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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 280183, 5 pages
doi:10.1155/2008/280183
Research Article
The Radius of Starlikeness of the Certain
Classesofp-ValentFunctionsDefinedby
Multiplier Transformations
Mugur Acu,
1
Yas¸ar Polato
˜
glu,
2
and Emel Yavuz
2
1
Department of Mathematics, ”Lucian Blaga” University of Sibiu, 5-7 Ion Ratiu Street,
Sibiu 550012, Romania
2
Department of Mathematics and Computer Science, TC
˙
Istanbul K
¨
ult
¨
ur University,
˙
Istanbul 34156, Turkey
Correspondence should be addressed to Emel Yavuz,
Received 12 November 2007; Accepted 02 January 2008
Recommended by Narendra Kumar K. Govil
The aim of this paper is to give the radius of starlikeness of the certain classes of p-valent functions
defined by multiplier transformations. The results are obtained by using techniques of Robertson
1953,1963 which was used by Bernardi 1970, Libera 1971, Livingstone 1966,andGoel1972.
Copyright q 2008 Mugur Acu 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.
1. Introduction
Let H be the class of analytic functions in the open unit disc
D {z ∈ C ||z| < 1} and Ha, n
be the subclasses of H consisting of the functions of the form fza a
n
z
n
a
n1
z
n1
···.
Let Ap, n denote the class of functions fz normalized by
fzz
p
∞
knp
a
k
z
k
p, n ∈
N :
1, 2, 3,
1.1
which are analytic in the open unit disc
D. In particular, we set
Ap, 1 : A
p
, A1, 1 : A A
1
. 1.2
If fz and gz are analytic in
D, we say that fz is subordinate to gz, written symbolically
as
f ≺ g or fz ≺ gzz ∈
D. 1.3
2 Journal of Inequalities and Applications
If there exists a Schwarz function wz which is analytic in
D with w00, |wz| < 1 such
that fzgwz, z ∈
D.
For two analytic functions fz and Fz, we say that Fz is superordinate to fz if
fz is subordinate to Fz.
For integer n ≥ 1, let Ωn denote the class of functions wz which are regular in
D
and satisfy the conditions w00, |wz| < 1, and wzz
n
φz for all z ∈ D,whereφz
is regular and analytic in
D and satisfies |φz| < 1 for every z ∈ D. Also, let P{p, n denote
the class of functions pzp
∞
kn
p
k
z
k
which are regular in D and satisfy the conditions
p0p,Repz > 0 for all z ∈
D.Wenotethatifpz ∈Pp, n,then
pzp
1 − wz
1 wz
1 − z
n
φz
1 z
n
φz
1.4
for some functions wz ∈ Ωn and every z ∈
D.
Definition 1.1. Let fz ∈Ap, n for m ∈
N
0
N ∪{0}, λ ≥ 0, l>0, one defines the multiplier
transformations I
p
m, λ, l on Ap, n by the following infinite series:
I
p
m, λ, lfz : z
p
∞
kpn
p λk − pl
p l
m
a
k
z
k
. 1.5
It follows that
I
p
0,λ,lfzfz,
p lI
p
2,λ,lfz
p1 − λl
I
p
1,λ,lfzλzI
p
1,λ,lfz
,
I
p
m
1
,λ,l
I
p
m
2
,λ,l
fz
I
p
m
2
,λ,l
I
p
m
1
,λ,l
fz
1.6
for all integers m
1
, m
2
.
Remark 1.2. This multiplier transformation was introduced by C
˘
atas¸ 1.Forp 1, l 0, λ ≥ 0,
the operator D
m
λ
: I
1
m, λ, 0 was introduced by Al-Oboudi 2 which reduces to the S
˘
al
˘
agean
differantial operator 3.Forλ 1, the operator I
m
l
: I
1
m, 1,l was studied recently by Cho
and Srivastava 4 and Cho and Kim 5. The operator I
m
: I
1
m, 1, 1 was studied by Urale-
gaddi and Somanatha 6 and the operator I
p
m, l : I
p
m, 1,l was investigated recently by
Sivaprasad Kumar et al. 7.
Definition 1.3 see 1.Letϕz be analytic in D and ϕ01. A function fz ∈Ap, n is said
to be in the class A
p
m, λ, l, n; ϕ if it satisfies the following subordination:
I
p
m 1,λ,lfz
I
p
m, λ, lfz
≺ ϕzz ∈
D. 1.7
Definition 1.4. The radius of starlikeness of the class A
p
m, λ, l, n, ϕ is defined by the following.
For each fz ∈A
p
m, λ, l, n; ϕ,letrf be the supremum of all numbers r such that
f
D
r
is starlike with respect to the origin. Then the radius of starlikeness for A
p
m, λ, l, n; ϕ is
r
st
A
p
m, λ, l, n; ϕ
inf
f∈A
p
m,λ,l,n,ϕ
rf. 1.8
Mugur Acu et al. 3
Theorem 1.5. Let fz ∈Ap, n and λ>0,thenfz belongs to the class A
p
m, λ, l, n; χ if and
only if Fz, defined by
Fz
p l
λz
p1−λl/λ
z
0
ζ
p1−λl/λ−1
fζdζ z
p
∞
kpn
p l
p l k − pλ
a
k
z
k
,
1.9
belongs to the class A
p
m 1,λ,l,n; χ.
This theorem was proved by C
˘
atas¸ 1.
2. Main result
Theorem 2.1. The radius of starlikeness of the class A
p
m, λ, l, n, φ is
r
st
⎛
⎜
⎝
p l
λp n
λ
2
p n
2
p l
p l − 2λp
⎞
⎟
⎠
1/n
. 2.1
This radius is sharp because the extremal function is
f
∗
z
λ
p l
z
p
c p c − pz
n
1 z
n
2p/n1
,c
p1 − λl
λ
. 2.2
Proof. If we take c p1 − λl/λ, then the function Fz in Theorem 1.5 can be written in
the form
Fz
p l
λz
c
z
0
ζ
c−1
fζdζ. 2.3
If we take the logarithmic derivative from 2.3 and after simple calculations, we g et
z
F
z
Fz
z
c
fz − c
z
0
ζ
c−1
fζdζ
z
0
ζ
c−1
fζdζ
. 2.4
Since Fz is starlike, hence there exists a function wz ∈ Ωn such that
z
F
z
Fz
z
c
fz − c
z
0
ζ
c−1
fζdζ
z
0
ζ
c−1
fζdζ
p
1 − wz
1 wz
. 2.5
Solving for fz,
fz
c pc − pwz
1 wz
z
c
z
0
ζ
c−1
fζdζ. 2.6
4 Journal of Inequalities and Applications
Taking the logarithmic derivative from 2.6,weget
z
f
z
fz
p
1 − wz
1 wz
b − 1
zw
z
1 wz
1 bwz
, 2.7
where b c − p/c p. To show that fz is starlike in |z| <r
0
, we must show that
Re
z
f
z
fz
> 0 2.8
for |z| <r
0
. This condition is equivalent to
1 − bRe
zw
z
1 wz
1 bwz
≤ Re
p
1 − wz
1 wz
. 2.9
On the other hand, we have the following relations:
Re
p
1 − wz
1 wz
p
1 −
wz
2
1 wz
2
,
1 − bRe
zw
z
1 wz
1 bwz
≤
1 − b
zw
z
1 wz
1 bwz
,
zw
z
≤
n|z|
n
1 −|z|
2n
1 −
wz
2
2.10
Golusin inequality, 8. Therefore, the inequality 2.9 will be satisfied if
n1 − b|z|
n
1 wz
1 bwz
1 −
wz
2
1 −|z|
2n
≤ p
1 −
wz
2
1 wz
2
. 2.11
Simplifying and writing |z| r,weobtain
n1 − br
n
1 − r
2n
≤ p
1 bwz
1 wz
. 2.12
Since |wz|≤|z|
n
r
n
, p|1 bwz/1 wz|≥p1 br
n
/1 r
n
so that 2.12 will be
satisfied if
n1 − br
n
1 − r
2n
<p
1 br
n
1 r
n
. 2.13
The inequality 2.13 can be written in the following form:
p − 1 − bp nr
n
− bpr
2n
> 0, 2.14
which gives the required root r
0
of the theorem.
To see that the result is sharp, consider the function Fzz
p
/1 z
n
2p/n
.Forthis
function, we have
f
∗
z
λ
p l
z
p
c pc − pz
n
1 z
n
2p/n1
,
z
f
∗
z
f
∗
z
p − 1 − bp nz
n
− pbz
2n
1 z
n
2p/n1
.
2.15
So that zf
∗
z/f
∗
z 0for|z| r
0
. Thus, fz is not starlike in any circle |z| <rif r>r
0
.
Mugur Acu et al. 5
Remark 2.2. If we give special values to m, λ, l, n, we obtain the radius of starlikeness for the
corresponding integral operators.
Acknowledgment
This paper was supported by GAR 20/2007.
References
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˘
atas¸, On certain classes of p-valent functions defined by multiplier transformations, in Proceedings
of the International Symposium on Geometric Function Theory and Applications,
˙
Istanbul, Turkey, August
2007.
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˘
al
˘
agean operator, International
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al
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agean, Subclasses of univalent functions, in Complex Analysis—Fifth Romanian-Finnish Seminar,
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ur Mathematik,VEBDeutscher
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