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
Classes of p-Valent Functions Defined by
Multiplier Transformations
Mugur Acu,1 Yaşar Polatog̃lu,2 and Emel Yavuz2
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 İstanbul Kültür University,
İstanbul 34156, Turkey
Correspondence should be addressed to Emel Yavuz, e.yavuz@iku.edu.tr
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, and Goel 1972.
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 fz a an zn an1 zn1 · · · .
Let Ap, n denote the class of functions fz normalized by
∞
fz zp ak zk
p, n ∈ N : 1, 2, 3, . . .
1.1
knp
which are analytic in the open unit disc D. In particular, we set
Ap, 1 : Ap ,
A1, 1 : A A1 .
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 ≺ gz z ∈ D.
1.3
2
Journal of Inequalities and Applications
If there exists a Schwarz function wz which is analytic in D with w0 0, |wz| < 1 such
that fz gwz, 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 w0 0, |wz| < 1, and wz zn φz for all z ∈ D, where φz
is regular and analytic in D and satisfies |φz| < 1 for every z ∈ D. Also, let P{p, n denote
k
the class of functions pz p ∞
kn pk z which are regular in D and satisfy the conditions
p0 p, Re pz > 0 for all z ∈ D. We note that if pz ∈ Pp, n, then
pz p
1 − wz 1 − zn φz
1 wz 1 zn φz
1.4
for some functions wz ∈ Ωn and every z ∈ D.
Definition 1.1. Let fz ∈ Ap, n for m ∈ N0 N ∪ {0}, λ ≥ 0, l > 0, one defines the multiplier
transformations Ip m, λ, l on Ap, n by the following infinite series:
Ip m, λ, lfz : zp ∞ p λk − p l m
ak zk .
p
l
kpn
1.5
It follows that
Ip 0, λ, lfz fz,
p lIp 2, λ, lfz p1 − λ l Ip 1, λ, lfz λzIp 1, λ, lfz ,
1.6
Ip m1 , λ, l Ip m2 , λ, l fz Ip m2 , λ, l Ip m1 , λ, l fz
for all integers m1 , m2 .
Remark 1.2. This multiplier transformation was introduced by Cătaş 1. For p 1, l 0, λ ≥ 0,
the operator Dλm : I1 m, λ, 0 was introduced by Al-Oboudi 2 which reduces to the Sălăgean
differantial operator 3. For λ 1, the operator Ilm : I1 m, 1, l was studied recently by Cho
and Srivastava 4 and Cho and Kim 5. The operator Im : I1 m, 1, 1 was studied by Uralegaddi and Somanatha 6 and the operator Ip m, l : Ip m, 1, l was investigated recently by
Sivaprasad Kumar et al. 7.
Definition 1.3 see 1. Let ϕz be analytic in D and ϕ0 1. A function fz ∈ Ap, n is said
to be in the class Ap m, λ, l, n; ϕ if it satisfies the following subordination:
Ip m 1, λ, lfz
≺ ϕz z ∈ D.
Ip m, λ, lfz
1.7
Definition 1.4. The radius of starlikeness of the class Ap m, λ, l, n, ϕ is defined by the following.
For each fz ∈ Ap m, λ, l, n; ϕ, let rf be the supremum of all numbers r such that
fDr is starlike with respect to the origin. Then the radius of starlikeness for Ap m, λ, l, n; ϕ is
rst Ap m, λ, l, n; ϕ inf
f∈Ap m,λ,l,n,ϕ
rf.
1.8
Mugur Acu et al.
3
Theorem 1.5. Let fz ∈ Ap, n and λ > 0, then fz belongs to the class Ap m, λ, l, n; χ if and
only if Fz, defined by
Fz z
pl
λzp1−λl/λ
ζp1−λl/λ−1 fζdζ zp 0
∞ kpn
pl
ak zk ,
p l k − pλ
1.9
belongs to the class Ap m 1, λ, l, n; χ.
This theorem was proved by Cătaş 1.
2. Main result
Theorem 2.1. The radius of starlikeness of the class Ap m, λ, l, n, φ is
⎛
⎞1/n
⎜
rst ⎝
pl
⎟
⎠ .
2
λp n λ2 p n p l p l − 2λp
2.1
This radius is sharp because the extremal function is
λ zp c p c − pzn
f∗ z ,
2p/n1
pl
1 zn
c
p1 − λ l
.
λ
2.2
Proof. If we take c p1 − λ l/λ, then the function Fz in Theorem 1.5 can be written in
the form
pl
Fz λzc
z
ζc−1 fζdζ.
2.3
0
If we take the logarithmic derivative from 2.3 and after simple calculations, we get
z c−1
c
F z z fz − c 0 ζ fζdζ
z
.
z
Fz
ζc−1 fζdζ
0
2.4
Since Fz is starlike, hence there exists a function wz ∈ Ωn such that
z c−1
c
F z z fz − c 0 ζ fζdζ
1 − wz
.
z
p
z
c−1
Fz
1 wz
ζ fζdζ
0
2.5
Solving for fz,
c p c − pwz
fz 1 wz zc
z
0
ζc−1 fζdζ.
2.6
4
Journal of Inequalities and Applications
Taking the logarithmic derivative from 2.6, we get
z
f z
1 − wz
zw z
p
b − 1 ,
fz
1 wz
1 wz 1 bwz
where b c − p/c p. To show that fz is starlike in |z| < r0 , we must show that
f z
>0
Re z
fz
for |z| < r0 . This condition is equivalent to
zw z
1 − wz
1 − bRe .
≤ Re p
1 wz
1 wz 1 bwz
2.7
2.8
2.9
On the other hand, we have the following relations:
2
1 − wz
1 − wz
Re p
p
,
1 wz
1 wz2
zw z
1 − bRe 1 wz 1 bwz
zw z ≤
1 − bzw z
,
≤
1 wz 1 bwz
2.10
2 n|z|n 1 − wz
2n
1 − |z|
Golusin inequality, 8. Therefore, the inequality 2.9 will be satisfied if
2
2
1 − wz
1 − wz
n1 − b|z|n
≤ p
.
1 wz 1 bwz 1 − |z|2n
1 wz2
2.11
Simplifying and writing |z| r, we obtain
1 bwz n1 − br n
≤
p
1 wz .
1 − r 2n
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 br n
<p
.
2.13
2n
1 rn
1−r
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 r0 of the theorem.
To see that the result is sharp, consider the function Fz zp /1 zn 2p/n . For this
function, we have
λ zp c p c − pzn
f∗ z ,
2p/n1
pl
1 zn
f z p − 1 − bp nzn − pbz2n
z ∗
.
2p/n1
f∗ z
1 zn
2.15
So that zf∗ z/f∗ z 0 for |z| r0 . Thus, fz is not starlike in any circle |z| < r if r > r0 .
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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2007.
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