
p−q
p−q !
p−q !

−n

p

zp

m−q

p

p m−q ! p
p!
z
m !
p−q !

m−q

.

m−q

p−q

n

zp

m−q

p!
zp−q 1 − αzm ,
p−q !
q

Dn 1 fm,α z

p m−q m
p!
z ,
zp−q 1 − α
p−q
p−q !
2.15


Journal of Inequalities and Applications

7

we obtain
q

Dn 1 fm,α z

1

1
γ

1

F z

1
γ

p!/ p − q ! zp−q 1 − α p

−p

q

1

q

Dn fm,α z

q

n

m − q / p − q zm

p!/ p − q ! zp−q 1 − αzm
n

1

where ζ

−p

a − αbζ
γ 1 − αζ

zm , a

−p

q

1 − αzm −p q n
γ 1 − αzm
1

n

n
2.16

m−q / p−q

1

q

m/ p − q , and

ϕ ζ ,

1, b

−p

ϕζ

For α > 1 we, have ϕ U

p

−p

q

n

a − αbζ
.
γ 1 − αζ

2.17

C∞ − D c, d , where D is the disc with the center

c

α2 b − a
γ α2 − 1

2.18

d

α b−a
.
γ 1 − α2

2.19

and the radius

We have F U
C∞ − D c 1, d where D c, d
{w : |w − c| < d} and we deduce that
Re F z > 0 for all z ∈ U does not hold.
n
∈ n
We have obtained that for α > 1, fm,α ∈ Nm,p q, γ , but fm,α / Em,p q, γ and in this case
n
n

⊆
Nm,p q, γ / Em,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
fm,α ∈ Nm,p q, γ .
We also obtain that f / Em,p q, γ like in Case 1.
∈ n


8

Journal of Inequalities and Applications

5 Let f fm,α be given by 2.11 , where α > |γ| − p
m/ p − q and |γ| − p q n 1 m/ p − q > 0. Then
∞

n

k−q
k!
k−q ! p−q

k p m

p

×α

>

p

m−q !

p

α

m !

p

p

m−q
p−q
−n

m−q
p−q

p!
p−q !

k−q
−p
p−q

n

−p

p

n

n

1 / |γ| − p

q

n

q

n

q

n

q

1

γ ak

m−q
−p
p−q

γ

p m−q !
p!
·
p m !
p−q !

m−q
−p
p−q

p!
γ
p−q !

q

q

2.20

γ

n

1

which implies that

fm,α / Km,p q, γ
∈ n

for m ∈ N, n ∈ N0 , γ ∈ −∞, 0 .

2.21

We have

F z

1

1
γ

q

Dn 1 fm,α z
q
Dn fm,α

z

−p

q

n

1

a − αbζ
γ 1 − αζ

1

ϕζ ,

2.22

where ϕ is given by 2.17 .
From ϕ U
D c, d where c and d are given by 2.18 and 2.19 , we obtain

Re F z ≥ 1

If γ ∈ −∞, p − q − n − 1 − m/ p − q
m/ p − q , 1 , then

α γ

b
γ α

αb

γ α

and α ∈

γ
1

a

a
.
1

|γ| − p

> 0,

2.23

q

n

1 / |γ| − p

q

n

1

2.24


Journal of Inequalities and Applications

9

and if
γ∈
α∈

γ −p
γ −p

q

q

1

n

n

m
,0 ,
p−q

p−q−n−1−

1

γ −p

,

m/ p − q

−p

n

q

q

m/ p − q

1

n

1
− γ

∩ 0, 1 ,
2.25

n

then 2.24 also holds. By combining 2.24 with 2.23 and the definition of Em,p q, γ , we
obtain that

n
fm,α ∈ Em,p q, γ

for α ∈

γ −p
γ −p

q

q

n

n

1

m/ p − q

1

,

γ −p
−p

q

n

m/ p − q − γ

q

1

n

1

∩ 0, 1 , γ ∈ −∞, 0 .
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

∞

zp

f z

ak zk

A.1

k p 1

which are analytic in U. For such a function f, we say that f ∈ Gn q, γ if it satisfies
1,p

Re 1

1
γ

Dn 1 f

q

Dn f q

z
z

−p

q

n

>0

z∈U

A.2

for some complex number γ with 0 < Re 1/γ < 1/ p − q − n − 1 .
If we define the function F for f ∈ Gn q, γ by
1,p

F z

1

1/γ

Dn 1 f

q

z /Dn f
1

q

z −p

1−p

q

q

n −i 1−p

n Re 1/γ

q

n Im 1/γ

,

A.3


10

Journal of Inequalities and Applications

then we know that F is analytic in U, F 0
1, and Re f z > 0 z ∈ U . Thus F is the
Carath´ odory function. Since the extremal function for the Carath´ odory function F is given
e
e
by
1 z
,
1−z

F z

A.4

we can write
1

1/γ

Dn 1 f

q

z /Dn f

z −p

q

1−p

1

q

q

n −i 1−p

q

n Im 1/γ

1 z
.
1−z

n Re 1/γ

A.5

This shows us that
Dn 1 f

q

Dn f q

z
z

−p

q

γ − iγ 1 − p

n

q

1
γ

n Im

1−p

γ 1

q

n Re

1
γ

1 z
.
1−z
A.6

Noting that
Dn 1 f

q

1
z Dn f
p−q

z

q

z

,

A.7

iγ 1 − p

q

2
1−z

1
,
z

we see that
Dn f q z
1
p − q Dn f q z
γ 1

−

1
p−q−n−γ
z

1−p

q

1
n Re
γ

1
γ

n Im

A.8

that is,
Dn f q z
1
p − q Dn f q z

1
z

−

1−p

2γ 1

q

1
γ

n Re

1
.
1−z

A.9

It follows from the above that
z
0

Dn f q t
1
p − q Dn f q t

−

1
t

dt

2γ 1

1−p

q

1
γ

n Re

z
0

1
dt.
1−t

A.10

Calculating the above integrations, we have that
1
log Dn f
p−q

q

z − log z

−2γ 1

1−p

q

n Re

1
γ

log 1 − z .

A.11


Journal of Inequalities and Applications

11

Therefore, we obtain that
Dn f

q

z

z

1/ p−q

1−z

2γ 1

,

1−p q n Re 1/γ

A.12

that is,
n

D f

q

z

z
1−z

2γ 1

1−p q n Re 1/γ

p−q

.

A.13

Consequently, the function f defined by the above is the extremal function for the class
n
Gn q, γ . But our class Em,p q, γ is defined with analytic functions f with negative
1,p
coefficients. Thus we do not know how we can consider the extremal function for this class.

References
ˇ aa
1 G. S. S˘ l˘ gean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish
Seminar Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, 1983.
2 M. A. Nasr and M. K. Aouf, “Starlike function of complex order,” The Journal of Natural Sciences and
Mathematics, vol. 25, no. 1, pp. 1–12, 1985.
3 T. Bulboac˘ , M. A. Nasr, and G. S. S˘ l˘ gean, “Function with negative coefficients n-starlike of complex
a
¸ aa
order,” Universitatis Babes-Bolyai. Studia Mathematica, vol. 36, no. 2, pp. 7–12, 1991.
¸
4 H. M. Srivastava, S. Owa, and S. K. Chatterjea, “A note on certain classes of starlike functions,”
Rendiconti del Seminario Matematico della Universit` di Padova, vol. 77, pp. 115–124, 1987.
a
5 H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical
Society, vol. 51, pp. 109–116, 1975.
6 R. Parvathan and S. Ponnusanny, Eds., “Open problems,” World Scientific, 1990, Conference on New
Trends in Geometric Function Theory and Applications.
7 S. Owa and G. S. S˘ l˘ gean, “Starlike or convex of complex order functions with negative coefficients,”
aa
Surikaisekikenkyusho K¯ kyuroku, no. 1062, pp. 77–83, 1998.
o ¯
¯
¯

8 S. Owa and G. S. S˘ l˘ gean, “On an open problem of S. Owa,” Journal of Mathematical Analysis and
aa
Applications, vol. 218, no. 2, pp. 453–457, 1998.