Volume 2008, Article ID 280183,5pages 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 Yas¸ar Polato ˜glu,2 and Emel Yavuz2
1Department of Mathematics, ”Lucian Blaga” University of Sibiu, 5-7 Ion Ratiu Street,
Sibiu 550012, Romania
2Department of Mathematics and Computer Science, TC ˙Istanbul K ¨ult ¨ur University, ˙Istanbul 34156, Turkey
Correspondence should be addressed to Emel Yavuz,[email protected]
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 ofp-valent functions defined by multiplier transformations. The results are obtained by using techniques of Robertson
1953,1963which was used by Bernardi1970, Libera1971, Livingstone1966, and Goel1972.
Copyrightq2008 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
LetHbe the class of analytic functions in the open unit discD{z∈C| |z|<1}andHa, n
be the subclasses ofH consisting of the functions of the formfz aanznan1zn1· · ·. LetAp, ndenote the class of functionsfznormalized by
fz zp
∞
knp
akzk
p, n∈N:1,2,3, . . . 1.1
which are analytic in the open unit discD. In particular, we set
Ap,1:Ap, A1,1:AA1. 1.2
Iffzandgzare analytic inD, we say thatfzis subordinate togz, written symbolically as
If there exists a Schwarz functionwzwhich is analytic inDwithw0 0,|wz| < 1 such thatfz gwz,z∈D.
For two analytic functionsfzandFz, we say that Fzis superordinate tofzif
fzis subordinate toFz.
For integern ≥ 1, let Ωn denote the class of functionswzwhich are regular in D and satisfy the conditionsw0 0,|wz| < 1, andwz znφzfor allz ∈ D, whereφz is regular and analytic inDand satisfies|φz| < 1 for everyz ∈ D. Also, letP{p, ndenote the class of functionspz p∞knpkzk which are regular inDand satisfy the conditions
p0 p, Repz>0 for allz∈D. We note that ifpz∈ Pp, n, then
pz p1−wz
1wz
1−znφz
1znφz 1.4
for some functionswz∈Ωnand everyz∈D.
Definition 1.1. Letfz∈ Ap, nform ∈N0 N∪ {0},λ ≥0,l >0, one defines the multiplier transformationsIpm, λ, lonAp, nby the following infinite series:
Ipm, λ, lfz:zp
∞
kpn
pλk−p l
pl
m
akzk. 1.5
It follows that
Ip0, λ, lfz fz,
plIp2, λ, lfz
p1−λ lIp1, λ, lfz λzIp1, λ, lfz
,
Ip
m1, λ, l
Ip
m2, λ, l
fzIp
m2, λ, l
Ip
m1, λ, l
fz
1.6
for all integersm1,m2.
Remark 1.2. This multiplier transformation was introduced by C˘atas¸1. Forp1,l0,λ≥0, the operatorDmλ :I1m, λ,0was introduced by Al-Oboudi2which reduces to the S˘al˘agean differantial operator3. Forλ 1, the operatorIml : I1m,1, lwas studied recently by Cho and Srivastava4and Cho and Kim5. The operatorIm:I1m,1,1was studied by Urale-gaddi and Somanatha6and the operatorIpm, l:Ipm,1, lwas investigated recently by Sivaprasad Kumar et al.7.
Definition 1.3see1. Letϕzbe analytic inDandϕ0 1. A functionfz∈ Ap, nis said to be in the classApm, λ, l, n;ϕif it satisfies the following subordination:
Ipm1, λ, lfz
Ipm, λ, lfz ≺ϕz z∈D. 1.7
Definition 1.4. The radius of starlikeness of the classApm, λ, l, n, ϕis defined by the following. For each fz ∈ Apm, λ, l, n;ϕ, letrf be the supremum of all numbersr such that
fDris starlike with respect to the origin. Then the radius of starlikeness forApm, λ, l, n;ϕis
rst
Apm, λ, l, n;ϕ
inf
f∈Apm,λ,l,n,ϕ
Theorem 1.5. Letfz ∈ Ap, nandλ > 0, thenfzbelongs to the classApm, λ, l, n;χif and only ifFz, defined by
Fz pl λzp1−λl/λ
z
0
ζp1−λl/λ−1fζdζzp
∞
kpn
pl
pl k−pλ akz
k,
1.9
belongs to the classApm1, λ, l, n;χ.
This theorem was proved by C˘atas¸1.
2. Main result
Theorem 2.1. The radius of starlikeness of the classApm, λ, l, n, φis
rst ⎛ ⎜
⎝ pl
λpn
λ2pn2 plpl−2λp ⎞ ⎟ ⎠
1/n
. 2.1
This radius is sharp because the extremal function is
f∗z pλlz
pcp c−pzn
1zn2p/n1 , c
p1−λ l
λ . 2.2
Proof. If we takec p1−λ l/λ, then the functionFzinTheorem 1.5can be written in the form
Fz pl λzc
z
0
ζc−1fζdζ. 2.3
If we take the logarithmic derivative from2.3and after simple calculations, we get
zF
z
Fz
zcfz−cz
0ζc−1fζdζ z
0ζc−1fζdζ
. 2.4
SinceFzis starlike, hence there exists a functionwz∈Ωnsuch that
zF
z
Fz
zcfz−cz
0ζc−1fζdζ z
0ζc−1fζdζ
p1−wz
1wz. 2.5
Solving forfz,
fz cp c−pwz
1wzzc
z
0
Taking the logarithmic derivative from2.6, we get
zf
z
fz p
1−wz
1wz b−1
zwz
1wz1bwz, 2.7
whereb c−p/cp. To show thatfzis starlike in|z|< r0, we must show that
Re
zf
z
fz >0 2.8
for|z|< r0. This condition is equivalent to
1−bRe
zwz
1wz1bwz ≤Re
p1−wz
1wz . 2.9
On the other hand, we have the following relations:
Re
p1−wz
1wz p
1−wz2
1wz2,
1−bRe
zwz
1wz1bwz ≤
1−bzwz
1wz1bwz,
zwz≤ n|z|n
1− |z|2n
1−wz2
2.10
Golusin inequality,8. Therefore, the inequality2.9will be satisfied if
n1−b|z|n
1wz1bwz1−wz 2
1− |z|2n ≤p
1−wz2
1wz2. 2.11
Simplifying and writing|z|r, we obtain
n1−brn
1−r2n ≤p
11bwwzz. 2.12
Since|wz| ≤ |z|nrn,p|1bwz/1wz| ≥p1brn/1rnso that2.12will be satisfied if
n1−brn 1−r2n < p
1brn
1rn . 2.13
The inequality2.13can be written in the following form:
p−1−bpnrn−bpr2n>0, 2.14 which gives the required rootr0of the theorem.
To see that the result is sharp, consider the function Fz zp/1 zn2p/n. For this function, we have
f∗z pλlz
pcp c−pzn
1zn2p/n1 ,
zf
∗z
f∗z
p−1−bpnzn−pbz2n
1zn2p/n1 .
2.15
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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